Unit 3-Student Copy

Date __________

1 Unit 3-The Derivative

Rules for Differentiation Finding the Derivative of a Product of Two Functions

Rewrite the function f(x) = (2x – 3)(x2 – 2x + 1) as a cubic function. Then, find ).(' xf What does this equation of )(' xf represent, again? Two men, Isaac Newton and Gottfried Leibniz, are credited for developing the study of calculus. In 1673, Leibniz published an article in which he derived what we know today as the Product Rule of Differentiation. Let’s write this rule together in the box below.

Product Rule of Differentiation To show that this rule works, let’s apply this rule to the function f(x) = (2x – 3)(x2 – 2x + 1) that we rewrote and differentiated as a polynomial above. Students often wonder why this rule is so important if we could just rewrite as a polynomial and easily differentiate it. The answer to that question is simple. If it is possible to rewrite as a polynomial, always do so.

But in the case of the function xxxg sin)( 2 , there is no way to rewrite as a polynomial. Apply the product

rule to find the slope of the normal line to the graph of xxxg sin)( 2 when x = π.

Date __________


Use the product rule to find the derivative of each of the following functions.

332)( 22 xxxxf

23)( 2 xxxxg

xxxf sin)( 3

xxxh cos)23()(

sin3)( xg

xxxh cossin)(

Find the equation of the line tangent to the graph of tttg cos)( 2 when t = 6 .

Date __________


There is a very valuable lesson that we must learn when we are introduced to the product rule.

What is the lesson to be learned from the algebraic analysis above? If )3)(1)(32()( xxxxg , what is the slope of the normal line to the graph of g(x) when x = 2?

On page 193, you were asked to find )(' xf by applying the product rule to the function

23)( 2 xxxxf . In the space below, write the result that you obtained.

Given the function 23)( 2 xxxxf . Rewrite the function in polynomial form. Then, find )(' xf .

Date __________


Below are graphs of two functions—f(x) and g(x). Let )()()( xgxfxP and let )()( 2 xgxxR . Use the graphs to answer the questions that follow. Graph of f(x) Graph of g(x)

If 2)4(' g , what is the value of )4(' P ?

If )2(' R = 20, what is the value of )2(' g ?

Find the equation of the line tangent to the graph of P(x) when x = –4.

Find the equation of the line tangent to the graph of R(x) when x = –2.

Date __________


Let f(x) and g(x) be differentiable functions such that the following values are true. Estimate the value of )5.3('f .

If )(4)(2)( xgxfxq , what is the value of )4('q ?

If )()(2)( xgxfxp , what is the value of )3('p ?

Find the equation of the line tangent to the graph of

)()( 3 xfxxv when x = –1.

If )(33)(2)( xgxfxk , what is the value of )3('k ?

x f(x) g(x) )(' xf )(' xg

4

1

7

2

–3

3

–2

–3

–4

2

–1

2

–2

1

–1

Date __________


Name_________________________________________Date____________________Class__________

Day #19 Homework In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified method.

1.

32)( 2 xxxxf

Rewrite f(x) as a polynomial first. Then apply the power rule

to find )(' xf .

2.

32)( 2 xxxxf

Apply the product rule to find)(' xf .

For exercises 3 – 5, find the derivative of each function.

3. xxxxf 22)( 22

5. 4)( 23 xxxf

4. 5323)( 23 xxxxxf

Date __________


Find the slope of the normal line drawn to the graph of each function at the indicated value of x.

6. xxxg sin)( when x = π 7. )cos(sinsin)( xxxxh when x = 4

For each of the functions below, find the equation of the tangent line drawn to the graph of g(x) at the indicated value of x.

8. 42)( 2 xxxg when x = 4

9. xxxg cos)( 2 when x = 2

Date __________


Use the table below to complete exercises 10 – 12. 10. If )()(2)( xgxfxH , what is the equation of the tangent line when x = –1?

11. If xxgxJ sin)()( , what is the value of )0('J ?

12. If 2)(2)(4)( xgxfxxK , what is the slope of the normal line when x = –2?

x f(x) f ’(x) g(x) g’(x)

–2

1

–1

2

4

–1

3

–2

1

1

0

–1

2

–2

–3

Date __________


Rules for Differentiation Finding the Derivative of a Quotient of Two Functions

Rewrite the function2

23 232)(

x

xxxf

as a function in polynomial form. Then, find ).(' xf

Just as Leibniz was the first to publish a proof of the Product Rule for Differentiation, Isaac Newton was the first to publish a proof of the Quotient Rule of Differentiation using the limit definition of the derivative. Let’s write this rule together in the box below.

