Calculus one math notes

Calculus I

Differential Calculus

Lecture Notes

Veselin Jungic & Jamie MulhollandDepartment of Mathematics

Simon Fraser University

c©Jungic/Mulholland, August 26, 2013License is granted to print this

document for personal/educational use.

Contents

Contents i

Preface iii

Greek Alphabet v

1 Review: Functions and Models 1

1.1 Exponential Functions & Inverse Functions and Logarithms . . . . . . . . . . . . . . . . . . . 2

2 Limits and Derivatives 11

2.1 The Tangent and Velocity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Calculating Limits Using the Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 The Precise Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Limits at Infinity: Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Derivatives and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.8 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Differentiation Rules 59

3.1 Derivatives of Polynomials and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 60

3.2 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.6 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.7 Rates of Change in the Natural and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . 90

3.8 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.9 Related rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.10 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

i

ii CONTENTS

4 Applications of the Derivative 113

4.1 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3 How Derivatives Affect the Shape of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.4 Indeterminate Forms and L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.5 Summary of Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.6 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.7 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.8 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5 Parametric Equations and Polar Coordinates 155

5.1 Curves Defined by Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.3 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.4 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6 Review Material 185

6.1 Midterm 1 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.2 Midterm 2 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.3 End of Term Review Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

6.4 Final Exam Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

6.5 Final Exam Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Bibliography 221

Preface

This booklet contains our notes for courses Math 150/151 - Calculus I at Simon Fraser University. Stu-dents are expected to use this booklet during each lecture by follow along with the instructor, filling in thedetails in the blanks provided, during the lecture.

Definitions of terms are stated in orange boxes and theorems appear in blue boxes .

Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet toaccompany the example - just like the ones used by your instructor in the lecture. Clicking the link abovewill take you to the following website containing all the applets:

http://www.sfu.ca/ jtmulhol/calculus-applets/html/appletsforcalculus.html

Try it now.

No project such as this can be free from errors and incompleteness. We will be grateful to everyonewho points out any typos, incorrect statements, or sends any other suggestion on how to improve thismanuscript.

Veselin JungicSimon Fraser [email protected]

Jamie MulhollandSimon Fraser Universityj [email protected]

August 26, 2013

iii

http://www.sfu.ca/~jtmulhol/calculus-applets/html/appletsforcalculus.html

Greek Alphabet

lowercase

capital name pronunciation lowercase

capital name pronunciation

α A alpha (al-fah) ν N nu (new)β B beta (bay-tah) ξ Ξ xi (zie)γ Γ gamma (gam-ah) o O omicron (om-e-cron)δ ∆ delta (del-ta) π Π pi (pie)ε E epsilon (ep-si-lon) ρ P rho (roe)ζ Z zeta (zay-tah) σ Σ sigma (sig-mah)η H eta (ay-tah) τ T tau (taw)θ Θ theta (thay-tah) υ Υ upsilon (up-si-lon)ι I iota (eye-o-tah) φ Φ phi (fie)κ K kappa (cap-pah) χ X chi (kie)λ Λ lambda (lamb-dah) ψ Ψ psi (si)µ M mu (mew) ω Ω omega (oh-may-gah)

v

Part 1

Review: Functions and Models

1

PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 2

1.1 Exponential Functions & Inverse Functions and Logarithms

(This lecture corresponds to Sections 1.5 and 1.6 of Stewart’s Calculus.)

1. Reminder. For all a ∈ (0, 1) ∪ (1,∞) and all x, y ∈ R:

(a) ax+y = ax · ay

(b) ax−y =ax

ay

(c) (ax)y

= axy

(d) (ab)x = ax · bx

2. Reminder. Sketch the graphs of the functions f(x) = 2x and g(x) = 3x.

3. Reminder. Sketch the graph of the function F (x) =

(1

2

)x.


4. Reminder. Evaluate f(2), f(−2), f( 12 ) and f( 3

2 ) if f(x) = 4x.

5. BIG Question. What is 4√

2?


6. Reminder. Napier’s constant:

e ≈ 2.718281828459045235360287471352

(John Napier, 1550-1617)


7. Definition. A function f is called a one-to-one function if it never takes on the same value twice;that is

if x1 6= x2 then f(x1) 6= f(x2) .

8. Example. Which of the following functions are one-to-one?

(a) f(x) = x2

(b) g(x) = x3

(c) h(x) = ex

(d) i(x) = sinx

(e) j(x) = sinx, x ∈[−π2 ,

π2

]


9. Horizontal Line Test. A function is one-to-one if and only if no horizontal line intersects its graphmore than once.

10. Definition. Let f be one-to-one function with domain A and range B. Then its inverse functionhas domain B and range A and is defined by

f−1(y) = x ⇔ f(x) = y

for any y ∈ B.


11. Example. Find a formula for the inverse of f(x) =x

3x+ 1.

12. Logarithmic Function. The inverse function of the exponential function f(x) = ax is called thelogarithmic function with base a.

13. All You Need To Know. For any a > 0, a 6= 1, any x > 0, and any y ∈ R

loga x = y ⇔ ay = x

14. Example. Determine log2 (16), log2 ( 18 ) and log2 (1).

15. Example. Can you find log2 (−32)?

16. Reminder. For all a ∈ (0, 1) ∪ (1,∞) and any positive x and y:

(a) loga(xy) = loga x+ loga y

(b) loga

(xy

)= loga x− loga y

(c) loga(xr) = r loga x (r is a real number)

17. Notation.log10 x = log x

loge x = lnx


18. Reminder. Sketch the graph of the function y = lnx

19. Example. Solve the equation ex3−3 − 9 = 0 for x.

20. Inverse Trig Functions. Here we will limit our discussion to sin.

21. Definition. The inverse function of the sine function f(x) = sinx, −π2 ≤ x ≤ π2 , is called arcsine

and is denoted by either sin−1 or arcsin.


22. All You Need To Know. For any −1 ≤ x ≤ 1, and any −π2 ≤ y ≤π2

sin−1 x = y ⇔ sin y = x

23. Determine the following.

(a) sin−1(√

32 )

(b) sin(sin−1( 13 ))

(c) sin−1(sin( 3π4 ))


24. Additional Notes

Part 2

Limits and Derivatives

11

PART 2: LIMITS AND DERIVATIVES LECTURE 2.1 TANGENT AND VELOCITY PROBLEMS 12

2.1 The Tangent and Velocity Problems

(This lecture corresponds to Section 2.1 of Stewart’s Calculus.)

1. Quote. If I were again beginning my studies, I would follow the advice of Plato and start with math-ematics.Galileo Galilei, Italian philosopher and astronomer, 1564-1642.

2. The Tangent Problem. Find an equation of the tangent line ` to a curve with equation y = f(x) ata given point P .

3. Three Questions.

(a) What is the tangent line ` to a curve with equation y = f(x) at a given point P ?

(b) If a curve with equation y = f(x) and a point P on the curve are given, does the tangent `exist?

(c) If a curve with equation y = f(x) and a point P = (x0, f(x0)) are given and if the tangent line `exists then an equation of ` is given by

y − f(x0) = m(x− x0) .

How do we calculate the slope m?

4. Hint. Find the slopes of the secant lines to the parabola y = x2 through the points (1, 1) and:

(a) (2, 4)


(b) (1.5, 1.52)

(c) (1.1, 1.12)

(d) (1.001, 1.0012)

5. BIG Question. What if the second point is VERY, VERY close to the point (1, 1)?

6. Velocity Problem. By definition

avarage velocity =distance traveled

time elapsed

What if the period of time elapsed is very small?

7. Example. The position of the car is given by the values in the table.

t 0 1 2 3 4 5s 0 10 32 70 119 178

where t is in seconds and s is in feet.

Find the average velocity for the time beginning when t = 2 and lasting

(a) 3 seconds


(b) 2 seconds

(c) 1 second

8. Question. What is the meaning of the number that we see on the car speedometer as we travel incity traffic?

9. Answer. The number represents the instantaneous velocity.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 16

2.2 The Limit of a Function



1. Quote. “Black holes are where God divided by zero.”Steven Wright, American comedian, 1955-

2. Problem. Let f(x) =x2 − x− 2

x− 2.

(a) Determine the domain of f .

(b) Complete the tablex f(x) x f(x)1 31.9 2.11.99 2.011.999 2.0011.9999 2.0001


3. Definition. We writelimx→a

f(x) = L

and say

”the limit of f(x), as x approaches a, equals L”

if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to besufficiently close to a (on either side of a) but not equal to a.

4. Example. Guess the value of

limx→0

sinx

x.

5. Problem. What can we say about

limx→0

|x|x

?

6. Definition. We writelimx→a+

f(x) = L

and say

”the right-hand limit of f(x), as x approaches a, equals L”

if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to besufficiently close to a and x greater than a.


7. Example. Sketch the graph of the function

f(x) =

x+ 1 if x ≤ −1x2 if x ∈ (−1, 0)1 if x = 0x2 if x ∈ (0, 1]x+ 1 if x > 1

Find

(a) limx→−1−

f(x)

(b) limx→−1+

f(x)

(c) limx→0−

f(x)

(d) limx→0+

f(x)

(e) limx→0

f(x)

8. Fact.limx→a

f(x) = L ⇐⇒ ( limx→a−

f(x) = L and limx→a+

f(x) = L)


9. Problem. Sketch the graph of f(x) =1

(x+ 1)2.

10. Definition. Let f be a function defined on both sides of a, except possibly at a itself. Then

limx→a

f(x) =∞

means that the values of f(x) can be made arbitrarily large (as large as we please) by taking xsufficiently close to a, but not equal to a.

11. Examples. Sketch the graph of the following function.

g(x) =x+ 3

x− 1

12. Read Example 10 in text regarding f(x) = tan (x)

13. Definition. The line x = a is called a vertical asymptote of the curve y = f(x) if at least one ofthe following statements is true:

limx→a

f(x) =∞ limx→a−

f(x) =∞ limx→a+

f(x) =∞limx→a

f(x) = −∞ limx→a−

f(x) = −∞ limx→a+

f(x) = −∞

PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 21

2.3 Calculating Limits Using the Limit Laws


1. Quote. “Laws are like sausages. It’s better not to see them being made.”Otto von Bismarck , German statesman, 1815 - 1898)

2. Example. Guess the value of limt→0

√t+ 9− 3

t.

3. Limit Laws. Suppose that c is a constant and the limits

limx→a

f(x) and limx→a

g(x)

exist. Then

(a) limx→a

(f(x) + g(x)) = limx→a

f(x) + limx→a

g(x)

(b) limx→a

(f(x)− g(x)) = limx→a

f(x)− limx→a

g(x)

(c) limx→a

(c · f(x)) = c · limx→a

f(x)

(d) limx→a

(f(x) · g(x)) = limx→a

f(x) · limx→a

g(x)

(e) limx→a

f(x)

g(x)=

limx→a f(x)

limx→a g(x)if limx→a g(x) 6= 0.

(f) limx→a

[f(x)]p/q = [ limx→a

f(x)]p/q

4. Two Special Limit Laws.

(a) limx→a c = c

(b) limx→a x = a


5. Example. Evaluate limx→2

(x3 + 3x2 − 4x+ 5).

6. Direct Substitution Property. If f is a polynomial or a rational function and a is in the domain off , then

limx→a

f(x) = f(a) .

7. Examples. Find the following limits.

(a) limx→−1

x+ 1

x3 + 1

(b) limt→0

√t+ 9− 3

t

(c) limh→0

f(x+ h)− f(x)

hif f(x) = x2


8. Example. Find limx→0

x2

|x|Reminder.

limx→a

f(x) = L ⇐⇒ ( limx→a−

f(x) = L and limx→a+

f(x) = L)

9. Theorem. If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both existas x approaches a, then

limx→a

f(x) ≤ limx→a

g(x) .

10. Squeeze Theorem. If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and

limx→a

f(x) = limx→a

h(x) = L

thenlimx→a

g(x) = L .


11. Example. Show that

limx→0

[x

(sin

1

x+ cos

1

x

)]= 0 .

PART 2: LIMITS AND DERIVATIVES LECTURE 2.4 THE PRECISE DEFINITION OF LIMIT 26

2.4 The Precise Definition of Limit


1. Quote.

“There’s a delta for every epsilon,It’s a fact that you can always count upon.

There’s a delta for every epsilonAnd now and again,There’s also an N.”

(Tom Lehrer, American singer-songwriter, satirist, pianist, and mathematician, 1928 - .)

2. The ε, δ Game. Consider the function f(x) = 3x − 1 and the point x = 1. There are two players inthis game: Player A and Player B. The game is played as follows. Player A chooses a number, say ε.The object of Player B is to find a number δ so that all values in the interval (1− δ, 1 + δ) have imagein the interval (f(1)− ε, f(1) + ε). The winner is determined as follows:

1) If Player A can pick a number ε such that Player B cannot find such a δ then Player A wins.2) If Player B can find a δ for any ε given by Player A then Player B wins.

Who wins Player A or Player B?


