The Difference Quotient A First Look at the Derivative · Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 121 Mark Sparks 2012 The Difference Quotient A

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 121

Mark Sparks 2012

The Difference Quotient

A First Look at the Derivative

Today we are introduced to the concept with which we will spend our greatest amount of time

investigating in Calculus AB—the derivative. Let’s draw a picture together.

What does the expression xhx

xfhxf

)(

)()(represent? What does this expression simplify to?

As h, the distance between the x – values, x and (x + h), approaches zero, what happens to the secant line?

What does the limit h

xfhxf

h

)()(lim

0

represent?

Suppose 14)( 2 xxxf . Findh

xfhxf

h

)()(lim

0

.

Your result to the previous limit is defined to be the derivative, )(' xf , of the function f(x). Now, let’s see

what this derivative represents in terms of the graph of f(x).


Mark Sparks 2012

Your result ofh

xfhxf

h

)()(lim

0

for 14)( 2 xxxf is a function in terms of x. The graph of f(x)

is pictured below. Complete the chart for the indicated x – values and )(' xf .

x – value Value of 42)(' xxf

–4

–2

–1

Now, use a ruler and draw a tangent line to the graph of f(x) on the grid above at x = –4, x = –2, and

x = –1. By investigating the graph, what does it appear that the derivative function 42)(' xxf

represents in terms of the graph at given values of x?

Definition of the Derivative and What It Represents Graphically

Find the equation of the tangent line to f(x) at each of the points below. Then, draw the graphs of the

tangent lines on the grid above where f(x) is graphed.

Equation of the tangent line at

x = –4


x = –2


x = –1


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When you hear “DERIVATIVE,” you think “SLOPE OF THE TANGENT

LINE.”

When you hear “SLOPE OF THE TANGENT LINE,” you think

“DERIVATIVE.”

Now that we understand what the derivative of a function represents graphically, let’s practice using the

limit of the difference quotient, h

xfhxf

h

)()(lim

0

, to find )(' xf for each of the functions below.

3)(3

2 xxf 32)( 2

2

1 xxxf

Notice that )(' xf for 3)(3

2 xxf was different than )(' xf for 32)( 2

2

1 xxxf . How are they

different and why do you suppose this is so?


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Find h

xfhxf

h

)()(lim

0

for the functions given below to find and use )(' xf .

2)( xxf

2

3)(

xxf

Find the equation of the line tangent to the graph of

2)( xxf at x = 7.

Find the equation of the line tangent to the graph of

2

3)(

xxf at x = 1.

Using a graphing calculator, graph each of the functions above and the equation of the tangent line that

you found to verify your work.


Mark Sparks 2012

Over the course of this lesson so far, you have found derivatives of several functions and evaluated that derivative at certain x – values. Look back at

your work and complete the table below.

Equation of

Function, f(x)

Equation of Derivative,

)(' xf Value of )(' xf at

the Indicated

value of x

Find the Value of the Limit

ax

afxf

ax

)()(lim , where a is the value of x.

14)( 2 xxxf

x = –1

2)( xxf

x = 7

What inference can you make that explains what the limit ax

afxf

ax

)()(lim represents?


Mark Sparks 2012

Complete the table below, stating what each of the indicated limits finds in terms of the derivative of a

function, f(x).

Definition of the

Derivative

h

xfhxf

h

)()(lim

0

Alternate Form

of the Definition

of the Derivative

ax

afxf

ax

)()(lim


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Name_________________________________________Date____________________Class__________

Day #12 Homework

Find h

xfhxf

h

)()(lim

0

for each of the functions below. Then, find the equation of the tangent line to

the graph of f(x) at the given value of x.

1. xxxf 2)( 3 3. xxf 3)(

2. Find the equation of the line tangent to the graph

of xxxf 2)( 3 at x = –1.

4. Find the equation of the line tangent to the graph

of xxf 3)( at x = –6.


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For problems 5 – 9, use the function2

)(

x

xxf .

5. Find )(' xf by finding h

xfhxf

h

)()(lim

0

.

6. Find the slope of the tangent line drawn to the

graph of f(x) at x = –2.

7. Find the slope of the tangent line drawn to the

graph of f(x) at x = –1.

8. Find the equation of the tangent line drawn to

the graph of f(x) at x = –1.

