CSE Calculus Ifelix077/download/calc1.pdf · Chapter 1 Preliminaries...

CSE Calculus IUniversity of Minnesota

CSE Calculus IUniversity of Minnesota

Bryan FélixUniversity of Minnesota

Last updated: September 23, 2017

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These are notes intended for the students of CSE Calculus I at the Uni-versity of Minnesota. By no means this a complete reference. The intentionis to compile a useful list of definitions, problems and Sage code that will beuseful to my students while exploring the course. The author is to blame forany typos and errors.

Contents

1 Preliminaries 11.1 Functions, Velocity and Average Velocity . . . . . . . . . . . . . 11.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . 21.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Worksheet 5911 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Worksheet 5912 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Derivatives 72.1 Secant and Tangent Lines . . . . . . . . . . . . . . . . . . . . . 72.2 Definition of the derivative . . . . . . . . . . . . . . . . . . . . 72.3 Constant Multiplication, Addition, Product and Quotient Rules 72.4 Derivative of a composite function . . . . . . . . . . . . . . . . 82.5 Derivative of implicit functions . . . . . . . . . . . . . . . . . . 82.6 Basic Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Toolbox 133.1 Function evaluation . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Factorization and Root Solver . . . . . . . . . . . . . . . . . . . 143.4 Secant and Tangent Lines . . . . . . . . . . . . . . . . . . . . . 143.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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viii CONTENTS

Chapter 1

Preliminaries

The following definitions intend to cover most of what high school level math-ematics that is used through Calculus.

1.1 Functions, Velocity and Average VelocityDefinition 1.1.1. A function is a collection of points in the plane with theproperty that no two points have the same first coordinates. The set of all firstcoordinates of these points is referred to as the domain of the function andthe set of all second coordinates of these points is referred to as the range ofthe function.

Definition 1.1.2. We say that a function f is well defined at the point a iff(a) exists.

Definition 1.1.3 (Well defined values of common functions). To figure thedomain of a function one can start by looking at the graph. However, we havethe following rules at hand

1. Polinomial and exponential functions are well defined in (−∞,∞).

2. Rational functions fg are well defined everywhere unless g(x) = 0 (we can

not divide by 0).

3. The logarithmic of a function log(f) is well defined whenever f > 0

4. The square root of a function√f is well defined whenever f ≥ 0 (as

opposed to logarithms, the square root may take the vlalue 0)

Definition 1.1.4. Two functions f and g are said to be equal at the point aif both f and g are well defined at a and f(a) = g(a).

Definition 1.1.5. A function f is called increasing on an interval I if

f(x1) < f(x2) whenever x1 < x2

A function f is called decreasing on an interval I if

f(x1) > f(x2) whenever x1 > x2

Definition 1.1.6. Given two functions f and g, the composite functionf ◦ g is defined as

f ◦ g = f(g)

1

2 CHAPTER 1. PRELIMINARIES

Definition 1.1.7. The slope of a line that intercepts the points (x1, y1) and(x2, y2) is given by

y2 − y1x2 − x1

Definition 1.1.8. An equation of a line passing through the point (x1, y1)with slope m is given by

y − y1 = m(x− x1)

Definition 1.1.9 (Laws of exponents). If a and b are positive numbers and xand y are any real numbers, then

1. ax+y = axay

2. ax−y = ax

ay

3. (ax)y

= ax·y

4. (ab)x = ax · bx

Definition 1.1.10 (Laws of logarithms). If x and y are positive numbers,them

1. loga(xy) = loga x+ loga y

2. loga

(xy

)= loga x− loga y

3. loga(xr) = r loga x

Definition 1.1.11. We refer to average velocity as the change in position(note that it could be negative) divided by the change in time.

average velocity =position1 − position2

time1 − time2

Definition 1.1.12. A linear function is a polinomial of the form

m · x+ b.

For example:

1. x+ 1

2. 3x+ π

3. 1000x+ 5.