3. Definition. Let f be a function defined on some open interval that contains the number a, exceptpossibly at a iself. Then we say that the limit of f(x) as x approaches a is L, and we write

limx→a

f(x) = L

if for every number ε > 0 there is a δ > 0 such that

|f(x)− L| < ε whenever 0 < |x− a| < δ .

[link to applet]

http://www.sfu.ca/~jtmulhol/calculus-applets/html/appletsforcalculus.html


4. Example. Prove the statement using the ε, δ definition.

limx→3

(2− 5x) = −13


5. To Be Continued ... ... in Math 242.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 30

2.5 Continuity


1. Quote. “If I were asked to name, in one word, the pole star round which the mathematical firmamentrevolves, the central idea which pervades the whole corpus of mathematical doctrine, I should pointto Continuity as contained in our notions of space, and say, it is this, it is this! ”(JJ Sylvester, English mathematician, 1814-1897)

2. Example. What is the difference between the two graphs?

3. Definition. A function f is continuous at a number a if

limx→a

f(x) = f(a) .

4. Note.

(a) a belongs to the domain of f(b) lim

x→af(x) exists

(c) limx→a

f(x) = f(a)


5. Definition. If

(1) f is defined on an open interval containing a, except perhaps at a, and(2) f is not continuous at a

we say that f is discontinuous at a.

6. Example. Where are each of the following functions discontinuous?

(a)

f(x) =

x2 − 4

x− 2if x 6= 2

5 if x = 2

(b)

g(x) =

1

x− 2if x 6= 2

5 if x = 2

(c)

h(x) =

1 if x ∈ [1, 2)2 if x ∈ [2, 3)


7. Definition. A function f is continuous from the right at a number a if

limx→a+

f(x) = f(a)

and f is continuous from the left at a if

limx→a−

f(x) = f(a) .

8. Definiton. A function f is continuous on an interval if it is continuous at every number in thatinterval. We understand continuous at the endpoint to mean continuous from the right or continuousfrom the left.

9. Example. Find the number c that makes f(x) continuous for every x.

f(x) =

x4 − 1

x3 − 1if x 6= 1

c if x = 1


10. Fact. The following types of functions are continuous on their domains:

(a) polynomials(b) rational functions(c) root functions(d) trigonometric functions(e) inverse trigonometric functions(f) exponential functions(g) logarithmic functions

11. More Facts. If f and g are continuous at a and c is a constant, then the following functions are alsocontinuous at a:

f + g, f − g, cf, fg, fg

if g(a) 6= 0 .

12. Example. For which a, b ∈ R is the function

f(x) =

√1− x− 1

axif x ∈ (0, 1]

1 if x = 0

bx4 + bx

x2 + xif x ∈ (−1, 0)

continuous on (−1, 1]?


13. Theorem. If f is continuous at b and limx→a

g(x) = b then

limx→a

f(g(x)) = f( limx→a

g(x)) = f(b) .

14. Example. Evaluate

limx→0

e√

1−x−1x .

15. Theorem. If g is continuous at a and f is continuous at g(a), then the composite function f g givenby (f g)(x) = f(g(x)) is continuous at a.


16. Intermediate Value Theorem. Suppose that f is continuous on the closed interval [a, b] and let Nbe any number between f(a) and f(b), where f(a) 6= f(b). Then there exists a number c in (a, b) suchthat f(c) = N .

17. Example. Use the Intermediate Value Theorem to prove that√

2 exists, i.e., prove that there is c ∈ Rsuch that c2 = 2.


18. Example. Use the Intermediate Value Theorem to show that the equation

ex = 2− x

has at least one real solution.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 38

2.6 Limits at Infinity: Horizontal Asymptotes


1. Quote. “Infinity is a floorless room without walls or ceiling. ”(Anonymous)

2. Problem. Sketch the graphs of the following functions

(a) f(x) =1

x

(b) g(x) = ex

(c) h(x) = tan−1 x

(d) i(x) =1

1 + x2


3. Definition. Let f be a function defined on some interval (a,∞). Then

limx→∞

f(x) = L

means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.


(a) limx→∞

1

x

(b) limx→−∞

1

x

(c) limx→−∞

ex

(d) limx→∞

tan−1 x

(e) limx→−∞

tan−1 x


5. Definition. The line y = L is called a horizontal asymptote of the curve y = f(x) if either

limx→∞

f(x) = L or limx→−∞

f(x) = L .

6. Fact. If r > 0 is a rational number, then

limx→∞

1

xr= 0 .

If r > 0 is a rational number such that xr is defined for all x then

limx→−∞

1

xr= 0 .


(a)

limx→∞

3x3 − 4x2 − 1

6x3 + x+ 2

(b)

limx→−∞

√3x2 − 5− 1

2x+ 5


(c)lim

x→−∞(√x2 + ax−

√x2 + bx)

8. Problem. Find the following limits.

(a) limx→∞

x2

(b) limx→∞

x2 + 2x− 1

x3 + 3

(c) limx→∞

x4 + 5x3 − 1

x2 + x+ 1

(d) limx→∞

ex

(e) limx→∞

ex

x2


9. Additional Notes

PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 43

2.7 Derivatives and Rates of Change


1. Quote. ”The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.”(Marcel Proust, French author, 1871- 1922)

2. Definition. The tangent line to the curve y = f(x) at the point P (a, f(a)) is the line through Pwith slope

m = limx→a

f(x)− f(a)

x− aprovided that this limit exists.

3. Note. If limx→a

f(x)− f(a)

x− aexists then

limx→a

f(x)− f(a)

x− a= limh→0

f(a+ h)− f(a)

h


4. Example.

(a) Find the slope of the tangent line to the graph of f(x) = x3 at the pointi. x = 1

ii. x = 2

(b) Find the equation of the tangent line at each of the points above.

5. Example.

(a) Find the slope of the tangent to the curve

y =1√x

at the point where x = a.


(b) Find the equation of the tangent line at the point (1, 1).

6. The Most Important Definition in This Course.

Definition of Derivative. The derivative of a function f at a number a, denoted by f ′(a), is

f ′(a) = limh→0

f(a+ h)− f(a)

h

if this limit exists.

7. Note. If limx→a

f(x)− f(a)

x− aexists then f ′(a) = lim

x→a

f(x)− f(a)

x− a= limh→0

f(a+ h)− f(a)

h

8. Example. Find the derivative of the function

y =1

x− 1

at the point where x = 3.

9. Example. The following limit represents the derivative of some function f at some number a. Statef and a.

limh→0

2h+3 − 8

h


10. Example. Let f(x) = |x|. Does f ′(0) exist?

11. Must Know! An equation of the tangent line to y = f(x) at (a, f(a)) is given by

y − f(a) = f ′(a)(x− a) .

12. Example. Find the equation of the tangent line to f(x) =1

x− 1at the point where x = 3.

13. Compare the derivatives at each of the points on the graph.


14. Reminder. By definition

average velocity =displacement

time

15. More Precisely... Suppose an object moves along a straight line according to an equation of motions = f(t), where s is the displacement of the object from the origin at time t.

The average velocity of the object in the time interval from t = a to t = a+ h is given by

average velocity =f(a+ h)− f(a)

h.

16. BIG Question. What if h is small?

17. Definition. We define the velocity (or instantaneous velocity) v(a) at time t = a as

v(a) = limh→0

f(a+ h)− f(a)

h.

18. Example. If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters)after t seconds is given by

H = 58t− 0.83t2 .

(a) Find the velocity of the arrow when t = a.


(b) When will the arrow hit the moon?

(c) With what velocity will the arrow hit the moon?

19. Rates of Change. Let f be a function defined on an interval I and let x1, x2 ∈ I. Then the incre-ment of x is defined as

∆x = x2 − x1

and the corresponding change in y is

∆y = f(x2)− f(x1) .

The average rate of change of y with respect to x over the interval [x1, x2] is defined as

∆y

∆x=f(x2)− f(x1)

x2 − x1.

20. Must Know! The instantaneous rate of change of y with respect to x is defined as

lim∆x→0

∆y

∆x= limx2→x1

f(x2)− f(x1)

x2 − x1.


21. Example. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottomof the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank aftert minutes as

V = 100, 000

(1− t

60

)2

0 ≤ t ≤ 60 .

Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of Vwith respect to t) as a function of t. What are the units?


22. Example. The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at aprice of p dollars per pound is Q = f(p).

(a) What is the meaning of the derivative f ′(8)? What are the units?

(b) Is f ′(8) positive or negative? Explain.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 52

2.8 The Derivative as a Function


1. Quote. “I turn away with fear and horror from this lamentable sore of continuous functions withoutderivatives.”(Charles Hermite, French mathematician, 1822-1901.)

2. Reminder. The derivative of a function f at a number a, denote by f ′(a), is

f ′(a) = limh→0

f(a+ h)− f(a)

h


3. Find the derivative of the function f(x) = x2 at

(i) x = 0,(ii) x = 1,

(iii) x = 2,(iv) x = 10.


4. Problem. If a function f : I → R is given, find the set J ⊂ I such that f ′(x) exists for each x ∈ J .If J 6= ∅ then this new function f ′ : J → R is called the derivative of f .

5. Example. Letf(x) = x

23 =

3√x2 .

(i) Determine the domain of f .

(ii) Determine the formula for f ′(x). What is the domain of f ′?


(iii) Sketch graphs of f and f ′.


6. The graph of f is given. Sketch the graph of f ′.

7. Notation. For y = f(x) it is common to write:

f ′(x) = y′ =dy

dx=df

dx=

d

dxf(x) = Df(x) = Dxf(x)

Also,

f ′(a) =dy

dx

∣∣∣∣x=a

=dy

dx

]x=a

.

8. Definition. A function is differentiable at a if f ′(a) exists. It is differentiable on an openinterval (a, b) [or (a,∞) or (−∞, a) or (−∞,∞)] if it is differentiable at every number in the interval.


9. Two Questions.

(i) Is every continuous function differentiable?

(ii) Is every differentiable function continuous?

10. Three Cases. A function f is not differentiable at a number a from its domain if:

(i) The graph of f has a corner at the point (a, f(a));

(ii) f is not continuous at a;

(iii) The graph of f has a vertical tangent line when x = a.


11. Higher Derivatives. Suppose that f is a differentiable function. The second derivative of f isthe derivative of f ′.Notation.

(f ′)′ = f ′′

(y′)′ = y′′

d

dx

(dy

dx

)=d2y

dx2

12. Example. Find f ′′(x) if f(x) = x2.

13. Acceleration. The instantaneous rate of change of velocity with respect to time is called theacceleration of the object.

a(t) = v′(t) = s′′(t).

14. Example. The figure shows the graphs of three functions. One is the position function of a particle,one is its velocity, and one is its acceleration. Identify each curve.

Part 3

Differentiation Rules

59

PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 60

3.1 Derivatives of Polynomials and Exponential Functions


1. Quote. “Young man, in mathematics you don’t understand things, you just get used to them.” (Johnvon Neumann, Hungarian mathematician and polymath, 1903-1957)

2. Reminder. The derivative of a function f is the function f ′ defined by

f ′(x) = limh→0

f(x+ h)− f(x)

h

for all x for which this limit exists. Recall that we also use the notation ddx (f(x)) = f ′(x) for the

derivative.

3. Must Know!

(a) Derivative of a Constant.d

dx(c) = 0

(b) We have already seen that the following are true:

d

dx(x) = 1,

d

dx(x2) = 2x,

d

dx(x3) = 3x2.

You may be able to see a pattern. In fact, we have the following rule.

The Power Rule. If n is any real number, then

d

dx(xn) = nxn−1


(c) Constant Multiple Rule. If c is a constant and f is a differentiable function, thend

dx(cf(x)) = c · d

dxf(x)

(d) Sum Rule. If f and g are differentiable functions, thend

dx(f(x) + g(x)) =

d

dxf(x) +

d

dxg(x)

(e) The Derivative of a Polynomial. If

p(x) = anxn + an−1x

n−1 + . . .+ a1x+ a0

where n is a nonnegative integer and an 6= 0 then

p′(x) = nanxn−1 + (n− 1)an−1x

n−2 + . . .+ a1 .


4. Example. Find an equation of the tangent line to the curve y = 2x3 − 7x2 + 3x+ 4 at the point (1, 2).

5. Example. Find an equation for the straight line that passes through the point (1, 5) and it is tangentto the curve y = x3.

6. Fact. If f(x) = ax, a > 0, a 6= 1, is an exponential function then

f ′(0) = limh→0

ah − 1

h

exists.

7. Fact It is straightforward to show that if f(x) = ax then

f ′(x) = f ′(0) · ax .


8. Must Know! e is is the number such that

limh→0

eh − 1

h= 1 .

e ≈ 2.71828

9. Derivative of the Natural Exponential Function. If f(x) = ex is the natural exponential functionthen

f ′(x) = f(x) .

Thusd

dx(ex) = ex .

10. Example. Differentiate the function

f(x) = 2x3 + 3x23 − ex+2 .