Quotient Rule of Differentiation

To show that this rule works, let’s apply this rule to the function 2

23 232)(

x

xxxf

that we rewrote and

differentiated as a polynomial-form above.

Find the equation of the tangent line drawn to the graph of3

12)(

x

xxg when x = –2.

Date __________


We will now use the quotient rule to derive the derivative formulas for the remaining trigonometric functions. Rewrite each function in terms of sine and/or cosine and differentiate using the Quotient Rule.

tan)( f cot)( f

sec)( f csc)( f

Find the equation of the normal line drawn to the graph of

cos

3)( f when .

Date __________


Find the derivative of each of the functions below by applying the quotient rule.

2

2)(

2

x

xxxf

2

tan)(

x

xxg

cos1

sin)(

h

5

3)(

1

xxf x

Date __________


Show, using the quotient rule, that if1

23)(

2

2

x

xxxf , then

2)1(

3)('

xxf .

Similar to the Product Rule, there is a very valuable lesson that we must learn when we are introduced to the

quotient rule. In the box below, first factor and simplify the function,1

23)(

2

2

x

xxxf , from above. Then,

differentiate using the quotient rule What is the lesson to be learned from the algebraic analysis above?

Date __________


Below are graphs of two functions—f(x) and g(x). Let )(

)()(

xg

xfxP and let

)(

sin)(

xf

xxR . Use the graphs to

answer the questions that follow. Graph of f(x) Graph of g(x)

Find )5('P .

Find )0('R

Find the equation of the line tangent to the graph of P(x) when x = 5

Find the equation of the line tangent to the graph of R(x) when x = 0.

Date __________


Let f(x) and g(x) be differentiable functions such that the following values are true. Estimate the value of )5.2('g .

If)(

)()(

xf

xgxp , what is the value of )4('p ? What

does this value say about the graph of p(x) when x = 4? Give a reason for your answer.

If

)(

)(2)( 2

xg

xfxxq , what is the value of )2('q ?

Find the equation of the line tangent to the graph of

)(

3)(

xg

xxv when x = 3.

x f(x) g(x) )(' xf )(' xg

4

1

7

8

–2

3

–5

–3

–4

6

2

2

–1

9

–1

Date __________


Name_________________________________________Date____________________Class__________

Day #20 Homework

For exercises 1 and 2, show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified method.

1.

2

23 332)(

x

xxxf

Rewrite f(x) in a polynomial-

form first. Then apply the power rule to find )(' xf .

2.

2

23 332)(

x

xxxf

Apply the quotient rule to find

)(' xf .

3. Find the equation of the line tangent to the graph of13

32)(

2

x

xxxg when x = –1.

Date __________


Find the derivative of each of the following functions.

4. 1

)(2

x

xxh

5. 1

)(

x

xxh

6. 3

cos)(

g

7.

cos2

)sin1(3)(

f

Date __________


Use the table below to complete exercises 8 – 10.

8. If )(

)(2)(

xg

xfxH , what is the equation of

the tangent line when x = –1?

9. If )(

cos3)(

xf

xxxJ

, what is the value of

)0('J ?

10. If )(3

)(4)(

xg

xfxxK

, what is the slope of the normal line when x = –2?

x f(x) f ’(x) g(x) g’(x)

–2

1

–1

2

4

–1

3

–2

1

1

0

–1

2

–2

–3

Date __________


11. If sincsc)( f , show that coscot)(' 2f . 12. Find the equation of the line tangent to the graph of sintan)( f when

4 .

Date __________


Rules for Differentiation Finding the Derivative of a Composite Function

Rewrite the function 3)32()( xxf as a function in polynomial form. Then, find ).(' xf Leibniz was the first of the two great calculus developers to use the Chain Rule to differentiate composite functions. Let’s write this rule together in the box below.

Chain Rule of Differentiation of Composite Functions

To show that this rule works, let’s apply this rule to the function 3)32()( xxf that we rewrote and differentiated as a polynomial-form above.

Find the slope of the normal line to the graph of 2sin)( f when 4

3 .

Date __________


Find the derivative of each of the following functions by applying the chain rule.