1.2 Limits and ContinuityDefinition 1.2.1. If f is a function and each of a and L are numbers, thenwe say that the limit of f as x approaches a is L if the values for f(x) getclose to the number L as the values for x get close to the number a. We writethis as

limx→a

f(x) = L

Theorem 1.2.2 (Limit Existence Theorem). If f is a function, then thelimx→a f(x) exists if and only if

limx→a−

f(x) and limx→a+

f(x)

exist and are equal.

1.3. PROBLEMS 3

Definition 1.2.3. If f is a function and a is in the domain of f then we saythat f is continuous at a if

limx→a

f(x) = f(a).

Theorem 1.2.4 (Continuous Functions Theorem).

1. Every polynomial is continuous on (−∞,∞).

2. Every trigonometric function is continuous on its domain.

3. The sum, product, and difference of continuous functions is continuous.

4. The quotient f/g of continuous functions f and g is continuous whereverg is not equal to 0

5. If g is continuous at a and f is continuous at g(a) then f ◦g is continuousat a.

Theorem 1.2.5 (The Intermediate Value Theorem). If f is a continuousfunction on an interval (a, b) and y is a number between f(a) and f(b), thenthere is a number x in [a, b] such that f(x) = y.

1.3 ProblemsProblem 1.3.1. State the values at which the following functions are welldefined.

1. f(x) = 1x2−1

2. g(x) =√

4− x

3. x = y2 + 1

Problem 1.3.2. If f(x) = x+√

2− x and g(u) = u+√

2− u, is it true thatf = g for all values? See Definition 1.1.4

Problem 1.3.3. If

f(x) =x2 − xx− 1

and g(x) = x

is it true that f = g for all values? See Definition 1.1.4

Problem 1.3.4. Consider the function f(x) = sin(x) on the interval [0, 2π].Describe the intervals where the function is increasing and decreasing.

Problem 1.3.5. From worksheet 5911. A line passes through the points(−2, 5) and (7, 20). What is the slope of this line? What is the equationof this line?

Problem 1.3.6. Sketch multiple linear functions with slope equal to 2. Finda general equation that represents all such functions.

Problem 1.3.7. Consider the functions f(x) = x3 and g(x) = x+3. Computethe composition f ◦ g and g ◦ f .

Problem 1.3.8. Use the Laws of exponents to simplify

1.(6y3)4

2y5


2.x2n · x3n−1

xn+2

3.

√a√b√

ab

Problem 1.3.9. Use the Laws of logarithms to find the exact value of

1. log2 6− log2 15 + log2 20

2. log10

√10

3. ln(1/e)

Problem 1.3.10. Prove that cos(arcsinx) =√

1− x2.

Problem 1.3.11. From Worksheet 5911. The position of a car traveling alonga straight road is given by

s(t) = 30t+ 12t2 − t3,

where t is measured in hours and s in miles.

1. By what milepost is the car when t = 1 hour?

2. By what milepost is the car when t = 5 hours?

3. What is the average velocity of the car during these 4 hours?

Problem 1.3.12. Explain the meaning of the limit

limx→1

f(x).

Provide several examples with a functions of your choice.

Problem 1.3.13. Explain what it means to say that

limx→1−

f(x) = 3 and limx→1+

f(x) = 7.

In this situation, is it possible that limx→1 f(x) exists? See Theorem 1.2.2

Problem 1.3.14. Use a graph to find

limx→0

1

1 + e1x

Problem 1.3.15. Let f and g be as in Problem 3 Explain the difference, andvalidity, of the two statements:

1. f(a) = g(a) for all values

2. limx→a

f(x) = limx→a

g(x) for all values

Problem 1.3.16. Explain what it means for a function to be continuous at apoint. Provide examples of functions that are continuous and discontinuous.

Problem 1.3.17. Describe the values for which the following functions arecontinuous and discontinuous.

1. x2 + 5x+ 3

2. 3x−12x+5

1.4. WORKSHEET 5911 5

3. x+ tan(x)

Problem 1.3.18. Explain The Intermediate Value Theorem. Provide an ex-ample where it works and one where it does not.