11. Example. At what point on the curve y = ex is the tangent line parallel to the line y = 2x?

PART 3: DIFFERENTIATION RULES LECTURE 3.2 THE PRODUCT AND QUOTIENT RULES 66

3.2 The Product and Quotient Rules


1. Quote. ”Five out of four people have trouble with fractions.”

(Steven Wright, American comedian, 1955-)

2. Problem. Suppose we have two functions f(x) =3√x2 and g(x) = ex and we want to compute the

derivative of their product

d

dx(

3√x2ex).

How do we do this?

3. Product Rule. If f and g are both differentiable, then

d

dx[f(x)g(x)] = f(x)

d

dx[g(x)] + g(x)

d

dx[f(x)] .

In Newton’s notation this is written as (fg)′ = f · g′ + g · f ′.

4. Examples.

(a) Differentiate f(x) =3√x2 · ex.


(b) Differentiate g(x) = (x+ 1)(2x2 − x+ 1).

5. Quotient Rule. If f and g are differentiable, then

d

dx

[f(x)

g(x)

]=g(x)

d

dx[f(x)]− f(x)

d

dx[g(x)]

[g(x)]2.

In Newton’s notation this is written as(f

g

)′=g · f ′ − f · g′

g2.

6. Examples.

(a) Differentiate y =2t2 − 1

t3 + 1.

(b) Differentiate f(x) = e−x.


(c) If f(3) = 4, g(3) = 2, f ′(3) = −6, and g′(3) = 5, find the following numbers.i. (f + g)′(3)

ii. (fg)′(3)

iii.(f

g

)′(3)

iv.(

f

f − g

)′(3)


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 70

3.3 Derivatives of Trigonometric Functions


1. Quote. ”Trigonometry is the mathematics of sound and music.”(Frank Wattenberg, American mathematician, 1952-)

2. Problem. What is the derivative of sinx?

3. Must Know!

(a)d

dx(sinx) = cosx

(b)d

dx(cosx) = − sinx

(c)d

dx(tanx) = sec2 x

(d)d

dx(secx) = secx tanx

(e)d

dx(cscx) = − cscx cotx

(f)d

dx(cotx) = − csc2 x


4. Problem. Prove thatd

dx(sinx) = cosx .

5. Trigonometric Limits. Above we used the very important results

limθ→0

sin θ

θ= 1 and lim

θ→0

cos θ − 1

θ= 0.

We now prove these results.


6. Examples.

(a) Differentiate y =1 + tanx

x− cotx.

(b) Find the points on the curvey =

cosx

2 + sinx

at which the tangent is horizontal.

7. A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladderand the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of theladder slides away from the wall, how fast does x change with respect to θ when θ = π/3?


8. Examples. Evaluate

(a) limx→0

sin 2x

x

(b) limθ→0

sin 2θ

cos θ − 1


9. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 75

3.4 Chain Rule


1. Puzzle. A duck before two ducks, a duck behind two ducks, and a duck in the middle. How manyducks are there?

2. Reminder. The composition of the functions f and g is defined by

(f g)(x) = f(g(x)) .

3. Example. Let f(u) = sinu and g(x) = 1 + x2. Find F = f g.

4. Chain Rule. If f and g are both differentiable and F = f g is the composite function defined bF (x) = f(g(x)), then F is differentiable and F ′ is given by

F ′(x) = f ′(g(x)) · g′(x) .

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

dy

dx=dy

du· dudx

.

5. Examples.

(a) Let f(u) = sinu and g(x) = 1 + x2 and let F = f g. Find the derivative of F .


(b) Find y′ =dy

dx. if

i. y = (2− 5x)3

ii. y = (x+ sinx)5(1 + ex)2

iii. Express the derivative dy/dx in terms of x if

y = u5 and u =(4x− 1)2

x.


6. Examples. Find f ′.

(a)f(x) =

√2 + 5x2

(b)f(x) = (tan (x2))3

(c)f(x) = ecos x

7. Must Know!d

dx(ax) = ax ln a .


8. Examples.

(a) A pebble droppped into a lake creates an expanding circular ripple. Suppose that the radius ofthe circle is increasing at the rate of 2 in./s. At what rate is its area increasing when its radiusis 10 in.?

(b) Suppose that f(0) = 0 and f ′(0) = 1. Calculate the derivative of f(f(f(x))) at x = 0.

(c) Under certain circumstances a rumor spreads according to the equation

p(t) =1

1 + ae−kt

where p(t) is the proportion of the population that knows the rumor at time t and a and k arepositive constants.

i. Find limt→∞ p(t).ii. Find the rate of spread of the rumor.


9. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 80

3.5 Implicit Differentiation


1. Dictionary. implicitadjectiveDefinition:1. implied: not stated, but understood in what is expressedAsking us when we would like to start was an implicit acceptance of our terms.2. absolute: not affected by any doubt or uncertaintyimplicit trust3. contained: present as a necessary part of somethingConfidentiality is implicit in the relationship between doctor and patient.

2. Problem. The curvex3 + y3 = 3xy

is called the folium of Descartes. Find the equation of the tangent line at the point(

32 ,

32

).

3. Implicitly Defined Function. An equation in two variables x and y may have one or more solutionsfor y in terms of x or for x in terms of y. These solutions are functions that are said to be implicitlydefined by the equation.

4. Example

(a) x2 + y2 = 1

(b) x3 + y3 = 3xy


5. Implicit Differentiation.

(a) Use the chain rule to differentiate both sides of the given equation, thinking of x as the indepen-dent variable.

(b) Solve the resulting equation for dydx .

6. Example. The curvex3 + y3 = 3xy

is called the folium of Descartes. Find the equation of the tangent line at the point(

32 ,

32

).

7. Example. Suppose that water is being emptied from a spherical tank of radius 10 ft. If the depth ofwater in the tank is 5 ft and is decreasing at the rate of 3 ft/sec, at what rate is the radius r of the topsurface of the water decreasing?


8. Differentiation of an Inverse Function. Suppose f is a one-to-one differentiable function and itsinverse function f−1 is also differentiable. Use implicit differentiation to show that

(f−1)′(x) =1

f ′(f−1(x))

provided that the denominator is not 0.

9. Must Know!

(a)d

dx(sin−1(x)) =

1√1− x2

(b)d

dx(cos−1(x)) = − 1√

1− x2

(c)d

dx(tan−1(x)) =

1

1 + x2

(d)d

dx(cot−1(x)) = − 1

1 + x2


10. Example. Determine the points on the circle

(x− 1)2 + (y − 2)2 = 4

where the tangent line is horizontal or vertical.

11. Example. For the curvex2 + y2 = 5

find y′′ by implicit differentiation.


12. Orthogonal Trajectories.

(a) Two curves are called orthogonal if at each point of intersection their tangent lines are perpen-dicular.

(b) Two families of curves are orthogonal trajectories of each other if every curve in one familyis orthogonal to every curve in the other family.

(c) Show that the given families of curves are orthogonal trajectories of each other:

x2 + y2 = ax and x2 + y2 = by .

PART 3: DIFFERENTIATION RULES LECTURE 3.6 DERIVATIVE: LOGARITHMS 86

3.6 Derivatives of Logarithmic Functions


1. Quote. “One real estate development company advertised that an investment with it would growlogarithmically.”(From Ed Barbeaus column, Fallacies, Flaws, and Flimflam, in College Math. Journal 36 (2005),394-396.)

2. Must Know!d

dx(loga x) =

1

x ln a

3. When a = e this becomesd

dx(lnx) =

1

x.

4. Examples. Differentiate

(a) y = log2(3x2 + ex)

(b) y = ln(x+√x2 − 1)


(c) y =√

lnx

(d) y = ln√x

(e) y = ln

(x2

(x+ 3)4

)

5. More Examples. Differentiate

(a)

y =4√x3 5√x3 + 1

(2x+ 1)3


(b)y = xx

2

(c)y = ln |x|

6. Must Know!limx→0

(1 + x)1x = e


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 90

3.7 Rates of Change in the Natural and Social Sciences


1. Quote. ”If you want to see practical applied mathematics, read chemical engineering; if you want tosee theoretical applied mathematics, read electrical engineering.And if you want to read pure math, read economics.”(Unknown blogger.)

2. Example (Physics). The equation of motion for a particle is given by

s = 2t3 − 3t2 − 12t, t ≥ 0

where s is in meters and t is in seconds.

(a) Find the velocity and acceleration as functions of t.(b) The graph of s = s(t) is shown. Sketch the graphs of the velocity and acceleration functions for

0 ≤ t ≤ 4.(c) When is the particle speeding up? Slowing down?(d) What does the expression s′′′(t) = a′(t) represent?

[link to applet]

http://www.sfu.ca/~jtmulhol/calculus-applets/GeoGebra-Worksheets/particle-motion-Applet.html


3. Exercise.Let v(t) be a function which gives the velocity of a particle at time t. Consider the speed functionw(t) = |v(t)|.

(a) The particle is speeding up when w′(t) > 0. Show that is equivalent to the condition that v(t)and a(t) have the same sign.

(b) Similarly, the particle is slowing down when w′(t) < 0. Show that is equivalent to the conditionthat v(t) and a(t) have opposite signs.

[Hint: Remove the absolute value signs by writing w(t) as a piecewise defined function. Then differ-entiate the piecewise function, paying careful attention to the conditions which define each piece.]


4. Example (Chemistry). If one molecule C is formed from one molecule of the reactant A and onemolecule of the reactant B, and the initial concentrations of A and B have a common value [A] =[B] = a moles/L then

[C] =a2kt

akt+ 1

where k is a constant.

(a) Find the rate of reaction at time t.(b) Show that if x = [C], then

dx

dt= k(a− x)2

(c) What happens to the concentration as t→∞?(d) What happens to the rate of reaction as t→∞?


5. Example (Economics). Suppose that the cost (in dollars) for a company to produce x pairs of newline of jeans is

C(x) = 2000 + 3x+ 0.01x2 + 0.0002x3

(a) Find the marginal cost function.(b) Find C ′(100) and explain its meaning. What does it predict?(c) Compare C ′(100) with the cost of manufacturing the 101st pair of jeans.


6. Example. The height of a certain cylinder is always twice its radius r. If its radius is changing, showthat the rate of change of its volume with respect to r is equal to its surface area.


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 96

3.8 Exponential Growth and Decay


1. Quote. “It’s the whole issue with exponential growth, it’s very slow in the beginning but over thelong term it gets ridiculous.”(Drew Curtis, Founder and chief administrator of Fark.com, 1973 - )

2. Exponential Growth and Decay: A quantity q is said to be growing (or decaying) exponentially if

q = Aekt

where A and k are constants.

3. Natural Growth Equation. The solution of the initial-value problem

dy

dt= ky, y(0) = y0

isy(t) = y0e

kt .


4. Example. Calculopolis had a population of 25000 in 1980 and the population of 30000 in 1990.What population can the Calculopolis planers expect in the year 2020 if the population grows at arate proportional its size?


5. Radioactive Decay. Radioactive material is known to decay at a rate proportional to the amountpresent. This means, if N is the amount (mass) of radioactive material at time t then it must satisfythe model:

dN

dt= −kN, k > 0.

6. Example. It takes 8 days for 20% of a particular radioactive material to decay. How long does it takefor 100 grams of material to decay to 50 grams? 40 grams? 0 grams?

7. Remark. Usually k is specified in terms of the half-life of the isotope

τ =ln 2

k.

This is the time required for half of any given quantity to decay.


8. Newton’s Law of Cooling and Heating. If a warm object is put in cooler surroundings its tempera-ture will steadily decrease. A law of physics known as Newtons law of cooling says that the rate atwhich the object cools is proportional to the difference between its temperature and the surroundingtemperature. This law is modeled by the differential equation:

dT

dt= k(T −M)

where

• T (t) is the temperature of the object at time t• M is the temperature of the surroundings (ambient temperature - which is constant)• k a constant (called the cooling constant)

9. Example. When a cold drink is taken from a refrigerator, its temperature is 5C. After 25 minutesin a 20C room its temperature has increased to 10C.

(a) What is the temperature of the drink after 50 minutes?(b) When will its temperature be 15C?


10. Continuously Compound Interest. Suppose an amount A0 is invested at an interest rate r. IfA(t) is the amount after t years then

A(t) = A0(1 + r)t .

If the interest is compound n times per year then after t years the investment is worth

A(t) = A0

(1 +

r

n

)nt.

The formula for continuous compounding is

A(t) = A0ert .

Example: If $1000 is borrowed at 19% interest, find the amounts due at the end of 2 years if theinterest is compounded

(a) annually(b) quarterly(c) monthly(d) daily(e) continuously

PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 102

3.9 Related rates


1. Quote. “If you want to increase your success rate, double your failure rate. ”(Thomas John Watson, Sr., Founder of IBM, 1874 - 1956.)

2. A spherical balloon is being inflated. The radius r of the balloon is increasing at the rate of 0.2 cm/swhen r = 5 cm. At what rate is the volume V of the balloon increasing at that moment?


3. The Method of Related RatesWhen two variables are related by an equation and both are functions of a third variable (such astime), we can find a relation between their rates of change. In this case, we say the rates are related,and we can compute one if we know the other.

We proceed as follows:

(a) Identify the independent variable (usually time) on which the other quantities depend and as-sign it a symbol, such as t. Also, assign symbols to the variable quantities that depend on t.

