Unit #7 – Advanced Integration and Applications
AP Calculus
Day
Objective
57 Understand and use the Second Fundamental Theorem of
Calculus and functions defined by integrals.
Assignment
Daily Lessons pages 591 – 595
Day #57 Homework
58
59
Finding integrals by substitution, including changing of variables Daily Lessons pages 599 – 602
and limits of integration.
Day #58 Homework
Find general and particular solutions for differential equations.
Daily Lessons pages 605 – 608
2000 AB #6 Parts a and b
60
Identify and create slope fields that represent the general
solutions of differential equations.
2008 AP #5
61
2010 (Form B) AP #5
Quiz #12
Finding the area between two curves.
62
Find volumes of solids of revolution using the disk method.
Day #59 Homework
2001 AB #6 Parts a and b
2002 (Form B) AB #5 Parts a
and b
Daily Lessons pages 611 – 618
Day #60 Homework
2004 AB #6 Parts a, b, and c
2004 (Form B) AB #5 Parts a,
b, and c
Daily Lessons pages 635 – 638
Finish note packet begun in
class.
Daily Lessons pages 639 – 641
Day #62 Homework
63
More work with volumes of solids of revolution and find
volumes of solids of known cross sections.
64
Quiz #13
2006 AP #3
2011 AB #3
65
Daily Lessons pages 645 – 647
Day #63 Homework
Daily Lessons pages 652 – 655
2004 AP #5
2011 AB #4
Finish any of the 4 Free
Responses not completed in
class
Study for Unit #7 Test
Unit #7 Test – Advanced Integration and Applications
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 590
The Second Fundamental Theorem of Calculus
Functions Defined by Integrals
Given the functions, f(t), below, use F ( x) 
x
1
f (t )dt to find F(x) and F’(x) in terms of x.
1. f(t) = 4t – t2
2. f(t) = cos t
Given the functions, f(t), below, use F ( x)  
x2
1
3. f(t) = t3
f (t )dt to find F(x) and F’(x) in terms of x.
4. f(t) =
6 t
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 591
Second Fundamental
Theorem of Calculus
Complete the table below for each function.
Function
F ' ( x) from page 584
Find F ' ( x) by applying the Second
Fundamental Theorem of Calculus
x


F ( x)   4t  t 2 dt
1
x


F ( x)   4t  t 2 dt
1
F ( x)   cos t dt
x
1
F ( x)   cos t dt
x
1
x2 3
t dt
1
F ( x)  
x2 3
t dt
1
F ( x)  
F ( x) 
F ( x) 
x2
1
x2
1
6 t dt
6 t dt
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 592
Find the derivative of each of the following functions.
F ( x) 
2x
 2
2  t 2 dt
G ( x) 
3
x
2
e cos t dt
H ( x) 
cos x
0
t 2 dt
Pictured to the right is the graph of g(t) and the function
f(x) is defined to be f ( x)  
2x
4
g (t )dt .
1. Find the value of f(0).
2. Find the value of f(2).
3. Find the value of f ' (1) .
4. Find the value of f ' (2) .
5. Find the value of f ' ' (2) .
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 593
Given to the right is the graph of f(t) which consists of three line segments and one semicircle.
Additionally, let the function g(x) be defined to be g ( x)  
x
1
f (t )dt .
1. Find g(–6).
2. Find g(6).
3. Find g ' (6) .
4. Find g ' (2) .
5. Find g ' ' (2) . Give a reason for your
answer.
6. Find g ' ' (4) . Give a reason for your
Answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 594
The continuous function f is defined on the interval –4 < x < 3. The graph consists of two quarter circles
x
and one line segment, as show in the figure above. Let g ( x)  1 x 2   f (t )dt .
2
0
Find the value of g(3).
Find the value of g(−4).
Find the value of g ' (3) .
Find the value of g ' ' (2) .
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 595
Name_________________________________________Date____________________Class__________
Day #57 Homework
Find the derivative of each of the following functions defined by integrals.
1. g ( x)  
3x
2
2. h( x)  