Problem 1.3.19. Use The Intermediate Value Theorem to prove that f(x) =sinx− x2 + x is equal to 0 somewhere in the interval (1, 2).

Problem 1.3.20. Is there a function that is discontinuous at every singlepoint?

1.4 Worksheet 59111. A line pases through the points (−2, 5) and (7, 20). What is the slope ofthis line?(a) What is the equation of this line?

2. A car is traveling along a road. At time t = 15 minutes the car is at milepost8. At time t = 100 minutes the car is at milepost 110. What is the averagevelocity of the car during these 85 minutes?

3. The position of a car traveling along a straight road is given by

s(t) = 30t+ 12t2 − t3,

where t is measured in hours and s in miles.(a) By what milepost is the car when t = 1 hour?(b) By what milepost is the car when t = 5 hours?(c) What is the average velocity of the car during these 4 hours?

1.5 Worksheet 59121. Simplify

xx+5 −

3x+1

x− 5.

2. Simplifyx− 3

x+2x+7 −

3x+3

.

3. Given

f(x) =10x −

12x2+3x

3x + 2

x+3

,

find f(8) and f(25).

4. Divide 2x3 − 14x2 + 27x− 35 by x− 5 using long division.

5. Solve the quadratic equation x2 − 2x− 15 = 0.

6. Consider the right triangle given below. If sin(A) = .6859, what is sin(B)?

Chapter 2

Derivatives

2.1 Secant and Tangent LinesDefinition 2.1.1 (Secant line). Given a function f(x) and two points on itsdomain x1 and x2, we define its secant to be the equation of the line thatpasses through the points (x1, f(x1)) and (x2, f(x2)).

Definition 2.1.2 (Tangent line). The tangent line of a function f(x) atx = a is the line that passes through the point (a, f(a)) and has slope equalto:

limx→a

f(a)− f(x)

a− xdefined only if the limit exists. See Theorem 1.2.2

2.2 Definition of the derivativeDefinition 2.2.1 (The definition of the derivative). Given a function f , its

derivative (denoted as f ′(x) ord

dx(f)) is defined as:

f ′(x) = limh→0

f(x+ h)− f(x)

h

Definition 2.2.2. We say that a function is differentiable at x if f ′(x) exists.

2.3 Constant Multiplication, Addition, Productand Quotient Rules

Theorem 2.3.1 (Constant Multiplication). If f is a differentiable function,and c is a constant, then

d

dx(c · f) = c · f ′.

Theorem 2.3.2 (Addition property). If f and g are two differentiable func-tions, then

d

dx(f + g) = f ′ + g′.

Theorem 2.3.3 (Product Rule). If f and g are two differentiable functions,then

d

dx(f · g) = f ′g + fg′.

7

8 CHAPTER 2. DERIVATIVES

Theorem 2.3.4 (Quotient Rule). If f and g are two differentiable functions,then

d

dx

(f

g

)=f ′g − fg′

g2.

2.4 Derivative of a composite functionOften refered to as chain rule.

Theorem 2.4.1 (The Chain Rule). If f and g are two differentible functions,then

d

dx(f ◦ g) = (f ′ ◦ g) · g′.

Recall that f ◦ g is the same as f(g(x)).

2.5 Derivative of implicit functionsDefinition 2.5.1. Given the function f(x), let y = f(x). The differentialdy of the dependent variable y is defined as

dy = f ′(x) dx.

Definition 2.5.2. Given a function y = f(x), the increment ∆y is definedas

∆y = f(x+ ∆x)− f(x).

2.6 Basic DerivativesTheorem 2.6.1 (The power Rule). If n is a real number, then

d

dx(xn) = nxn−1.

Theorem 2.6.2 (Exponential derivatives). If a is a real number, then

d

dx(ax) = ln(a)ax,

in particulard

dx(ex) = ex.

Theorem 2.6.3 (Logarithm derivatives). If a is a real number, then

d

dx(loga(x)) =

1

ln(a)x,

in particulard

dx(ln(x)) =

1

x.