(b) Find an equation that relates the dependent variables.(c) Differentiate both sides of the equation with respect to t (using the chain rule if necessary).(d) Substitute the given information into the related rates equation and solve for the unknown rate.


4. A rocket is launched vertically and is tracked by a radar station located on the ground 5 km from thelaunch pad. Suppose that the elevation angle θ of the line of sight to the rocket is increasing at 3 persecond when θ = 60. What is the velocity of the rocket at that instant?[link to applet]

http://www.sfu.ca/~jtmulhol/calculus-applets/GeoGebra-Worksheets/RelatedRates-Rocket/rocket.html


5. A man 6 ft tall walks with a speed of 8 ft/s away from a street light that is atop an 18-ft pole. Howfast is the tip of his shadow moving along the ground when he is 100 ft from the light pole?[link to applet]

http://www.sfu.ca/~jtmulhol/calculus-applets/GeoGebra-Worksheets/RelatedRates-HomersShadow/HomersShadow.html


6. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shorelineand its light makes four revolutions per minute. How fast is the beam of light moving along theshoreline when it is 1 km from P ?[link to applet]

http://www.sfu.ca/~jtmulhol/calculus-applets/GeoGebra-Worksheets/RelatedRates-Lighthouse/LighthouseBeam.html


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 108

3.10 Linear Approximation and Differentials


1. Quote. It is the mark of an instructed mind to rest satisfied with the degree of precision to whichthe nature of the subject admits and not to seek exactness when only an approximation of the truthis possible.(Aristotle, Greek philosopher, 384 BC - 322 BC.)

2. Problem. If f(1) = 4 and f ′(1) = 1 use the linear approximation to f(x) at x = 1 to approximatef(2).

3. Idea. Instead of evaluating f(x) evaluate L(x) where L is the tangent line to the graph of y = f(x)at a known point (a, f(a)) that is close to the point (x, f(x)).

4. Linear Approximation. The linear function

L(x) = f(a) + f ′(a)(x− a)

is called the linearization of f at a.For x close to a we have that

f(x) ≈ L(x) = f(a) + f ′(a)(x− a)

and this approximation is called the linear approximation of f at a.


5. Example. If f(1) = 4 and f ′(1) = 1 use the linear approximation to f(x) at x = 1 to approximatef(2).

6. Example. Use linear approximation to approximate√

37. What is the accuracy of this approxima-tion?


7. Differential. Let f be a function differentiable at x ∈ R. Let ∆x = dx be a (small) given number.The differential dy is defined as

dy = f ′(x)∆x .

8. Important!

f(a+ dx) ≈ L(a+ dx)

f(a+ dx) ≈ f(a) + f ′(a)(a+ dx− a)

f(a+ dx) ≈ f(a) + f ′(a)dx = f(a) + dy

dy ≈ f(a + dx)− f(a)

Small differential means good approximation.


9. Example. The equatorial radius of the earth is approximately 3960 mi. Suppose that a wire iswrapped tightly around the earth at the equator. Approximately how much must this wire be length-ened if it is to be strung all the way around the earth on poles 10 ft above the ground. (1 mi = 1760yards = 1760 · 3 ft.)

Part 4

Applications of the Derivative

Image Source: http://commons.wikimedia.org/wiki/File:Mount_Everest_as_seen_from_Drukair.jpg

113

http://commons.wikimedia.org/wiki/File:Mount_Everest_as_seen_from_Drukair.jpg

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 114

4.1 Maximum and Minimum Values


1. Quote. “I feel the need of attaining the maximum of intensity with the minimum of means. It is thiswhich has led me to give my painting a character of even greater bareness.”(Joan Miro, Catalan-Spanish artist, 1893 - 1983)

2. Definition. A function f has an absolute maximum at c if

f(c) ≥ f(x) for all x ∈ D, the domain of f .

The number f(c) is called the maximum value of f on D.A function f has an absolute minimum at c if

f(c) ≤ f(x) for all x ∈ D, the domain of f .

The number f(c) is called the minimum value of f on D.

3. Definition. A function f has a local maximum at c if

f(c) ≥ f(x) for all x in an open interval, in the domain, containing c .

A function f has a local minimum at c if

f(c) ≤ f(x) for all x in an open interval, in the domain, containing c .

4. Extreme Value Theorem. If f is continuous on a closed interval [a, b], then f attains an absolutemaximum value f(c) and an absolute minimum value f(d) at some numbers c, d ∈ [a, b].


5. Fermat’s Theorem. If f has a local maximum or minimum at c, and f ′(c) exists, then f ′(c) = 0.

6. Examples. Find all local extrema of

(a) f(x) = 3x4 − 16x3 + 18x2, − 1 ≤ x ≤ 4

(b) f(x) = |x|, − 1 < x < 1


7. Definition. A critical number of a function f is a number c in the domain of f such that eitherf ′(c) = 0 or f ′(c) does not exist.

8. Problem. Find the maximum and minimum values of the function

f(x) = x2 + 4x+ 7, −3 ≤ x ≤ 0

9. Closed Interval Method. To find the absolute maximum and minimum values of a continuousfunction f on a closed interval [a, b]:

(a) Find the values of f at the critical numbers of f in (a, b).(b) Find the values of f at the endpoints of the interval.(c) The largest of the values from Step (a) and Step (b) is the absolute maximum value; the smallest

of these values is the absolute minimum value.

10. Examples. Find the maximum and minimum values of the given functions on the indicated closedintervals.

(a) f(x) = x+ 4x , x ∈ [1, 4]


(b) g(x) = 2− 3√x, x ∈ [−1, 8]

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 119

4.2 The Mean Value Theorem


1. Quote. “The Mean Value Theorem is the midwife of calculus - not very important or glamorous byitself, but often helping to deliver other theorems that are of major significance.”(Edwin Purcell and Dale Varberg, American mathematicians)

2. Rolle’s Theorem. (Michel Rolle, French mathematician, 1652-1719) Let f be a function that satisfiesthe following three hypotheses:

(a) f is continuous on the closed interval [a, b].(b) f is differentiable on the open interval (a, b).(c) f(a) = f(b).

Then there is a number c in (a, b) such that f ′(c) = 0.

3. Example. Check if the following functions satisfy the hypotheses of Rolle’s theorem.

(a) f(x) = x1/2 − x3/2 on [0, 1].


(b) f(x) = 1− x2/3 on [−1, 1].

4. The Mean Value Theorem. Let f be a function that satisfies the following hypotheses:

(a) f is continuous on the closed interval [a, b].(b) f is differentiable on the open interval (a, b).

Then there is a number c in (a, b) such that

f ′(c) =f(b)− f(a)

b− a

or, equivalently,f(b)− f(a) = f ′(c)(b− a).


5. Example. A car is driving along a rural road where the speed limit is 70 km/h. At 3:00 pm itsodometer reads 18075 km. At 3:18 its reads 18100 km. Prove that the driver violated the speed limitat some instant between 3:00 and 3:18 pm.

6. Show that the equation x4 = x+ 1 has exactly one solution in the interval [1, 2].


7. Must Know! If f ′(x) = 0 for all x in an interval (a, b), then f is constant on (a, b).

8. Fact. If f ′(x) = g′(x) for all x in an interval (a, b), then f − g is constant on (a, b); that is,

f(x) = g(x) + c

where c is a constant.

9. Example. Prove the identity

arcsin

(x− 1

x+ 1

)= 2 arctan (

√x)− π

2

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 124

4.3 How Derivatives Affect the Shape of a Graph


1. Quote. “The spread of civilization may be likened to a fire; First, a feeble spark, next a flickeringflame, then a mighty blaze, ever increasing in speed and power.”(Nikola Tesla, American inventor and engineer, 1856 - 1943)

2. Increasing/Decreasing Test.

(a) If f ′(x) > 0 on an interval, then f is increasing on that interval.(b) If f ′(x) < 0 on an interval, then f is decreasing on that interval.


3. Example. Find the open intervals on the x-axis on which the function

f(x) = 3x4 − 4x3 − 12x2 + 5

is increasing and those on which is decreasing.

4. The First Derivative Test. Suppose that c is a critical number of a continuous function f .

(a) If f ′ changes from positive to negative at c, then f has a local maximum at c.(b) If f ′ changes from negative to positive at c, then f has a local minimum at c.(c) If f ′ does not change sign at c, then f has no local minimum or maximum at c.

5. Example. Find all local extrema of the function

f(x) = 3x4 − 4x3 − 12x2 + 5


6. Examples. Find all local extrema off(x) = |x|

7. Definition. If the graph of f lies above all of its tangent lines on an interval I, then it is calledconcave upward on I. If the graph of f lies below all of its tangents on I, it is called concavedownward on I.


8. Concavity Test.

(a) If f ′′(x) > 0 for all x ∈ I, then the graph of f is concave upward on I.(b) If f ′′(x) < 0 for all x ∈ I, then the graph of f is concave downward on I.

9. Definition. A point P on a curve y = f(x) is called an inflection point if f is continuous there andthe curve changes from concave upward to concave downward or from concave downward to concaveupward at P .


10. The Second Derivative Test. Suppose f ′′ is continuous near c.

(a) If f ′(c) = 0 and f ′′(c) > 0 then f has a local minimum at c.(b) If f ′(c) = 0 and f ′′(c) < 0 then f has a local maximum at c.

11. Example. Sketch the graph of the function

f(x) = 3x4 − 4x3 − 12x2 + 5

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.4 L’HOSPITAL’S RULE 130

4.4 Indeterminate Forms and L’Hospital’s Rule


1. “Proof”. Let a = b.a2 = ab (Multiply both sides by a.)a2 + a2 − 2ab = ab+ a2 − 2ab (Add a2 − 2ab to both sides.)2(a2 − ab) = a2 − ab (Factor the left, and collect like terms on the right.)2 = 1 (Divide both sides by a2 − ab.)

2. Indeterminate Forms.

(a) Indeterminate form of type 00 .

Example. Evaluate limx→0

sin kx

xfor k ∈ R.

(b) Indeterminate form of type ∞∞ .

Example. Evaluate limx→∞

ax+ 1

bx+ 1for a, b ∈ R.


3. L’Hospital’s Rule. Suppose that f and g are differentiable and g′(x) 6= 0 near a (except possibly ata.) Suppose that

limx→a

f(x) = 0 and limx→a

g(x) = 0

or thatlimx→a

f(x) = ±∞ and limx→a

g(x) = ±∞

Thenlimx→a

f(x)

g(x)= limx→a

f ′(x)

g′(x)

if the limit on the right side exists (or is∞ or −∞).

4. Examples. Find

(a) limx→0

ex − 1

sin 2x

(b) limx→∞

ex

x2 + x

(c) limx→∞

lnx√x


5. Indeterminate Form 0 · ∞. Find limx→∞

x ln

(x− 1

x+ 1

).

6. Indeterminate Form∞−∞. Find limx→0

(1

x− 1

sinx

).

7. Indeterminate Form 00,∞0,1∞.

8. Examples. Find

(a) limx→0

(cosx)1/x2


9. Additional Notes

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHING 134

4.5 Summary of Curve Sketching


1. Puzzle. Connect the nine dots with four (only four) straight lines without ever lifting your pen orpencil from the paper.

2. Guidlines

(a) Domain(b) Intercepts: For the x-intercepts set y = 0 and solve for x. For the y-intercept calculate f(0).(c) Symmetry:

i. Even function - symmetric about the y-axis.ii. Odd function - symmetric about the origin.

iii. Periodic functions.(d) Asymptotes:

i. Horizontal Asymptotes: y = L if limx→∞

f(x) = L or if limx→−∞

f(x) = L.

ii. Vertical Asymptotes: x = a if at least one of the following is true

limx→a+

f(x) =∞ limx→a−

f(x) =∞

limx→a+

f(x) = −∞ limx→a−

f(x) = −∞.

iii. Slant Asymptotes: y = mx+ b if

limx→∞

(f(x)− (mx+ b)) = 0.

(e) Intervals of Increase and Decrease:f ′(x) > 0 on an interval I means f increasing on I.f ′(x) < 0 on an interval I means f decreasing on I.

(f) Local Maximum and Minimum Values:- First Derivative Test- Second Derivative Test

(g) Concavity and Points of Inflection:f ′′(x) > 0 on an interval I means f concave up ( on I.f ′′(x) < 0 on an interval I means f concave down ) on I.

(h) Sketch the curve.


3. Examples. Sketch graphs of the following functions.

(a) f(x) =2 + x− x2

(x− 1)2


(b) f(x) = x2ex


4. In this example we will focus just on asymptotes in the guidelines outlined in (2).

Determine the asymptotes of the function f(x) =x2 + x− 1

x− 1.


5. Additional Notes

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.6 OPTIMIZATION PROBLEMS 139

4.6 Optimization Problems



1. Quote. “There is no branch of mathematics, however abstract, which may not someday be applied tothe phenomena of the real world.”(Nikolai Lobachevski, Russian mathematician, 1792 - 1856)

2. Examples.

(a) Find the dimensions of the right circular cylinder with greatest volume that can be inscribed ina right circular cone of radius 8 cm and height 12 cm.


