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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).
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 122
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
Equation of the tangent line at
x = –2
Equation of the tangent line at
x = –1
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 123
Mark Sparks 2012
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?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 124
Mark Sparks 2012
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 125
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?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 126
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 127
Mark Sparks 2012
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 128
Mark Sparks 2012
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 129
Mark Sparks 2012
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 130
Mark Sparks 2012
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 131
Mark Sparks 2012
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 132
Mark Sparks 2012
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?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 133
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 134
Mark Sparks 2012
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 135
Mark Sparks 2012
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 136
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 137
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 138
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 139
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 140
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)?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 141
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)(
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 142
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 143
Mark Sparks 2012
14. Find the equation of the line tangent to the graph of 4 3
2)(
x
xg when x = 1.
15. The line defined by the equation )3(232
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.
16. The line defined by the equation )3(33
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 144
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 145
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 146
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?
Justify your answer.
Graph of f(x)
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 147
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 148
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 149
Mark Sparks 2012
2. f(x) = 3x5 – 5x3
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 150
Mark Sparks 2012
3. f(θ) = θ + 2sinθ on (0, 2π)
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 151
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 152
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 153
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?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 154
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 155
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
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 156
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).
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 157
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?
Justify your answer.
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 158
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?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 159
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?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 160
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.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 161
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.