9. Find ax

afxf

ax

)()(lim , where a = –1.


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Understanding the Derivative from a Graphical and Numerical Approach

So far, our understanding of the derivative is that it represents

the slope of the tangent line drawn to a curve at a point.

Complete the table below, estimating the value of )(' xf

at the indicated x – values by drawing a tangent line and

estimating its slope.

x –

Value

Estimation of Derivative

Is the function

Increasing,

Decreasing or at a

Relative Maximum

or Relative

Minimum

Equation of the tangent line at this

value of x.

–7

–6

–4

–2

–1

1

3

5

7


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Based on what you observed in the table on the previous page, what inferences can you make about the

value of the derivative, )(' xf , and the behavior of the graph of the function, f(x)?

Numerically, the value of the derivative at a point can be estimated by finding the slope of the secant line

passing through two points on the graph on either side of the point for which the derivative is being

estimated.

x –

Value

Estimation of Derivative

Is the function

Increasing,

Decreasing or at a

Relative Maximum

or Relative

Minimum

Equation of the tangent line at this

value of x.

0

1

4

6

x –3 0 1 4 6 10

f(x) 2 1 –3 0 –7 2


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The graph of a function, g(x), is pictured to the right. Identify the following characteristics about the

graph of the derivative, )(' xg . Give a reason for your answers.

Definition of the Normal Line

Pictured to the right is the graph of 4)1()( 2

2

1 xxf .

Draw the tangent line to the graph of f(x) at x = 1. Then, estimate

the value of )1('f .

Find the equation of the tangent line to the graph of f(x) at x = 1.

The normal line is the line that is perpendicular to the tangent line at the point of tangency. Draw this line

and find the equation of the normal line.

The interval(s) where

)(' xg < 0

The interval(s) where

)(' xg > 0

The value(s) of x

where )(' xg = 0


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The graph of the derivative, )(' xh , of a function h(x) is pictured below. Identify the following

characteristics about the graph of h(x) and give a reason for your responses.

The interval(s) where h(x) is increasing

The interval(s) where h(x) is decreasing

The value(s) of x where h(x) has a

relative maximum.

The value(s) of x where h(x) has a

relative minimum.

If h(–1) = ½, what is the equation of the

tangent line drawn to the graph of h(x) at

x = –1?

If h(2) = –3, what is the equation of the

normal line drawn to the graph of h(x) at

x = 2?


Mark Sparks 2012

Name_________________________________________Date____________________Class__________

Day #13 Homework

1. The line defined by the equation )3(323

2 xy is tangent to the graph of g(x) at x = –3. What is

the value of 3

)3()(lim

3

x

gxg

x? Show your work and explain your reasoning.

Use the graph of f(x) pictured to the right to perform the actions in exercises 2 – 6. Give written

explanations for your choices.

2. Label a point, A, on the graph of y = f(x) where the derivative is negative.

3. Label a point, B, on the graph of y = f(x) where the value of the function is negative.

4. Label a point, C, on the graph of y = f(x) where the derivative is greatest in value.

5. Label a point, D, on the graph of y = f(x) where the derivative is zero.

6. Label two different points, E and F, on the graph of y = f(x) where the values of the derivative

are opposites.


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7. Match the points on the graph of g(x) with the value of )(' xg in the table.

8. The function to the right is such that h(4) = 25 and )4('h = 1.5. Find the

coordinates of A, B, and C.

For exercises 9 – 11, use the function1

1)(

xxf .

9. Find )(' xf . 10. Find the equation of the tangent line drawn

to the graph of f(x) at x = 0.

11. Find the equation of the normal line drawn

to the graph of f(x) at x = 0.

Value of

)(' xg

Point on

g(x)

–3

–1

0

½

1

2


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12. Given below are graphs of four functions—f(x), g(x), h(x), and p(x). Below those graphs are

graphs of their derivatives. Label the graphs below as )(' xf , )(' xg , )(' xh , and )(' xp .

The table below represents values on the graph of a cubic polynomial function, h(x). Use the table to

complete exercises 22 – 24.

x –3 –2 –1 0 1 2 4

h(x) –24 0 8 6 0 –4 18

13. Two of the zeros of h(x) are listed in the table. Between which two values of x does the Intermediate

Value Theorem guarantee that a third value of x exists such that h(x) = 0? Explain your reasoning.