(b) A painting in an art gallery has height 3 m and is hung so that its lower edge is about 1 m abovethe eye of an observer. How far from the wall should the observer stand to get the best view?(i.e. the observer wants to maximize the angle θ subtended at the eye by the painting.)[link to applet]For an interesting paper connecting this problem to exponential functions click here. [2]

http://www.sfu.ca/~jtmulhol/calculus-applets/GeoGebra-Worksheets/optimization-art-gallery/art-gallery.html

http://myweb.facstaff.wwu.edu/curgus/Papers/ExcExpFinal.pdf


(c) The frame for a kite is to be made from six pieces of wood. The four exterior pieces have beencut with the lengths indicated in the figure. To maximize the area of the kite, how long shouldthe diagonal pieces be?


(d) Maya is 2 km offshore on a boat and wishes to reach a coastal village which is 6 km down thestraight shoreline form the point on the shore nearest to the boat. She can row at 2 km/hour andrun 5 km/hour. Where she should land her boat to reach the village in the least amount of time?[link to applet]

http://www.sfu.ca/~jtmulhol/calculus-applets/GeoGebra-Worksheets/optimization-row-run/row-run.html


3. Additional Notes

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.7 NEWTON’S METHOD 144

4.7 Newton’s Method


1. Quote. “Sometimes, close enough is good enough.”(Math Girl, Episode I - Differentials Attract)

2. Newton’s Method.

(a) Problem. Find a solution, say x = r, to

f(x) = 0.

(b) Idea.i. Let x1 be a ”good” estimate of r.

ii. Consider the tangent line L to the curve y = f(x) at the point (x1, f(x1)). Look at the x-intercept of L, call it x2. If x1 is close to r then x2 seems to be even closer to r, and we use x2

as a second approximation to r.How do we find x2 in terms of x1 and f?

iii. Repeat this procedure to get a third approximation x3 from our second approximation x2:

If we keep repeating this process we obtain a sequence of approximations for r:

x1, x2, x3, . . . .

iv. If the numbers xn become closer and closer to r as n becomes large then we say that thesequence converges to r and we write

limn→∞

= r.


(c) Method.i. Begin with an initial guess x1.

ii. Calculatex2 = x1 −

f(x1)

f ′(x1).

iii. If xn is known then

xn+1 = xn −f(xn)

f ′(xn).

iv. If xn and xn+1 agree to k decimal places then xn approximates the root r up to k decimalplaces and f(xn) ≈ 0.

3. Example. Solve5x+ cosx = 5

for x ∈ [0, 1] correct to 6 decimal places.


4. Example. Use Newton’s method to find√

2 accurate to eight decimal places.

5. Example. Use Newton’s method to solve x1/3 = 0 by taking x0 = 1.


6. Example. Let f(x) = x3 + 3x+ 1.

(a) Show that f has exactly one root in the interval (− 12 , 0). Explain.

(b) Use Newton’s method to approximate the root that lies in the interval (−12 , 0). Stop when the

next iteration agrees with the previous one at two decimal places.


7. Additional Notes

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.8 ANTIDERIVATIVES 149

4.8 Antiderivatives


1. Quote. “In school I had a friend named Cos Davis. We nicknamed him the Antiderivative of SinDavis. He actually liked it.”(Posted by SLD on May 14, 2004, 10:39 PM, at Stupid math or science jokes thread (again),

http://www.iidb.org/vbb/archive/index.php/t-85561&e=747

”Wouldn’t the antiderivative of Sin Davis be Negative Cos Davis?”(Posted by LVLLN on May 15, 2004, 03:30 PM)

2. Problem. For a given function f find all functions F with the property that

F ′ = f .

3. Definition. A function F is called an antiderivative of f on an interval I if

F ′(x) = f(x)

for all x ∈ I.

4. Example: Find an antiderivative of the following.

(a) f(x) = x2

(b) f(x) =1

x

5. Example. Show that all antiderivatives of f(x) = xn, n 6= −1, are given by

F (x) =1

n+ 1xn+1 + C

where C is an arbitrary constant.


6. Theorem. If F is an antiderivative of f on an interval I, then the most general antiderivative of fon I is

F (x) + C

where C is an arbitrary constant.

7. Table of Antiderivative Formulas.(f = Function, F = Particular antiderivative)

f F f F

cg(x) cG(x) sinx − cosx

g(x) + h(x) G(x) +H(x) sec2 x tanx

xn, n 6= −1 1

n+ 1xn+1 secx tanx secx

1/x ln |x| 1√1− x2

sin−1 x

ex ex1

1 + x2tan−1 x

cosx sinx


8. Example. Find all antiderivatives of

f(x) = x3 + 3√x− 4

x2

9. Example.(a) Find f if f ′(t) = 2 cos (3t) + 5 sin (4t) + 3et.(b) Which of the functions in (a) satisfies f(0) = 0?

10. Example. Find f such that f ′′(x) = 2e−2x + 5, f(0) = 3/2 and f ′(0) = 1.


11. Rectilinear Motion.

(a) Position function - s = f(t)

(b) Velocity function - v(t) = s′(t)

(c) Acceleration function - a(t) = v′(t)

12. Example. The skid marks made by an automobile indicate that its brakes are fully applied for adistance of 160 ft before it came to a stop. Suppose that the car in question has a constant decelerationof 20 ft/s2 under the condition of the skid. How fast was the car traveling when its brakes wereapplied?

PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 4.8 ANTIDERIVATIVES 154

Part 5

Parametric Equations and PolarCoordinates

155

PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.1 PARAMETRIC EQUATIONS 156

5.1 Curves Defined by Parametric Equations

(This lecture corresponds to Sections 10.1 & 10.2 (Tangents part only) of Stewart’s Calculus.)

1. Quote. “When you get to the top of the mountain, keep climbing.”(Zen proverb)

2. Motivation: Particle moving in plane

3. Problem. In the xy-plane draw the set P = (t2, t) : t ∈ R.


4. Vocabulary. Let I be an interval and let f and g be continuous on I.

(a) The set of points C = (f(t), g(t)) : t ∈ I is called a parametric curve.(b) The variable t is called a parameter.(c) We say that the curve C is defined by parametric equations

x = f(t), y = g(t).

(d) We say that x = f(t), y = g(t) is a parametrization of C.(e) If I = [a, b] then (f(a), g(a)) is called the initial point of C and (f(b), g(b)) is called the terminal

point of C.

5. Example. Find two parametrizations of the unit circle x2 + y2 = 1 .

6. Example. Find parametric equations for the ellipsex2

a2+y2

b2= 1 .


7. Example. Sketch the graph of the curve defined by parametric equations

x = sin t, y = sin2 t, −∞ < t <∞

8. Some neat examples:

Lissajous Curve:x = cos at, y = sin bt, t ∈ [0, 2π]

Knotty Curve:x = t+ 2 sin 2t, y = t+ 2 cos 5t, t ∈ [−2π, 2π]


9. Cycloid. The curve traced by a point P on the edge of a rolling circle is called a cycloid. The circlerolls along a straight line without slipping or stopping. Find parametric equations for the cycloid ifthe line along which the circle rolls is the x-axis but always tangent to it, and the point P begins atthe origin.

10. Derivatives of Parametric Curves. The derivative to the parametric curve x = f(t), y = g(t) isgiven by

dy

dx=

dydtdxdt

=g′(t)

f ′(t).


11. Example. Find the slope of the tangent line to the curve

x = cos t, y = sin t, 0 ≤ t ≤ 2π.

at the point corresponding to t = π/4.

12. Example. Determine the points on the cycloid where the tangent line is horizontal or vertical.

PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 161

5.2 Polar Coordinates


1. Ancient math joke:Q: What’s a rectangular bear?A: A polar bear after a coordinate transform.

2. Motivation: Given a point in the plane how can we describe its position?

3. Polar Coordinate System.

(a) Choose a point in the plane. Call it O, which we also call the pole.(b) Choose a ray starting at O. Call it the polar axis. (Usually taken as positive x axis.)(c) Take any point P , except O, in the plane. Measure the distance d(O,P ) and call this distance r.(d) Measure the angle between the polar axis and the ray starting at O and passing through P going

from x in counterclockwise direction. Let θ be this measure in radians.(e) There is a bijection between the plane and the set

R+ × [0, 2π) = (r, θ) : r ∈ R+ and θ ∈ [0, 2π)

This means that each point in the plane is uniquely determined by a pair (r, θ) ∈ R+ × [0, 2π).(f) r and θ are called polar coordinates of P .


4. Example. Plot the points whose polar coordinates (r, θ) are given.(a) (1, π/4) (b) (2, 5π/4) (c) (2,−π/3) (d) (−1, 5π/6)

5. Example. Plot the three points whose polar coordinates are (1, π/2), (1, 5π/2), and (−1, 3π/2).

6. Example. Plot the point given by the polar coordinates (3, π/3). Then find two other pairs of polarcoordinates of this point, one with r > 0 and one with r < 0.

7. Example. Find the connection between polar and Cartesian coordinates.


8. Example. Convert the point (2, π/6) from polar to Cartesian coordinates.

9. Example. Plot the point whose polar coordinates are (2√

2, 3π/4). Find the Cartesian coordinates ofthis point.

10. Example. Represent the point with Cartesian coordinates (−1, 1) in terms of polar coordinates.


11. Example. The cartesian coordinates of a point are (−2√

3,−2). Find polar coordinates (r, θ) of thispoint, where r > 0 and 0 ≤ θ < 2π).

12. Example. Sketch the region in the plane consisting of points whose polar coordinates satisfy:

1 ≤ r < 2,π

4≤ θ < 3π

4.

13. Polar Curves: The graph of a polar equation r = f(θ) consists of all points P that have at least onepolar representation (r, θ) whose coordinates satisfy the equation.

Some examples of polar curves are:


14. Example. What curve is represented by the polar equation r = 3?

15. Example. Sketch the graph of the curve r = 2 sin θ.


16. Example. Sketch the graph of the curve r = 2 cos 3θ.

17. Derivatives of Polar Curves. Suppose that r = f(θ) is a differentiable function of θ. Then fromthe parametric equations

x = r cos θ y = r sin θ

it follows that

dy

dx=

dy

dθdx

dθ

=

dr

dθsin θ + r cos θ

dr

dθcos θ − r sin θ

18. Example. Find an equation of the tangent line to the curve r = 2 cos 3θ when θ = 2π/3.


19. Example. Find an equation of the tangent line to the curve r = 1 + cos θ if θ = π/6.


20. Consider the curve given by the polar equation

r = 1 + 2 sin θ.

Find the point(s) on the curve which are furthest from the x-axis.

PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 170

5.3 Conic Sections


1. Quote. “If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, butif I already know it what does the formula teach me? ”(St. Augustine)

2. Objective:We are going to give geometric descriptions of parabolas, el-lipses and hyperbolas and derive their equations in Cartesiancoordinates. These types of curves are called conic sections,or just conics, because they arise from intersecting a cone anda plane.

3. Reminder.

• The distance d between two points (x1, y1) and (x2, y2) is given by

d =√

(x2 − x1)2 + (y2 − y1)2.

• The midpoint M between two points (x1, y1) and (x2, y2) is given by

M =

(x1 + x2

2,y1 + y2

2

).

4. Geometric Definition of a Parabola.A parabola is the set of points in a plane that are equidistant from a fixed point F and a fixed line.

5. Terminology:

• The point F is called the focus and the line is called the directrix.• The line through the focus perpendicular to the directrix is called the axis of the parabola.• The intersection of the parabola and the axis is called the vertex. (Note: The vertex is halfway

between the focus and the directrix.)


6. Deriving the Equation for a Parabola:We get a simple equation for a parabola in Cartesian coordinates, if we start by making the axis ofthe parabola the y-axis.Then we place the vertex of the parabola at the origin O, and the x-axis parallel to the directrix.With the focus at (0, p), the equation of the directrix is .


7. Theorem. An equation of the parabola with focus (0, p) and directrix y = −p is

y = ax2

where a =1

4p.

When p > 0 the parabola opens upward, and when p < 0 it opens downward.

8. If we interchange the x and y in the equation x2 = 4py, we obtain y2 = 4px which is the equation of aparabola opening to the right if p > 0, or to the left if p < 0. (Interchanging x and y means reflectingabout the line y = x.)All four cases for the parabola and the equation are shown below.


9. Example. Find the vertex, focus, and directrix of the parabola and sketch its graph.

2y2 = 5x

10. Example. Find an equation of the parabola. Then find the focus and directrix.

11. Geometric Definition of an Ellipse.An ellipse is the set of points in a plane the sum of whose distances from two fixed points F1 and F2

is a constant. (F1 and F2 are called foci.)


12. Deriving the Equation for an Ellipse:To get a simple equation for an ellipse, let the foci be at (−c, 0) and (c, 0).This means the origin is halfway between the foci.Let the sum of the distances from a point on the ellipse to both foci be the constant 2a > 0.

x 2 1 F (!c, 0)

P (x, y)

y

F (c, 0)


13. Theorem. The ellipsex2

a2+y2

b2= 1, a ≥ b > 0

has foci (±c, 0), where c2 = a2 − b2, and vertices (±a, 0).