32 23)( xxf 52)( xxg

3 2)2()( xxh 3 2 25)( xxxF

xxG 3cos)( 2 )12(sin)( 2 xxh

Date __________


Now that you know “THE BIG THREE” rules of differentiation—product, quotient, and chain—let’s see how the three can be incorporated with each other. Find the derivative of each of the following functions.

35)( xxxf

3

12sin)(

x

xxg

Date __________


3

52)(

x

xxh

Given the graph of H(x) pictured to the right, find the equation of the

tangent line to the graph of )()( xHxP when x = –4.

Date __________


Let f(x) and g(x) be differentiable functions such that the following values are true. Is the graph of ))(()( xgfxh increasing, decreasing or at a relative maximum or minimum when x = 3? Give a reason for your answer.

If )2()( xgxp , what is the value of )1('p ?

If )()()( xgxfxq , what is the value of )4('q ? What does this value say about the graph of q(x) when

x = 4? Give a reason for your answer.

x f(x) g(x) )(' xf )(' xg

4

1

7

8

–2

3

–5

4

–4

6

2

2

–1

0

–1

Date __________


Name_________________________________________Date____________________Class__________

Day #21 Homework In exercises 1 – 6, find the derivative of each of the following functions.

1. 3

2 2

5)(

x

xxf

2. 2

32)(

x

xxf

3. 13)( 2 xxxh 4. 3 2 49)( xxg

Date __________


5. 21)( xxxf 6. 1

)(2

x

xxp

For exercises 7 and 8, find the value of the derivative of the function at the given point. 7. 2sin)( 2

41g when 8. 2cos2sin)( f when

4

Date __________


The graph of the function 225)( xxf is pictured to the right. Use this function to complete exercises 9 – 11. 9. Find the values of )3(f and )3('f . 10. Find the equation of the line tangent to the graph of f(x) when x = 3 and graph this line on the grid with f(x). 11. Find the equation of the normal line to the graph of f(x) when x = 3 and graph this line on the grid with f(x). 12. At what value(s) of x does the graph of 12)( 2

21 xxxh have a horizontal tangent? Show your

work and give a reason for your answer.

Date __________


Use the table below to complete exercises 13 – 14.

13. If )()()( xgxfxH , is the graph of H(x) increasing or decreasing when x = –1? Give a

reason for your answer.

14. If 32

)()(2)( xgxfxP , what is the value of )0('P ?

15. Find the equation of the normal line to the graph of )3tan()( xxh when

12x .

x f(x) f ’(x) g(x) g’(x)

–2

1

–1

2

4

–1

3

–2

1

1

0

–1

2

–2

–3

Date __________


Problems to Discuss before Quiz #4

Problem #1 Find the following limit. Explain the reasoning that you used to arrive at your answer.

h

xhx

h

3cos)(3coslim

0

Problem #2

Find the equation of the tangent line to the graph of the given function when 3

x .

xxxf cos3)(

Date __________


Problem #3 Find the equation of the normal line to the graph of the function below when x = –2.

3 23)( xxf

Problem #4

At what point on the graph of the function 23)( xxf is the normal line perpendicular to the line defined

by the equation 341 xy ?

Date __________


Rules for Differentiation

Finding the Derivative of the Natural Exponential and Logarithmic Functions

Differentiation Rule for Natural Exponential Functions Find the derivative of each of the following functions.

xexf sin)( 32)( xexf

xexf 23)( xexxf 3)32()(

xexxf 22)( 62)( xexf

Date __________


2

5

3)(

x

exf

x

Differentiation Rule for Natural Logarithmic Functions Find the derivative of each of the following functions.

)32ln()( xxf xxxf 23ln)( 2

)ln(cos)( xxf 42ln)( xxf

Date __________


Finding Values of Derivatives Using the Graphing Calculator For each of the functions below, find the value of )(' xf at the indicated value of x using the graphing calculator. Then, determine if the function is increasing, decreasing, has a horizontal tangent or has a vertical tangent. Give a reason for your answer.

Function Value of )(' af Is f(x) increasing or decreasing, or does f(x) have a horizontal or a vertical tangent?

1.

xexf x sin3)(

a = –2

2.

xexf x sin3)(

a = 1

3.

2

)ln(cos)(

x

xxf

a = 3

4.

2

)ln(cos)(

x

xxf

a = π

5.