14. Estimate the value of )5.1('h . Based on this 15. Estimate the value of )75.1(' h . Based on

value, describe the behavior of h(x) at x = 1.5. this value, describe the behavior of h(x) at

Justify your reasoning. x = –1.75. Justify your reasoning.


Mark Sparks 2012

Analytically Finding the Derivative of Polynomial, Polynomial Type, Sine, and Cosine Functions

Consider the function f(x) = 3. What does the graph of this function look like? If a tangent line were

drawn to f(x) at any value of x, what would the slope of that tangent line be?

Based on this though process, if f(x) = c, where c is any constant, then ___________)(' xf .

Shown below are 6 different polynomial, or polynomial–type, functions. Watch as I find the derivative of

each function. See if you can figure out the algorithm that I am using for each function.

Function, f(x) Derivative, )(' xf

323)( 2 xxxf

1325)( 23 xxxxf

43 636)( xxxf

21 32)( xxxf

246)( 32

xxxf

21

21

36)( xxxf

Based on what you have seen in the table above, you should now be able to infer how to complete the

following Power Rule for Differentiation.

__________________nxdx

d


Mark Sparks 2012

In order to apply the Power Rule for Differentiation, the equation must be written in “polynomial form.”

To what do you suppose “polynomial form” refers?

Find )(' xf for each of the following functions. Leave your answers with no negative or rational

exponents and as single rational functions, when applicable.

3

24

2)( x

xxf

x

xxxxf

233)(

24

)12)(2)(3()( xxxxf

5

23 5)(

x

xxxf

3 2

3)(

x

xxf

41

43

24)( xxxf


Mark Sparks 2012

Remember two trigonometric identities that we will use to find the derivatives of the sine and cosine

functions.

cos(a + b) = _________________________________________

sin(a + b) = _________________________________________

Use h

xfhxf

h

)()(lim

0

to find )(' xf for each of the following functions. Your results will show the

derivative of the sine and cosine functions.

f(x) = sin x f(x) = cos x

__________________sin xdx

d __________________cos x

dx

d


Mark Sparks 2012

For each of the following functions, find the equation of the tangent line to the graph of the function at the

given point.

2)1)(12()( xxxf when x = –1 sin4)(f when θ = 2

cos32)( g when θ = π

3

2)(

x

xxh when x = 2


Mark Sparks 2012

Given the equation of a function, how might you determine the value(s) at which the function has a

horizontal tangent? Explain your reasoning.

At what value(s) of x will the function xxxf 3)( have a horizontal tangent?

At what value(s) of θ at which the function sin)( f has a horizontal tangent on the interval

[0, 2π)?


Mark Sparks 2012

Name_________________________________________Date____________________Class__________

Day #14 Homework

For exercises 1 – 12, find the derivative of each function. Leave your answers with no negative or

rational exponents and as single rational functions, when applicable.

1. 32 325)( xxxf 2.

x

xxxxh

232)(

23

3. 7

3)(

xxh 4.

8

52)(

x

xxg

5. cos3)( 2 f 6. 3 2)( xxh

7. sin2)( g 8. xxxp 2

3

2)(


Mark Sparks 2012

9. 2)12)(3()( xxxg 10.

3

2 22)(

x

xxxh

11. 3

3)(

x

xxf 12. xxxh cos36)(

13. For what value(s) of x will the slope of the tangent line to the graph of xxh 4)( be –2? Find the

equation of the line tangent to h(x) at this/these x – values. Show your work.


Mark Sparks 2012

14. Find the equation of the line tangent to the graph of 4 3

2)(

x

xg when x = 1.


1 yx is the line tangent to the graph of a function

f(x) when x = a. What is the value of )(' af ? Show your work and explain your reasoning.


2 xy is the line tangent to the graph of a function

f(x) at the point (–3, 3). What is the equation of the normal line when x = –3. Explain your

reasoning.

17. Determine the value(s) of x at which the function 28)( 24 xxxf has a horizontal tangent.

18. Determine the value(s) of θ at which the function cos23)( f has a horizontal tangent on


Mark Sparks 2012

the interval [0, 2π).