14. Terminology:

• The points (a, 0) and (−a, 0) are both on the ellipse, and are called vertices.• The line segment joining the vertices is called the major axis. The length is 2a.• The line segment joining the center to one of the vertices is called the semimajor axis. The

length of the semimajor axis is a which is also called the long radius of the ellipse.

15. If we interchange x and y then the equation for the ellipse becomes

x2

b2+y2

a2= 1, where a ≥ b > 0.

The foci are (0,±c), and the vertices are (0,±a).(Remember, interchanging x and y is the same as reflecting in the diagonal line y = x.)

x

(0, %)

(0, c)

(!(, 0) ((, 0)

(0, !c)

y

(0, !%)


16. Example. Find the vertices and the foci of the ellipse and sketch its graph.

100x2 + 36y2 = 225

17. Geometric Definition of a Hyperbola.A hyperbola is the set of points in a plane the difference of whose distances from two fixed points F1

and F2 is a constant. (F1 and F2 are called foci.)

P

1 2F (c, 0)(!c, 0)F

Notice the similarity to the definition of the ellipse. The only change here is that the sum of distanceshas become a difference.


18. This similarity to the ellipse means that the method of construction of an equation for the hyperbolais very similar to that of the ellipse.If we let the foci lie on the x-axis at (±c, 0), and the distance difference be 2a then the equation of thehyperbola is:

x2

b2− y2

a2= 1,

where c2 = a2 + b2.The x-intercepts (±a, 0) are called the vertices and the lines y = ±(b/a)x are the asymptotes.

(!c, 0)

(', 0)(!', 0)

(c, 0)

19. If the foci of the hyperbola are on the y-axis, then we switch x and y to get the equation

y2

a2− x2

b2= 1,

with foci (0,±c) where c2 = a2 + b2 and vertices (0,±a) and asymptotes y = ±(a/b)x.

(0, !c)

(0, ')

(0, !')

(0, c)


20. Conics That Are Shifted: So far, we have been constructing formulas for the conics by placing theorigin at the simplest possible location. What happens if we shift the conics?In general, we replace x and y by x− h and y − k , for constants h and k.

21. Example. Find an equation of the ellipse with foci (2,−2) and (4,−2), and with vertices (1,−2) and(5,−2).

PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.4 CONIC IN POLAR COORDS. 180

5.4 Conic Sections in Polar Coordinates


1. Math Joke.Q: Why do pirates work in polar coordinates?A: So their work involves lots or AARRR!!!!

2. Objective: We’re going to look at parabolas, ellipses and hyperbolas again, but this time in polarform. To begin with, we can give a more unified description of all three types of conic sections interms of a focus and directrix. This will allow us to see that conic sections can be easily described inpolar coordinates.

3. Definition.Let F be a fixed point, called the focus, and l be a fixed line, called the directrix. Suppose e is afixed positive number, called the eccentricity.

The set of all points P in the plane such that|PF ||Pl|

= e is a conic section.

If e < 1 the conic is an ellipse.If e = 1 the conic is a parabola.If e > 1 the conic is a hyperbola.

This definition of the conic sections is equivalent to the geometric definitions given earlier. The proofof this is in the text, and involves some algebra.Notice that when e = 1, we have |PF | = |Pl| for the definition of the parabola, as before.

4. Using the definition for conic sections given above find equations in polar coordinates which describethe curves.

P

x = d

F x

y

d


5. If the directrix is chosen to be on the left of the focus, at x = −d, then the conic section is

r =ed

1− e cos θ.

See the first figure below.

! # !$

!F

%

% # !$ directrix

!F

%

% # $ directrix

%

F !

directrix

6. If the directrix is chosen parallel to the x-axis then the conic section is as shown in the second twofigures of the diagram above.

7. Theorem. A polar equation of the form

r =ed

1± e cos θor r =

ed

1± e cos θ

represents a conic section with eccentricity e. The conic is an ellipse if e < 1, a parabola if e = 1, or ahyperbola if e > 1.

8. Example. Find a polar equation for an ellipse, with eccentricity 12 , that has its focus at the origin

and whose directrix is the line y = 4.


9. Example. A conic is given by the polar equation

r =3

2 + 2 cos θ.

Find the eccentricity, identify the conic, locate the directrix, and sketch the conic.

10. Example.

PART 6: REVIEW MATERIAL CONIC IN POLAR COORDS. 184

Part 6

Review Material

185

PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 186

6.1 Midterm 1 Review Package

1. Make sure you know the definitions of the terms: function, one-to-one, inverse of a function, compo-sition, even function, odd function, limit of a function, continuous function, asymptote, squeeze law,intermediate value property, derivative, etc.

2. Write sin2 (ex2−x) as a composition of (elementary) functions.

3. Compute the following limits.

(a) limx→2

(x2 − 1)3

(b) limx→0

ex2−1 + sinx

x+ 1

(c) limx→3−

x2 − 9

|x− 3|


(d) limx→0

1−√

1− x2

x

(e) limh→0

f(4 + h)− f(4)

hwhere f(x) =

1√x

(f) limx→∞

ln

(∣∣∣∣ 2− 3x3

x3 + 5x− 4

∣∣∣∣)


4. True or False. Justify your answers.

(a) If f(s) = f(t) then s = t.

(b) If f is an odd function and f(3) = 6 then f(−3) = −6.

(c) If x1 < x2 and g is a decreasing function then g(x1) > g(x2).

(d) If f and g are functions then f g = g f .

(e) limx→4

(2x

x− 4− 8

x− 4

)= limx→4

(2x

x− 4

)− limx→4

(8

x− 4

).

(f) If limx→5

f(x) = 2 and limx→5

g(x) = 0, then limx→5

f(x)

g(x)does not exist.


(g) If limx→5

f(x) = 0 and limx→5

g(x) = 0, then limx→5

f(x)

g(x)does not exist.

(h) If g(1) = −1 and g(2) = 5 then there exists a number c between 1 and 2 such that g(c) = 0.

(i) If 1 ≤ f(x) ≤ x2 + 2x+ 2 for all x near −1, then limx→−1

f(x) = 1.

(j) If the line x = 1 is a vertical asymptote of y = f(x), then f is not defined at 1 .

(k) The equation x+ ln (x+ 1) = x4 − 1 has a root in the interval (0, 2).

(l) If f is continuous on [1, 5] such that f(2) = 8 and the only solutions of the equation f(x) = 6 arex = 1 and x = 4 then f(3) > 6.

(m) If f ′(r) exists, then limx→r f(x) = f(r).


5. Determine the constant c that makes f continuous on (−∞,∞).

f(x) =

c2 + sin (xπ) if x < 2

cx2 − 4 if x ≥ 2.


6. Is there a number b such that

limx→−2

3x2 + bx+ b+ 3

x2 + x− 2

exists? If so, find the value of b and the value of the limit.

7. Evaluate limx→2

√6− x− 2√3− x− 1


8. Let f(x) =1

x2 − x. Calculate f ′(x) directly from the definition of derivative. Find the tangent line to

the curve y = f(x) at the point (2, f(2)).

9. For what value of x does the graph of f(x) = ex − 2x have a horizontal tangent?

10. Where does the normal line to the parabola y = x − x2 at the point (1, 0) intersect the parabola asecond time? Illustrate with a sketch.


Answers:Only answers are provided here. You are expected to provide fully worked out solutions. If you need helpwith solving any of these problems please visit the Calculus Workshop.

1. Know the precise statements of definitions and theorems as found in the textbook and the notes.

2. f g h k where f(x) = x2, g(x) = sinx, h(x) = ex, h(x) = x2 − x.

3. (a) 27

(b) e−1 = 1e

(c) −6

(d) 0

(e) −116

(f) ln 3

4. (a) F (b) T (c) T (d) F (e) F (f) T (g) F (h) F (i) T (j) F (k) T (l) T (j) T.

5. c = 2

6. b = 15; limit is −1

7. 12

8. (a) −3/4 (b) y = (−3/4)x+ 2

9. x = ln 2

10. (−1,−2)


6.2 Midterm 2 Review Package

1. Compute the following derivatives. You do not need to simplify your answers.

(a) f ′(x) if f(x) = (2x6 − 4x+ 3)4.

(b) g′(x) if g(x) =secx

xex.

(c) h′(x) if h(x) =3 + 2 sinx

x3 + 1

(d) y′ if y = x2 log3 (x2/3)


(e)ds

dtif s = 2t

2

(f) h(51)(t) if h(t) = ln (t2). (Compute the first few derivatives to find a pattern.)

(g)dy

dx

∣∣∣∣x=0

if 2

(x

y

)− ln (x+ y) = 0

(h) y′ if y = xcos x.


2. Suppose f is a differentiable such that f(g(x)) = x and f ′(x) = 1 + [f(x)]2. Show that g′(x) =1

1 + x2.

3. True or False. Justify your answers.

(a) If f and g are differentiable then the derivative of f(x)g(x) is f ′(x)g′(x).

(b) The function f(x) = |x| is differentiable for all real numbers.

(c) If f is differentiable, thend

dx

√f(x) =

f ′(x)

2√x

.

(d)d

dx(10x) = x10x−1.


4. The graph of f is given below. Sketch the graph of f ′ and f ′′(x) on coordinate axes below. You do NOTneed to justify your answers.


5. If a hemispherical bowl of radius 10 in. is filled with water to a depth of x inches, the volume of wateris given by V = π(10− x

3 )x2. Find the rate of increase of volume per inch increase in depth.

6. Use logarithmic differentiation to find the derivative y′ of the following function (you do not need tosimplify your answer)

y =

√x2 + 1 (3− 4x)5

2(3x− 1)1/4 (x− 2)4


7. Consider the curve defined by y2 = x3 + 5x2. The graph of the curve is shown below.

(a) Show that the point (−1, 2) is on the curve.

(b) Use implicit differentiation to finddy

dx.

(c) Find the equation of the tangent line to the curve at the point (−1, 2).


8. The position of a particle moving along a straight line is given by the function

s(t) = t3 − 15

4t2 + 3t+ 2, t ≥ 0

(a) What is the particles starting position?

(b) When is the particle speeding up? When is it slowing down? When is it stopped?

(c) Find the total distance the particle travels in the time interval 0 ≤ t ≤ 4.


9. The position at time t ≥ 0 of a particle moving along a coordinate line is

s(t) = 10 cos (t+ π/4).

(a) What is the particles starting position (t = 0)?

(b) What are the points farthest to the left and right of the origin reached by the particle.

(c) Find the particles velocity and acceleration at the points in (b).

(d) When does the particle first reach the origin? What are its velocity and acceleration at thispoint?


10. At 12:00 an apple pie is removed from the oven and placed on a table to cool. The temperature of theroom is 24C. At 12:20 the temperature of the pie is 36C and is decreasing at a rate of 2/min. Whatwas the temperature of the pie when it was brought out of the oven?

11. The designer of a 30-ft-diameter spherical hot-air balloon wishes to suspend the gondola 8 ft belowthe bottom of the balloon with suspension cables tangent to the surface of the balloon. Two of thecables are shown running from the top edges of the gondola to their points of tangency, (−12,−9) and(12,−9). How wide must the gondola be?


12. A girl facing North is standing next to a river which flows East. She tosses a stick into the waterexactly 4 meters North of where she stands. The river carries the stick East at the constant rate of 3m/s. How fast is the stick moving away from the girl after 2 seconds?

13. A cup of coffee, cooling off in a room at temperature 20C, has cooling constant k = 0.09min−1.

(a) How fast is the coffee cooling (in degrees per minute) when its temperature is T = 80C?(b) Use linear approximation to estimate the change in temperature over the next 6 seconds when

T = 80C.(c) The coffee is served at a temperature of 90. How long should you wait before drinking it if the

optimal temperature is 65C? Note: You may leave your answers in the exact form, i.e.,as an expression that contains powers of e and/or logarithms.


Answers:Only answers are provided here. You are expected to provide fully worked out solutions. If you need helpwith solving any of these problems please visit the Calculus Workshop.

1. (a) f ′(x) = 4(2x6 − 4x+ 3)3(12x5 − 4)

(b) g′(x) =( secx

x2ex

)(x tanx− x− 1)

(c) h′(x) =2(x3 + 1) cosx− (3x2)(3 + 2 sinx)

(x3 + 1)2

(d) y′ = 2x log3 (x2/3) +

(2

3 ln 3

)x =

(2x

3 ln 3

)(2 lnx+ 1)

(e)ds

dt= 2 ln (2)t2t

2

(f) h(51)(t) = 2 · (50!)t−51

(g)dy

dx

∣∣∣∣x=0

= 1

(h) xcos x(cosx

x− ln (x) sinx

)2. Use implicit differentiation on f(g(x)) = x and solve for g(x).