Theorem 2.6.4 (Trigonometric derivatives).

• ddx sin(x) = cos(x)

• ddx cos(x) = − sin(x)

• ddx tan(x) = sec2(x)

2.7. PROBLEMS 9

2.7 ProblemsProblem 2.7.1. From worksheet 5916. Evaluate the following limits:

1. limx→−2

4x2 + 11x+ 6

3x2 + 14x+ 16

2. limx→2

x− 25

x+3 − 1

Problem 2.7.2. From worksheet 5916. Let f(x) =x4 − 1√x− 1

for x 6= 1. Find

f(0.9) f(0.99) f(0.999)

f(1.1) f(1.01) f(1.001)

Based on this data, what would you gess is the value of the following limit:

limx→1

x4 − 1√x− 1

Problem 2.7.3. From worksheet 5914. Consider the parabola which is thegraph of the equation y = x2−3x−10. Find the equation of Secant line whichintrsects this parabola at the two points where x = −3 and where x = 4.

Problem 2.7.4. What is the Secant line of a linear function?

Problem 2.7.5. Explain, graphically, the definition of Tangent line. Providean example of a tangent line that fails to exist.

Problem 2.7.6. Explain the process of finding the tangent line given a func-tion f , and its derivative f ′

Problem 2.7.7. Use The definition of the derivative to compute the derivativeof the constant functions:

1. f(x) = 12

2. f(x) = 5

3. f(x) = 0

What can you conclude?

Problem 2.7.8. Use The definition of the derivative to compute the derivativeof the linear functions:

1. f(x) = 5x

2. f(x) = x+ 3

3. f(x) = 7x+ 5


Problem 2.7.9. Use The definition of the derivative to compute the derivativeof f(x) = xn where n could be any number.

an − bn = (a− b)(an−1 + an−2b+ an−3b2 + · · · abn−2 + bn−1)

Problem 2.7.10. Given the derivative of a function; i.e. given f ′(x). Is itpossible to recover f(x)?


Problem 2.7.11. Let f(x) = |x− 2|.

1. Graph f .

2. Does f ′(2) exists?

3. What is the derivative of f?

Problem 2.7.12. Provide examples where a function fails to be differentiableat x = 0 where:

1. the point is not on its domain

2. the left and right hand limits in Definition 2.2.1 fail to agree

Problem 2.7.13. Use the theorems in Section 3 together with The powerRule to prescribe the derivative of any given polynomial. Provide examples.

Problem 2.7.14. Compute the derivative of

1. x2+x+1√x

2. (x+1)(x+3)x

3. x1/2+x1/3+x−1/4

x2/5

Problem 2.7.15. Let f(x) = 2x −

x4

2 . Compute the second derivative of f ;i.e. the derivative of f ′

Problem 2.7.16. Sketch the graph of 2x3−3x2−12x+2 and use its derivativeto locate the points at which the graph has an horizontal tangent line.

Problem 2.7.17. Use Product Rule to prescribe a formula to compute thederivative of the product of three functions f, g, h; i.e. find (f · g · h)′.

Problem 2.7.18. Use Quotient Rule and the identity tan(x) = sin(x)/ cos(x)to prove that

d

dxtan(x) = sec2(x).

Problem 2.7.19. Analogous to Problem 2.7.18 find the derivatives of thefunctions sec(x), csc(x) and cot(x).

Problem 2.7.20. Compute the derivative of the following

1. x2 + ex + sin(x)

2. sin(x) cos(x)

3. x100ex

4. ln(x)x

Problem 2.7.21. From worksheet 5925. Suppose the function f(x) is de-fined as f(x) = x3g(x). If g(−2) = 5, g′(−2) = −3, g(3) = 8, g′(3) = 6, findf(−2), f ′(−2), f ′(3).

Problem 2.7.22. From worksheet 5925. Suppose

f(x) =x3 − 2x

g(x).

If g(2) = 8, g′(2) = 6, g(5) = 10, and g′(5) = 15, find f(2), f ′(2) and f ′(5).