)34.0tan()( xexf

a = 0

6.

)(lnsin5)( 2 xxf

a = 1

Date __________


When the value of the derivative of a function is positive, we say that the function is increasing. When the value of the derivative of a function is negative, we say that the function is decreasing. When speaking of quantities increasing or decreasing, they do so at a certain rate. We already understand the derivative to be the SLOPE OF THE TANGENT LINE. Slope is a rate. Therefore, the derivative of a function actually represents the RATE AT WHICH A FUNCTION IS CHANGING.

7.

The number of people entering a concert can be modeled by the function tetf sin560)( , where t represents the number of hours after the gates are open.

a.

Find the values of 21f and

21'f . Using correct units, explain what each value represents in the

context of this problem.

b.

How many people have entered the concert 2 hours after the gates are opened? Is the number of people entering increasing or decreasing at this time? Justify your answer.

8. After being poured into a cup, coffee cools so that its temperature, T(t), is represented by the

function 211070)(t

etT , where t is measured in minutes and T(t) is measured in degrees

Fahrenheit.

a. What is the temperature of the coffee 5 minutes after it has been poured into the cup?

b.

Is the temperature decreasing faster 1 minute after it is poured or 3 minutes after it is poured? Give a reason for your answer.

Date __________


Name_________________________________________Date____________________Class__________

Day #23 Homework

In exercises 1 – 10, find the derivative of the function. Express your answer in simplest factored form.

1. xexxF 23)( 2. 22)( xexP

3. xxexH ln)( 4. xexxg 32)( 2

5. 1ln)( 2 xexJ 6. )23ln()( xxF

Date __________


7. 25ln)( xxK 8. xexxF 42)(

9. 2

ln)(

x

xxT 10.

3

2)(

x

exP

x

11. Find the equation of the tangent line to the graph of x

xy

4

ln when x = 1.

Date __________


The Relationship between Continuity and Differentiability In this lesson, our goal is to establish a relationship between a function being continuous at a value of x and a

function being differentiable at the same value. In other words, if a function is continuous at a particular value

of x, does that imply that it is also differentiable. Or, if a function is differentiable, does that mean that it must

also be continuous. Let’s investigate three functions.

Consider the function 4)( 2 xxf at x = 2. Answer the questions that follow. On the grid to the right, sketch a graph of f(x) from your graphing calculator. Based on the graph, if f(x) continuous at x = 2? Explain your reasoning. Find the value of )2('f to determine if f(x) is differentiable at x = 2.

Date __________


Consider the function 2)( 31

xxf at x = 0. Answer the questions that follow. On the grid to the right, sketch a graph of f(x) from your graphing calculator. Based on the graph, if f(x) continuous at x = 0? Explain your reasoning. Find the value of )0('f to determine if f(x) is differentiable at x = 0.

Consider the function 2)( 32

xxf at x = 0. Answer the questions that follow. On the grid to the right, sketch a graph of f(x) from your graphing calculator. Based on the graph, if f(x) continuous at x = 0? Explain your reasoning. Find the value of )0('f to determine if f(x) is differentiable at x = 0. Based on what we have seen, does continuity imply differentiability or does differentiability imply continuity?

Date __________


In order for a function to be differentiable at a value of x, then two things must be true: 1.___________________________________________________________________________________ 2.___________________________________________________________________________________

Consider the function

53,5

30,1)(

xx

xxxg to answer the following questions.

Is g(x) continuous at x = 3? Show the complete analysis. Is g(x) differentiable at x = 3? Show the complete analysis.

Date __________


For what values of k and m will the function below be both continuous and differentiable at x = 3?

53,2

30,1)(

xmx

xxkxh

For what values of a and b will the function below be differentiable at x = 1?

1,34

1,123)(

24

2

xxbxax

xbxaxxf

Date __________


Name_________________________________________Date____________________Class__________

Day #24 Homework

Use the graph of H(x), pictured to the right, to complete exercises 1 – 4. 1. Graphically, identify a value of x at which the function is continuous but not differentiable. Give a reason for your answer. 2. Write an equation of H(x) and show analytically that H(x) is, in fact, continuous at the x – value that you identified in exercise 1. Show and explain your work. 3. Show analytically that H(x) is, in fact, not differentiable at the x – value that you identified in exercise 1. Show and explain your work. 4. Given the graph of H(x) pictured above, find the equation of the tangent line to the graph of

)()( xHxP when x = 3.