3. (a) False (b) False (c) False (d) False

5.dV

dx= πx(20− x)

6. y′ =

√x2 + 1 (3− 4x)5

2(3x− 1)1/4 (x− 2)4

(x

x2 + 1− 20

3− 4x− 3

4(3x− 1)− 4

x− 2

)

7. (b) y′ =x(3x+ 10)

2y(c) y =

−7

4x+

1

4

8. (a) 2(b) stopped at t = 1/2 and 2. Speeding up when

1

2< t <

5

4or 2 < t. Slowing down when 0 < t <

1

2or

5

4< t < 2.

(c) 155/8

9. (a) 10/√

2(b) Point furthest to the left are s = −10 and these occur when t = (2n+ 1)π − π

4 , where n ∈ Z. Pointfurthest to the right are s = 10 and these occur when t = 2nπ − π

4 , where n ∈ Z.(c) For leftmost points v = 0 and a = 10. For rightmost points v = 0 and a = −10. (d) t = π/4.v(π/4) = −10 and a(π/4) = 0.

10. 12e10/3 + 24 degrees.

11. 3 feet

12. 9√13≈ 2.5 m/s

13. (a) −5.4C/min (b) −0.54C (c) t = −10.09 ln

(4570

)

PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 205

6.3 End of Term Review Notes

1. Special Limits:

limx→0

sinx

x= 1 lim

x→0(1 + x)1/x = e lim

n→∞

(1 +

1

n

)n= e

2. Definition of Derivative: The derivative of a function f at a number a, denote by f ′(a), is

f ′(a) = limh→0

f(a+ h)− f(a)

h


3. Differentiation Rules: (Part 3)

(a) General Formulas:

ddx (c) = 0 d

dx [cf(x)] = cf ′(x)

ddx [xn] = nxn−1 d

dx [f(x)± g(x)] = f ′(x)± g′(x)

ddx [f(x)g(x)] = f(x)g′(x) + f ′(x)g(x) (product rule) d

dx

[f(x)g(x)

]= f ′(x)g(x)−f(x)g′(x)

[g(x)]2 (quotient rule)

ddx [f(g(x))] = f ′(g(x))g′(x) (chain rule)

(b) Exponential and Logarithmic Functions:

ddx (ex) = ex d

dx (ax) = ax ln a

ddx (ln |x|) = 1

xddx (loga x) = 1

x ln a

(c) Trigonometric functions:

ddx (sinx) = cosx d

dx (cosx) = − sinx ddx (tanx) = sec2 x

(d) Inverse Trigonometric Functions:

ddx (sin−1 x) = 1√

1−x2

ddx (cos−1 x) = − 1√

1−x2

ddx (tan−1 x) = 1

1+x2

(e) Hyperbolic Functions:

ddx (sinhx) = coshx d

dx (coshx) = sinhx ddx (tanhx) = sech2x

(f) Inverse Hyperbolic Functions:

ddx (sinh−1 x) = 1√

1+x2

ddx (cosh−1 x) = 1√

x2−1ddx (tanh−1x) = 1

1−x2


4. Natural Growth Equation: (Lecture 3.8) The solution of the initial-value problem

dy

dt= ky, y(0) = y0

isy(t) = y0e

kt .

Radioactive Decay problems: Usually k is specified in terms of the half-life of the isotope

τ =ln 2

k.

This is the time required for half of any given quantity to decay.Newton’s Law of Cooling/Heating problems: The temperature T of an object is modeled by:

dT

dt= k(T −M) −→ T (t) = Aekt +M

where

• M is the temperature of the surroundings (ambient temperature - which is constant)• k a constant (called the heating/cooling constant)

5. Linear Approximation and Differentials: (Lecture 3.10) The linear function

L(x) = f(a) + f ′(a)(x− a)

is called the linearization of f at a.For x close to a we have that

f(x) ≈ L(x) = f(a) + f ′(a)(x− a)

and this approximation is called the linear approximation of f at a.The differential dy is defined as

dy = f ′(x)∆x = f ′(x)dx .


6. L’Hospital’s Rule: (Lecture 4.4) Suppose that f and g are differentiable and g′(x) 6= 0 near a(except possibly at a.) Suppose that

limx→a

f(x) = 0 and limx→a

g(x) = 0

or that

limx→a

f(x) = ±∞ and limx→a

g(x) = ±∞

Then

limx→a

f(x)

g(x)= limx→a

f ′(x)

g′(x)

if the limit on the right side exists (or is∞ or −∞).

7. Example. Compute the following limit:

limx→0

(lnx)2

x

8. Newton’s Method for approximating solutions to f(x) = 0: (Lecture 4.8)

i. Begin with an initial guess x1.

ii. Calculate

x2 = x1 −f(x1)

f ′(x1).

iii. If xn is known then

xn+1 = xn −f(xn)

f ′(xn).

iv. If xn and xn + 1 agree to k decimal places then xn approximates the root r up to k decimal placesand f(xn) ≈ 0.

9. Example. (a) Show that the equation ex = 5x has exactly two solutions.

(b) Use Newton’s Method to find the two solutions to the equation in (a) to three decimal places.


10. Increasing/Decreasing Test.

(a) If f ′(x) > 0 on an interval, then f is increasing on that interval.(b) If f ′(x) < 0 on an interval, then f is decreasing on that interval.

11. Definition. A critical number of a function f is a number c in the domain of f such that eitherf ′(c) = 0 or f ′(c) does not exist.

12. The First Derivative Test. Suppose that c is a critical number of a continuous function f .

(a) If f ′ changes from positive to negative at c, then f has a local maximum at c.(b) If f ′ changes from negative to positive at c, then f has a local minimum at c.(c) If f ′ does not change sign at c, then f has no local minimum or maximum at c.

13. Concavity Test.

(a) If f”(x) > 0 for all x ∈ I, then the graph of f is concave upward ( on I.(b) If f”(x) < 0 for all x ∈ I, then the graph of f is concave downward ) on I.

14. Definition. A point P on a curve y = f(x) is called an inflection point if f is continuous therethe curve changes from concave upward ( to concave downward ) or from concave downward ) toconcave upward ( at P .

15. The Second Derivative Test. Suppose f” is continuous near c.

(a) If f ′(c) = 0 and f”(c) > 0 then f has a local minimum at c.(b) If f ′(c) = 0 and f”(c) < 0 then f has a local maximum at c.

16. The Mean Value Theorem. (Lecture 4.2) ) Let f be a function that satisfies the following hypothe-ses:

(a) f is continuous on the closed interval [a, b].(b) f is differentiable on the open interval (a, b).

Then there is a number c in (a, b) such that

f ′(c) =f(b)− f(a)

b− a

or, equivalently,f(b)− f(a) = f ′(c)(b− a).

17. Closed Interval Method for finding Absolute Extrema: To find the absolute maximum andminimum values of a continuous function f on a closed interval [a, b]:

(a) Find the values of f at the critical numbers of f in (a, b).(b) Find the values of f at the endpoints of the interval.(c) The largest of the values from Step 1 and Step 2 is the absolute maximum value; the smallest of

these values is the absolute minimum value.

18. Example. Find the absolute maximum and absolute minimum values of f(x) = e−x − e−2x on theinterval [0, 1].


19. Derivatives of Parametric Curves: (Lecture 5.1) The derivative to the parametric curve x = f(t),y = g(t) is given by

dy

dx=

(dydt

)(dxdt

) =g′(t)

f ′(t).

The second derivative is given by

d2y

dx2=

d

dx

(dy

dx

)=

ddt

(dydx

)dxdt

.

20. Derivative of Polar Curves: (Lecture 5.2) Suppose that r = f(θ) is a differentiable function of θ.Then from the parametric equations

x = r cos θ y = r sin θ

it follows that

dy

dx=

dy

dθdx

dθ

=

dr

dθsin θ + r cos θ

dr

dθcos θ − r sin θ

21. Example. True or False. Justify your answers.

(a) If f(x) is differentiable at x = a then f(x) is continuous at x = a.

(b) If f ′(c) = 0 then f has a local max/min at c.

(c) If f ′(x) < 0 for 1 < x < 6, then f is decreasing on (1, 6).

(d) If f ′′(2) = 0 then (2, f(2)) is an inflection point of f .

(e) If f ′(x) = g′(x) for 0 < x < 1, then f(x) = g(x) for 0 < x < 1.

(f) There exists a function f such that f(x) > 0, f ′(x) < 0, and f ′′(x) > 0 for all x.

(g) If f is periodic then f ′ is periodic.


(h) If f is even then f ′ is even.

(i) If the parametric curve x = f(t), y = g(t) satisfies g′(1) = 0, then it has a horizontal tangent linewhen t = 1.

(j) The equationsr = 2,

x2 + y2 = 4

andx = 2 sin (3t), y = 2 cos (3t) (0 ≤ t ≤ 2π

3)

all have the same graph.

(Answers: 7 0 21 (a) T (b) F (c) T (d) F (e) F (f) T (g) T (h) F (i) F (j) T )

PART 6: REVIEW MATERIAL FINAL EXAM CHECKLIST 211

6.4 Final Exam Checklist

(Page numbers refer to Stewart’s Calculus [4].)

Definitions

Chapter 2:1. Limit (page 87)

2. Left-hand (right-hand) limit (page 92)

3. Infinite limit (pages 93 and 94)

4. Vertical asymptote (page 94)

5. Function continuous at a number (page 118)

6. Function continuous from the right (left) at a number (page 120)

7. Function continuous on an interval(page 120)

8. Limit at infinity (page 131)

9. Horizontal asymptote (page 131)

10. Tangent line (page 143)

11. The derivative of a function at a number (page 146)

12. Function differentiable on a set (page 157)

13. Vertical tangent line (page 159)

Chapter 3:14. The number e (page 179)

15. Linear approximation (page 251)

16. Differential (page 253)

17. Hyperbolic functions (page 257)

Chapter 4:18. Absolute maximum/minimum (page 274)

19. Local maximum/minimum (page 274)

20. Critical number (page 277)

21. Function concave upward/downward (page 293)

22. Inflection point (page 294)

23. Antiderivative of a function (page 344)

Chapter 10:24. Parameter, parametric equations, parametric curve (page 636)

25. Initial and terminal points (page 637)

26. The Cycloid (page 639)


27. Polar coordinate system, pole, polar axis, polar coordinates (page 654)

Theorems/ Formulas/ Procedures

Chapter 2:1. lim

x→af(x) = L if and only if lim

x→a−f(x) = L and lim

x→a+f(x) = L’ (page 92, page 104)

2. Limit Laws (Page 99-106)

3. Direct Substitute Property (page 101)

4. Theorem about the monotonicity and limits. (page 105)

5. The Squeeze Theorem (page 105)

6. Continuity and combinations of functions (page 121)

7. Continuity of polynomials and rational functions (page 122)

8. Continuity and the combination of functions (page 123)

9. Continuity and the composition of functions (page 124, 125)

10. The Intermediate Value Theorem (page 125)

11. Limits at infinity of the power functions with negative rational exponents (page 133)

12. Relationship between differentiable and continuous functions (page 158)

Chapter 3:13. Derivative of a constant function (page 174)

14. Power rule (pages 175 and 176)

15. The Constant Multiple Rule (page 177)

16. The Sum Rule (page 177)

17. The Difference Rule (page 178)

18. Derivative of the natural exponential function (page 180)

19. The Product Rule (page 185)

20. The Quotient Rule (page 187)

21. Derivatives of Trigonometric Functions (pages 193 and 194)

22. The Chain Rule (page 199)

23. Derivative the exponential function f(x) = ax (page 203)

24. Derivatives of Inverse Trigonometric Functions (page 214)

25. Derivatives of Logarithmic Functions (page 218 and 220)

26. e as a limit (page 222)

27. The solution of the initial-value problem (page 237)

28. Hyperbolic Identities (page 258)

29. Derivatives of Hyperbolic Functions (page 259)

30. Derivatives of Inverse Hyperbolic Functions (page 261)


Chapter 4:31. The Extreme Value Theorem (page 275)

32. Fermat’s Theorem (page 276)

33. The Closed Interval Method (page 278)

34. Rolle’s Theorem (page 284)

35. The Mean Value Theorem (page 285)

36. The relationship between two functions implied by the equality of their derivatives (pages 288)

37. Increasing/Decreasing test (page 290)

38. The First Derivative Test (page 291)

39. Concavity Test (page 293)

40. The Second Derivative Test (page 295)

41. L’Hospital’s Rule (page 302)

42. Newton’s Method (page 339)

43. The relationship among antiderivatives of a function (page 344)

Examples

Chapter 2:1. Function with neither left-hand nor right-hand limits at the given point (page 90)

2. Function with the left-hand and right hand limits that are not equal (page 91)

3. Function with infinite left-hand and right hand limits (page 93)

4. Function with an infinite number of vertical asymptotes (page 95)

5. Function F = f ·g so that the limits of F and f at a exist and the limit of g at a does not exist. (page106)

6. Function F = f + g so that the limit of F at a exists and the limits of f and g at a do not exist.

7. Function F = fg so that the limit of F at a exists and the limits of f and g at a do not exist.

8. Function with a removable (infinite, jump) discontinuity. (page 119-120)

9. Function with the graph that intersects its horizontal asymptote (page 131)

10. Function with two horizontal asymptotes (page 132)

11. Function that is continuous but not differentiable (page 157)

12. Function with a vertical tangent line (page 159)

13. Function with a “corner” (page 159)

Chapter 3:14. Function that is the same as its derivative (page 180) [Can you find THREE different functions

with this property?]