19. For what value(s) of k is the line y = 4x – 9 tangent to the graph of f(x) = x2 – kx?

Connections between F(x) and F’(x) for Polynomial and Trigonometric Functions


Mark Sparks 2012

Pictured below is the graph of a function f(x). Answer the following questions about the derivative.

F’(x) F(x)

Is = 0

Is > 0

Is < 0

Changes from positive to negative

Changes from negative to positive

Graph of f(x) Possible Graph

of )(' xf


Mark Sparks 2012

1. Approximate the value of )4('f .

2. At what value(s) of x is )(' xf = 0. Justify

your answer.

3. On what open interval(s) is )(' xf < 0? Justify your answer.

4. On what open interval(s) is )(' xf > 0? Justify your answer.

5. At what value(s) of x does the graph of )(' xf go from being below the x – axis to above the x – axis?

Justify your answer.

6. At what value(s) of x does the graph of )(' xf go from being above the x – axis to below the x – axis?


Graph of f(x)


Mark Sparks 2012

Pictured below is the graph of )(' xf on the interval [–3, 4]. Answer the following questions about f(x).

1. On what open interval(s) is the graph of f(x)

increasing? Justify your reasoning.

2. On what open interval(s) is the graph of f(x) decreasing? Justify your answer.

3. At what value(s) of x does the graph of f(x) have a horizontal tangent? Justify your answer.

4. At what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer.

5. At what value(s) of x does the graph of f(x) have a relative minimum? Justify your answer.

6. What is the slope of the tangent line to the graph of f(x) at x = 0? Justify your reasoning.

7. What is the slope of the normal line to the graph of f(x) at x = 4? Justify your reasoning.

Graph of )(' xf


Mark Sparks 2012

For each of the given functions, determine the interval(s) on which f(x) is increasing and/or decreasing.

Find all coordinates of the relative extrema. Unless otherwise noted, perform the analysis on all values on

, . Provide justification for your answers.

1. 16)( 3 xxxf


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2. f(x) = 3x5 – 5x3


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3. f(θ) = θ + 2sinθ on (0, 2π)


Mark Sparks 2012

Name_________________________________________Date____________________Class__________

Day #15 Homework

For exercises 1 – 3, determine on what intervals the given function is increasing or decreasing. Also,

identify the coordinates of any relative extrema of the function. Show your work and justify your

reasoning.

1. f(x) = 2x3 + 3x2 – 12x

2. g(x) = x3 – 6x2 + 15


Mark Sparks 2012

3. h(x) = (x + 2)2(x – 1)

4. Pictured to the right is the graph of )(' xf . On what interval(s)

is the graph of f(x) increasing or decreasing? Justify your

reasoning.

5. Pictured to the right is the graph of )(' xf . At what value(s)

of x does the graph of f(x) have a relative maximum/minimum?

Justify your reasoning.


Mark Sparks 2012

Solidifying the Concept of the Derivative as the Tangent Line

Pictured to the right is the graph of a quadratic function,

4)2()( 221 xxg .

1. Find )4(' g and explain what this value represents in terms of

the graph of the function g(x).

2. Find the equation of the tangent line drawn to the graph of g(x) at x = –4. Sketch a graph of this

tangent line on the grid with the graph of g(x) above.

3. Using the equation of the tangent line, find the value of y when x = –3.9. Then, find the value of

g(–3.9).

4. What do you notice about the values of these two results from question 3? What does this imply about

how the equation of the tangent line might be used?


Mark Sparks 2012

Pictured to the right is the graph of the function 22)( xxxg .

Use the graph and the equation to answer questions 5 – 9.

5. Based on the graph, at what value(s) does the graph of g(x) have a

horizontal tangent? Give a reason. Show an algebraic analysis that

supports your answer.

6. On what interval(s) is )(' xg < 0? Give a reason for your answer.

7. On what interval(s) is )(' xg > 0? Give a reason for your answer.

8. For what value(s) of x is the slope of the tangent line equal to 2? Show your work.

9. Find an equation of the tangent line drawn to the graph of g(x) when x = 4. Then, draw the tangent

line on the grid above.


Mark Sparks 2012

The table of values below represents values on the graph of the derivative, )(' xh , of a polynomial function

h(x). The zeros indicated in the table are the only zeros of the graph of )(' xh . Use the table to answer

questions 10 – 15.