2.7. PROBLEMS 11

Problem 2.7.23. From worksheet 5926. Given f(x) =√

3x+ 10 find the

difference quotientf(x)− f(2)

x− 2. Simplify the difference quotient. Use this to

find f ′(2).

Problem 2.7.24. Write the following functions as a composition f ◦ g; e.g.

tan(x2) = tan(x) ◦ x2

1. sin(ex)

2. ecos2(x)

3. sin(x2)

4.√

10 + ecos(x)

5. sin(10x)

6. ecos(x2)

7. (tan(x2))5/2

Problem 2.7.25. Use The Chain Rule to compute the derivative of the fol-lowing composite functions. Clearly state what the functions f and g are.

1. (8x3 + x)4

2. sin(√x)

3.√

sin(x)

4. etan(x)

Problem 2.7.26. Use The Chain Rule to compute the derivative of the com-position of the three function f, g and h. In other words, compute the derivativeof a triple composition f ◦ (g ◦h). Use it to compute the derivative of sin(e

√x).

Problem 2.7.27. Recall that ddx (ex) = ex. Use The Chain Rule on the left

hand side and The power Rule on the right hand side to compue the derivativeof the equation

eln(x) = x.

Prove that (ln(x))′ =1

x.

Problem 2.7.28. Explain the process of finding the derivative of an arbitrarynumber of compositions. Provide examples.

Problem 2.7.29. Consider the infinite polynomial with unknown coefficients

f(x) = 1 + c1x+ c2x2 + c3x

3 + c4x4 + · · ·

Compute it’s derivative and find the values of the coefficients so that f(x) =f ′(x). The best approach is to set the coefficients of f and f ′ equal to eachother (according to the power of x).

Problem 2.7.30. Using the polynomial in the previous exercise (with the co-efficients you found substituted in) graph the following functions and comparethem.

1. 1 + c1x


2. 1 + c1x+ c2x2

3. 1 + c1x+ c2x2 + c3x

3

4. 1 + c1x+ c2x2 + c3x

3 + c4x4

5. ex


Problem 2.7.31. From Worksheet 5941. Consider the equation y2 +6y+9 =2x+ 8.

1. When x = 4, what are the corresponding values of y?

2. When x = 14, what are the corresponding values of y?

3. The equation y2 + 6y − 2x+ 1 = 0 defines y as two functions of x. Findan explicit expression for these two functions

4. For what values of x are these functions defined?

Problem 2.7.32. Graph the equation 3(x2 + y2)2 = 100xy. Determine itsslope at the point (3, 2).

Problem 2.7.33. Determine dydx for the equation sin(y) = x. Wirte dy

dx both,as a function of y and a function of x.

Problem 2.7.34. Given x2 + y2 = 25, find d2 ydx2 ; i.e. the second derivative of

y with respect to x.

Problem 2.7.35. Find the tangent line to the graph given by x2(x2+y2) = y2

at the point (√22 ,√22 ). Provide a graph.

Problem 2.7.36. For the following equations, sketch their graph and find theslope of the tangent line at the given point.

1. The Astroid. x2/3 + y2/3 = 5 at (8, 1).

2. Bifolium. (x2 + y2)2 = 4x2y at (1, 1).

3. Witch of Agnesi. (x2 + 4)y = 8 at (2, 1).

4. Folium of Descartes. x3 + y3 − 6xy = 0 at ( 43 ,

83 ).

5. Lemniscate. 3(x2 + y2)2 = 100(x2 − y2) at (4, 2)

6. Kappa curve. y2(x2 + y2) = 2x2 at (1, 1)

Problem 2.7.37. Given the circle (x + 2)2 + (y − 3)2 = 37. Find both dydx

and dxdy . Then, find the points at which dy

dx = 0 and dxdy = 0. Provide a graph.

What can you say about the tangent lines at the points you found?

Chapter 3

Toolbox

This section is aimed to provide a toolboox of common computations seen incalculus I. When using these codes, be carefull with the syntax. With that Imean, use parenthesis properly and use * for all multiplications.