Date __________


A continuous function on the interval −4 < x < 5, h(x), is described in the table below. Use the information to complete exercises 5 – 8. 5. Sketch a graph of h(x).

6. Estimate the value of )2(' h . Does this value support the claim that h(x) is increasing on the interval –4 < x < 0? Give a reason for your answer. 7. There are three x – values in the domain of h at which h(x) is not differentiable. What are these three values and give a reason for why h(x) is not differentiable at these values. 8. On what interval(s) of x is ?0)(' xh Give a reason for your answer.

x

–4

– 2

–1

0

–4 < x < 0

1

3

0 < x < 3

3 < x < 5

5

h(x)

–5

–4

–2

1

Increasing &

Concave Up

–1

–2

Decreasing &

Concave Up

Increasing &

Concave Up

0

Date __________


9. At what value(s) of x will the graph of f(x) = 2e2x – 3x have a tangent line whose slope is 1? 10. The graph of x – 2y = 9 is parallel to the normal line to the graph of f(x) when x = 5. What is the value of )5('f ? Justify your answer.

11. Let f be defined by the function

1,

1,3)(

2 xbxax

xxxf .

a. If the function is continuous at x = 1, what is the relationship between a and b? Explain your reasoning using limits. b. Find the unique values of a and b that will make f both continuous and differentiable at x = 1. Show your analysis using limits.

Date __________


Date __________


Unit #3 Problems for Review

1. Find the value of )0('f if f(x) = 32

x + 1. Then, sketch a graph of the function f(x) and use the graph to explain your analytical result for )0('f .

2. Find the value of )2('f if xxf 24)( . Then, sketch a graph of the function f(x) and use the graph to explain your analytical result for )2('f . 3. Find each of the indicated limits below. (Hint, remember the definition and alternate definition of a limit and what each tells you about a function.)

a. h

xhxh

ln)ln(lim

0

b. h

xhxh

33

0

323)(2lim

c.

2

2

2

sinsinlim

x

x

x d.

h

h

h

44

0

coscoslim

4. The equation of the normal line to the graph of y = e2x when 2dxdy is…

A. 121 xy B. 12

1 xy

C. y = 2x + 1 D. 222ln

21 xy

E. 22 22ln xy

Date __________


5. Given that f(x) = x2ex, what is an approximate value of f(1.1) if you use the equation of the

tangent line to the graph of f at x = 1?

A. 3.534

B. 3.635

C. 7.055

D. 8.155

E. 10.244

6. If )(cos5)( 2 xxf , then 2' f is …

A. 0

B. 32

C. 32

D. 65

E. 65

7. For what value(s) of k does the graph of g(x) = ke2x + 3x have a normal line whose slope is 51

when x = 1?

Date __________


The graph of the first derivative, )(' xf , of a polynomial function, f(x), is

pictured to the right. Use the graph to answer questions 8 – 11.

8. What type of polynomial function is f(x)? Give a reason for

your answer.

9. At what value(s) of x does the graph of f(x) have a horizontal

tangent? Give a reason for your answer.

10. On what interval(s) of x would the graph of f(x) be increasing? Give a reason for your answer.

11. On what interval(s) of x would the graph of f(x) be decreasing? Give a reason for your answer.

12. Find two values on the interval (0, 2π) where the slope of the tangent to the graph of f(x) = cos 2x

is equal to 3 .

Graph of )(' xf

Date __________


13. Consider the piece-wise defined function below to answer the questions that follow.

2,

2,2)(

2

xbax

xbxaxxf

a. If a = –3 and b = 4, will f(x) be continuous at x = 2? Justify your answer.

b. If a = –3 and b = 4, will f(x) be differentiable at x = 2? Justify your answer.

c. For what value(s) of a and b will f(x) be both continuous and differentiable at x = 2?

Show your work.

Date __________


14. A rodeo performer spins a lasso in a circle perpendicular to the ground. The height from the ground of the knot, measured in units of feet, in the lasso is modeled by the function

5cos3)(3

5 ttH ,

where t is the time measured in seconds after the lasso begins to spin. a. Find the value of H(0.75). Using correct units, explain what this value represents in the context of this problem. b. Find the value of )75.0('H . Using correct units, explain what this value represents in the context of this problem.

Unit 3-Student Copy

Documents

Transcript of Unit 3-Student Copy