Chapter 4:15. Function f with no minimum, no maximum but such that f ′(a) = 0 for some a. (page 275)

16. Function with a local minimum that is not the global minimum. (page 275)

17. Function(s) that does not satisfied the hypothesis of the Extreme Value Theorem. (page 275-276)

18. Function with a critical number but no maximum or minimum. (page 277)

19. Sketch the graph of a function that is increasing for x < 0, decreasing for 0 < x < 1, and increas-ing for x > 1. Also, suppose this function has a horizontal asymptote when x → −∞ and a slantasymptote when x→∞.

20. Function that is concave upward (concave downward) (page 293)

21. Function with an inflection point at which the first derivative equals 0.

22. Function with a local minimum at which the second derivative equals 0.

LimitsAre you able to do most of the following:

1. Exercises 2.3: 1-32, 37-50

2. Exercises 2.6: 1-57

3. Exercises 3.3: 39-48


DifferentiationAre you able to do most of the following:


2. Exercises 2.8: 21-32, 37-40, 43-46, 50-57

3. Exercises 3.2: 3-35, 41-50

4. Exercises 3.3: 1-34, 49-51

5. Exercises 3.4: 1-54, 59-78

6. Exercises 3.5: 1-30, 35-40, 42-60, 73-80

7. Exercises 3.6: 1-34, 37-50

8. Exercises 3.11: 30-47


Applications

1. Tangent lines (throughout the text)

2. Velocity (throughout text)

3. The Intermediate Value Theorem (Section 2.5: Exercises 48-54)

4. Rates of change (Section 2.7: Exercises 41-48, Section 3.7: Exercises 7-26)

5. Exponential growth and decay (Section 3.8: all exercises 1-20)

6. Related rates (Section 3.9: all exercises)

7. Linear Approximation and Differentials (Section 3.10: Exercises 11-22, 33-40)

8. Maximum and minimum values (Section 4.1: Exercises 47-62, 69-71)

9. Mean Value Theorem (Section 4.2: Exercises 15-31)

10. Graphs of functions (Section 4.3 and Section 4.5)

11. Optimization (Section 4.7)

12. Newton’s method (Section 4.8: Exercises 5-8, 11-22)

13. Antiderivative (Section 4.9: Exercises 25-48, 65-69)

Miscellaneous

1. Section 10.1: Exercises 1-28

2. Section 10.2: Exercises 1-30

3. Section 10.3: Exercises 15-46, 55-64 (page 648-649)

PART 6: REVIEW MATERIAL FINAL EXAM PRACTICE QUESTIONS 216

6.5 Final Exam Practice Questions

Please refer to the final exam checklist (section 6.4) for an indication of what you will need to know forthe final exam (in short, the material from any section we covered this term will be on the exam, with theexception of lectures 2.4, 5.3 and 5.4).

To prepare for the exam you should: (i) read all sections of the textbook again; (ii) go through all thehomework questions again and make sure you can do every single one of them, (iii) work through moreproblems from the textbook; (iv) work through the final exam checklist (this can be done in conjunctionwith (i) through (iii)). Only once you have done all this should you attempt the following questions.

The following is a list of practice exam questions. This will give you an idea of the types of questions youwill be asked on the exam.

Instructions: No calculators, books, papers, or electronic devices shall be allowed within the reach of astudent during the examination. Leave answers in ”calculator ready” expressions: such as 3 + ln 7 or e

√2.

Questions 1-3 are on the statements of definitions and theorems and on your ability to give ex-amples of functions with specified properties..

1. Define the following terms.

(a) Limit(b) Function continuous at a number(c) Function continuous on an interval(d) Tangent line(e) The derivative of a function at a number(f) Function differentiable a on a set(g) The number e(h) Differential(i) Absolute maximum and absolute minimum(j) Local maximum and local minimus

(k) Critical number(l) Function concave upward and concave downward

(m) Inflection point(n) Antiderivative of a function

2. State the following theorems.

(a) The Squeeze Theorem(b) The Intermediate Value Theorem(c) Fermat’s Theorem(d) Extreme Value Theorem(e) Rolle’s Theorem(f) The Mean Value Theorem(g) L’Hospital’s Rule

3. Give an example for each of the following.

(a) Function with an infinite number of vertical asymptotes.


(b) Function F = f · g so that the limits of F and f at a exist and the limit of g at a does not exist.(c) Function with a removable discontinuity.(d) The most general form of a function with the property that its second derivative is the zero

function.(e) Function that is continuous but not differentiable at a point.(f) Function with a critical number but no maximum or minimum.(g) Function with a local minimum at which the second derivative equals 0.

Questions 4-16 are short answer questions. The questions are given in no particular order.

4. Find the derivative y′ =dy

dxof each of the following:

(a) y = cos−1 (x2)− ln (1 + x3) [Note: Another notation for cos−1 is arccos.](b) y = xsin(x).

(c) arctan(yx

)= 1

2 ln (x2 + y2).

5. Let f(x) = tanx. Find f ′′(x), the second derivative of f .

6. Find the tangent line to the curve y + x ln y − 2x = 0 at the point (1/2, 1).

7. Evaluate limx→∞

2x3 + 3x− 1

1− 2x2 + 5x3.


x2 − 1

x2 − 3x+ 2.

9. Let f(x) =2x

x2 + 3.

(a) Find the equation of the tangent line to the curve y = f(x) at x = 1.(b) Use linear approximation to give an approximate value for f(1.2).

10. A particle moves along the x-axis so that its position at time t is given by x = t3 − 4t2 + 1.

(a) At t = 2, what is the particle’s speed?(b) At t = 2, in what direction is the particle moving?(c) At t = 2, is the particle’s speed increasing or decreasing?

11. The curve y = xe−x has one inflection point. Find the x-coordinate of this point.

12. Find an equation of the slant asymptote to the curve y =x3 + 2x2

x2 + 3x+ 2.

13. Find a number x0 between 0 and π such that the tangent line to the curve y = sinx at x = x0 isparallel to the line y = −x/2.


x sin1

x.


ex − 1− xx2

.


xsin x.

Questions 17-33 are full-solution problems. Justify your answers and show all your work. Thequestions are given in no particular order.


17. Let f(x) =5

3x− 1. Calculate f ′(2) directly from the definition of derivative.

18. A water-trough is 10m long and has a cross-section which is the shape of an isosceles trapezoid thatis 30cm wide at the bottom, 80cm wide at the top, and has height 50cm. If the trough is being filledwith water at the rate of 0.2 m3/min, how fast is the water level rising when the water is 30cm deep?

[Recall: The area of an isosceles trapezoid as shown in the diagram is A = 12 (a+ b)h.]

19. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to ahemispherical dome with diameter 50m.

20. At 2:00 p.m.a car’s speedometer reads 30 mi/h. At 2:10 p.m. it reads 50 mi/h. Show that at some timebetween 2:00 and 2:10 the acceleration is exactly 120 mi/h2.

21. A piece of wire 10m long is cut into two pieces. One piece is bent into a square and the other is bentinto an circle. How should the wire be cut so that the total area enclosed is minimum.

22. A turkey is put into an oven that has a constant temperature of 200C. A thermometer embedded inthe turkey registers its temperature. When the turkey is put into the oven, the thermometer reads20C, and 30 minutes later it reads 30C. The turkey will be ready to eat when the thermometer reads80C. How many minutes after being put into the oven will the turkey be ready to eat? Assume thatthe turkey’s temperature satisfies Newton’s law of cooling/heating.

23. Sketch the graph of the function

f(x) =−2x2 + 5x− 1

2x− 1.

24. Let f(x) = 2x3 − 6x2 + 3x+ 1.

(a) First show that f has at least one zero in the interval [2, 3] and then use the first derivative of fto show that there is exactly one root of f between 2 and 3.

(b) Use Newton’s method to approximate the root of f in the interval [2, 3] by starting with x1 = 5/2and finding x2.

25. Find the dimensions of the largest rectangle that can be inscribed inside a semicircular region ofradius 5 such that one side of the rectangle is parallel to the base of the semicircular region.

26. (a) A metal storage tank with fixed volume V is to be constructed in the shape of a right circularcylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?

(b) Suppose the metal for the hemisphere costs twice as much as the metal for the lateral sides. Whatare the dimensions for the tank that minimizes cost?

(Recall: The volume of a sphere of radius r is 43πr

3 and the surface area is 4πr2.)

27. (a) Show that Newton’s Method applied to the equation x2 − a = 0 yields the iterative formula

xn+1 =1

2

(xn +

a

xn

)and thus provides a method for approximating the square root

√a which uses only addition and

multiplication.

(b) Approximate√

3 by taking x1 = 3/2 and calculating x2.


28. Find f if f ′′(x) = 2 + cosx, f ′(0) = −1 and f(π/2) = 0.

29. Sketch the curve which is given by the parametric equations

x = cos (πt), y = sin (πt), 1 ≤ t ≤ 2.

Clearly label the initial and terminal points and describe the motion of the point (x(t), y(t)) as t variesin the given interval (i.e. indicate the direction the point is traveling).

30. A curve called the folium of Descartes is defined by the parametric equations

x =3t2(t+ 1)

3t2 + 3t+ 1, y =

−3t(t+ 1)2

3t2 + 3t+ 1, −∞ < t <∞.

(a) Show that a Cartesian equation of this curve is x3 + y3 = 3xy.(b) Find the point on the curve corresponding to t = −1/2.(c) Find the equation of the tangent line to the curve at the point corresponding to t = −1/2.(d) Find the values of the parameter t which correspond to the point (0, 0) on the curve.(e) Find equations of the tangent lines to the curve at the point (0, 0).

31. Consider the curve given by the parametric equations

x = 2 sin t, y = 4 + cos t, 0 ≤ t ≤ 2π.

Determine the points on the curve which are closest to the origin and those which are furthest away.

32. Sketch the curve with the polar equation r = 2 cos 4θ.

33. Consider the curve given by the polar equation

r = 1 + 2 sin (3θ), 0 ≤ θ ≤ 2π.

(a) Determine the points on the curve which are furthest from the origin.(b) Find the slope of the tangent line to each of the points found in part (a).


Answers:

4. (a) y′ = − 2x√1− x4

− 3x2

1 + x3(b) y′ = xsin x( sin x

x+ lnx cosx) (c) y′ =

x+ y

x− y5. f ′′(x) = 2 sec2 (x) tan (x)

6. y = 43x+ 1

3

7. 2/5

8. −29. (a) y = 1

4x+ 1

4(b) f(1.2) ≈ 1

4(1.2) + 1

4= 11

20= 0.55

10. (a) −4 (b) to the left (c) speed is decreasing

11. x = 2

12. y = x− 1

13. 2π/3

14. 0 (Hint: use Squeeze Theorem since sin(1/x) is bounded)

15. 1/2

16. 1

17. −3/518. 1

30m/min = 10

3cm/min

19. 58π ≈ 2 m3

20. Hint: use Mean Value Theorem

21. The length of wire used to make the square should be 40π+4

22. t = ln (2/3)2 ln (17/18)

≈ 3.55 hours ≈ 3 hours 33 minutes

24. (b) x2 = 16/7 ≈ 2.285714286

25. base is 5√2 and height is 5√

2

26. (a) hemisphere (b) r = 12

3√

3V/π and h = 3√

3V/π.

27. (b) 74

28. f(x) = − cosx+ x2 − x+ π/2(1− π/2)29. Consists of points on the unit semicircle in quadrants 3 and 4. Points are moving counterclockwise along curve

with initial point (−1, 0) and terminal point (1, 0).

30. (b) (3/2, 3/2) (c) y = −x+ 3 (d) t = −1, 0 (e) There are two lines: the first is when t = −1 and the tangent isthe horizontal line y = 0, the second occurs when t = 0 and is the vertical line x = 0.

31. closest point is (0, 3), furthest point is (0, 5)

33. (a) The three points are (r, θ) = (3, π6), (3, 5π

6), (3, 3π

2). (b) slopes of tangents at these points are −

√3,√3, and 0,

respectively.

Bibliography

[1] C. Adams, A. Thompson, and J. Hass. How to Ace Calculus: The Streetwise Guide. W.H. Freeman and Company,1998.

[2] B. Curgus. An exceptional exponential function. The College Mathematics Journal, 37(5):344–354, 2006.http://myweb.facstaff.wwu.edu/curgus/Papers/ExcExpFinal.pdf.

[3] D. Ebersole, D. Schattschneider, A. Sevilla, and K. Somers. A Companion to Calculus. Brooks/Cole SengageLearning, 2nd edition, 2005.

[4] J. Stewart. Calculus: Early Transcendentals. Brooks/Cole Sengage Learning, 7th edition, 2011.

221

http://myweb.facstaff.wwu.edu/curgus/Papers/ExcExpFinal.pdf

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