3.1 Function evaluationTo evaluate a function with Sage, first define your function f(x) and then, toevaluate at a point, say x = 1 type f(1).

f(x)=1/(1+e^(1/x)) ;#Functionshow(f(x)) ;#Display your function to catch typosf(.5) ;#Value to evaluate

The following code expands by evaluating at a list of values, given a step size,an initial value and a final value.

f(x)=1/(1+e^(1/x)) ;#Functionshow(f(x)) ;#Display your function to catch typosxmin=1 ;#Initial pointxmax=8 ;#Final points=.5 ;#Step sizefor i in ellipsis_range(xmin ,Ellipsis ,xmax ,step=s):

print 'x=',i,'f(x)=',f(i)

3.2 PlotsTo plot a function, simply define it and set some constraints. (Click Evaluate

to generate the output).

f(x)=x^2*sin(x) ;#Write your functionxMin=-5 ;#Lower bound for the domainxMax=5 ;#Upper bound for the domianyMin=-5 ;#Lower bound for the rangeyMax=5 ;#Upper bound for the rangeshow(f(x)) ;#This line displays your equation to detect

typosplot(f(x),xmin=xMin ,xmax=xMax ,ymin=yMin ,ymax=yMax);#This

line generates the plot

To plot more than one function at once input your functions as a touple(f(x),g(x)) inside plot, as shown below.

13

14 CHAPTER 3. TOOLBOX

f(x)=x^2 ;#function 1g(x)=x^3 ;#function 2xMin=-5 ;#lower bound for xxMax=5 ;#upper bound for xyMin=-5 ;#Lower bound for the rangeyMax=5 ;#Upper bound for the rangeplot((f(x),g(x)),xmin=xMin ,xmax=xMax ,ymin=yMin ,ymax=yMax ,legend_label='automatic ')

#Here we generate labels to avoid confusion

3.3 Factorization and Root Solver

You may use the following comand to factor a function

f(x)=x^12-1 ;#As always , define your functionF=factor(f(x)) ;#factorshow(F) ;#Display the factors 'nicely '

The following code can be used to find the roots of a function. The output is atouple, where the first coordinte is the actual root, and the second coordinateis the multiplicity of the root.

f(x)=(x^2-1)*(x-2)*(x-3)*(x-4) ;#FunctionR=f.roots() ;#Root findingshow(R) ;#Display 'nicely '

3.4 Secant and Tangent Lines

The next code spitls out the secant line of a function given two points on itsdomain. It also prints the slope of the secant line.

f(x)=x^2 #Functionx1 = 0 #First valuex2 = 2 #Second valuexMin=-5 #Lower bound of the plotxMax=5 #Upper bound of the plot

##CODE BELOW##y1 = f(x1)y2 = f(x2)

slope = (y2-y1)/(x2-x1)

p = plot(f(x),x,xMin ,xMax)pt1 = point((x1, y1), rgbcolor =(1,0,0), pointsize =30)pt2 = point((x2, y2), rgbcolor =(1,0,0), pointsize =30)

l = plot(slope*(x-x1)+y1,x,xMin ,xMax , rgbcolor =(1,0,0))

t1 = text("(%s,␣%s)" % (float(x1), float(y1)), (x1+0.5,y1 -0.8), rgbcolor =(1,0,0))

t2 = text("(%s,␣%s)" % (float(x2), float(y2)), (x2+0.5,y2 -0.8), rgbcolor =(1,0,0))

t3 = text("slope:␣%s" % float(slope), ((x1+x2)/2-0.5,(y1+y2)/2), rgbcolor =(1,0,0))

3.5. LIMITS 15

g = p+pt1+pt2+l+t1+t2+t3g.show(xmin=xMin , xmax=xMax)

3.5 LimitsThe following code takes a function f(x) and a value a to compute lim

x→af(x)

f(x)=1/x #Your functiona=0 #Valuelimit(f(x), x=a)

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