10. On what interval(s) is the function h(x) increasing and decreasing? Give reasons for your answers.

11. At what x – value(s) does the graph of h(x) have a relative maximum? Justify your answer.

12. At what x – value(s) does the graph of h(x) have a relative minimum? Justify your answer.

13. If h(3) = 2, what is the equation of the tangent line to the graph of h(x) at x = 3? What is the

equation of the normal line to the graph of h(x) at x = 3?

14. Find the tangent line approximation of h(3.1).

15. Find the value of each of the following limits:

)(lim xhx

)(lim xhx

x −8 −5 −2 0 3 5 7 10 12

h’(x) 11 5 0 −1 −3 −1 0 −3 −9


Mark Sparks 2012

The derivative of a polynomial function, f(x), is given by the equation )3)(2()(' xxxxf . Use this

equation to answer questions 16 – 20.

16. On what intervals is f(x) increasing? Decreasing? Justify your answers.

17. At what value(s) of x does the graph of f(x) reach a relative minimum? Justify your answers.

18. At what value(s) of x does the graph of f(x) reach a relative maximum? Justify your answers.

19. If f(4) = –1, what is the equation of the tangent line drawn to the graph of f(x) at x = 4?

20. Approximate the value of f(4.1). Explain why this is a good approximation of the true value of f(4.1).


Mark Sparks 2012

Pictured to the right is a graph of )(' xp , the derivative of a

polynomial function, p(x). Use the graph to answer the questions

21 – 25.

21. On what interval(s) is the graph of p(x) decreasing?


22. On what interval(s) is the graph of p(x) increasing? Justify

your answer.

23. At what value(s) of x does the graph of p(x) reach a relative maximum? Justify your answer.

24. At what value(s) of x does the graph of p(x) reach a relative minimum? Justify your answer.

25. Approximate the value of p(1.8) using the tangent line approximation if p(2) = –3.


Mark Sparks 2012

Name_________________________________________Date____________________Class__________

Day #16 Homework

1. If )1()3()(' 2 xxxg , determine on what intervals the graph of g(x) is increasing or decreasing

and identify the value(s) of x at which g(x) has a relative maximum or minimum. Justify your

reasoning and show your work.

For exercises 2 – 4, use the graph of t function, h(x), pictured to the right. Use the graph to identify the

following. Provide written justification.

2. On what interval(s) is )(' xh < 0?

3. On what interval(s) is )(' xh > 0?

4. At what value(s) of x does )(' xh change from positive to negative? From negative to positive?


Mark Sparks 2012

Consider the quadratic function 4)( 2

21 xxxf .

5. Sketch an accurate graph of the function.

6. Find )(' xf and use it to find the absolute

maximum of the graph of f(x).

7. Estimate the value of )0('f and explain what this value represents in terms of the graph of f(x).

8. Find the equation of the tangent line to the graph of f(x) at x = 0. Draw a graph of this line.

9. Sketch a graph of the normal line to the tangent line at x = 0. What is the equation of this line?

10. Use the equation of the tangent line to approximate f(0.1). Then, find f(0.1) using the equation

of f(x). Is the approximation an under or over approximation of the actual value of f(0.1)? Based

on the graph of f(x), why do you suppose this is true?


Mark Sparks 2012

11. For what function does h

xhx

h

sin2)sin(2lim

0

give the derivative? Find the limit.

12. Find h

xhx

h

55

0

)(lim

. 13. Find

h

hxx

h

0lim .

14. If x

xxf

3)( , what is the slope of the normal line to the graph of f(x) when x = 4?

15. If 2x – 3 = 5(y + 1) is the equation of the normal line to the graph of f(x) when x = a, find the

value of )(' af . Show your work and explain your reasoning.


Mark Sparks 2012

16. On the interval [0, 2π), find the coordinates of the relative minimum(s) of sin23)( f .

The derivative of a function f(x) is )5()3()(' 2 xxxf . Use this derivative for exercises 17 and 18.

17. At what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer.

18. Use the equation of the tangent line to approximate the value of f(2.1) if f(2) = –3.

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The Difference Quotient A First Look at the Derivative · Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 121 Mark Sparks 2012 The Difference Quotient A