4. H ( x)  

1 2
t  2t dt
2x
3. f ( x)  
5. P( x)  
x 2  2x
2
x4
(2t  3)dt
3t  2dt
2
3 t dt
cos x
5
2t 2 dt

e t  t dt
ln x
6. f ( x)  
2
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 596
Pictured to the right is the graph of f(t)and F ( x) 
2x
 6f (t )dt . Use the
graph and F(x) to answer the questions 7 – 11.
7. Find the value of F(0).
8. Find the value of F  1 .
2
 
9. Find the value of F ' (2) .
10. Find the value of F ' (2.5) .
Pictured to the right is the graph of f and G ( x) 
11. Find the value of F ' ' (0)
x
 2f (t)dt . Use the graph to answer questions 12 – 15.
12. Find the value of G(3).
13. Find the value of G(–4).
14. Find the value of G ' (2) .
15. Find the value of G ' ' (5) .
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 597
x
If g ( x)   t 3e t dt , find each of the following values in questions 16 – 17.
0
16. Find the value of g ' (1).
If h( x)  
2
x2
17. Find the value of g ' ' (1).
1  t 4 dt , find each of the following values in questions 18 – 19.
18. Find h' ( x).
19. Find h' ' (1).
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 598
Integration of Composite Functions
For each of the functions given below, find both f ' ( x) and
 f ' ( x)dx .
f ' ( x)
f(x)
 f ' ( x)dx
f ( x)  sin 3x
f ( x)  e cos x


f ( x)  ln x 2  3
f ( x) 
x2  3
Anti-differentiation by Pattern Recognition
d
 f ( g ( x))  ____________________________________
dx
 f ' ( g ( x))  g ' ( x) dx  _____________________________
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 599
Find each of the following indefinite integrals by pattern recognition.
 3 cos 3x dx
 2 sin(2x  3) dx
 cos(3x  2)dx
 5e
3x
dx
 2x
3
x 2  5 dx
 x2x22x dx
2
 3x

x 2  2 dx
3x
2x 2 3
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
dx
Page 600
Ant-differentiation by U–Substitution
In each of the eight examples above, the g ' ( x) , or “license to integrate,” existed in the integrand of
 f ' ( g ( x))  g ' ( x) dx or g ' ( x) was attainable by multiplying by a constant. The g ' ( x) does not always
exist and there are times when it is not attainable by multiplication of a constant. Consider the example
below.
 x(2x  1)
3
dx
Identify the “inner function,” g(x): _________________________
What is g ' ( x) ? ___________________
Is g ' ( x) part of the integrand? _____________________
Is g ' ( x) attainable by multiplying the integrand by a constant? ________________________
In this case, we must find the anti-derivative by a method known as U-Substitution. Here is how it works.
1. Let u = the inner function, g(x).
4. Rewrite the entire integrand as a polynomial or
polynomial type of function in terms of u. Then,
anti-differentiate.
2. Find du and solve the equation for dx.
3. Find an expression for x in terms of u.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 601
 2xx14 dx
1. Let u = the inner function, g(x).
4. Rewrite the entire integrand as a polynomial or
polynomial type of function in terms of u. Then,
anti-differentiate.
2. Find du and solve the equation for dx.
3. Find an expression for x in terms of u.
4
Find the value of
 x 2x  1 dx
. Then, check the result using the graphing calculator.
0
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 602
Name_________________________________________Date____________________Class__________
Day #58 Homework
In problems 1 – 6, find the indefinite integral.
1.
 

2.
 x 
3.
x 3 sin x 4 dx
 
4.

 1 x  dx
5.
 5x 1  x 2 dx
6.
u
3
x 3 x 4  3 dx
3
1  2 x 2 dx
x3
4 2
2
u 3  2du
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 603
For problems 7 and 8, find the indefinite integral by using substitution.
7.
x
2 x  1 dx
9. Find the value of
8.

5
1
x
2x  1
 x  1
2  x dx
dx . Show your work and then check using a graphing calculator.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 604
Solving Differential Equations
Examples of Variable Separable Differential Equations
Given below are differential equations with given initial condition values. Find the particular solution for
each differential equation.
1. dy  6 x 2  6 x  2 and f(–1) = 2
3.
dx
2. dy  1  12 x
dx
2 x
dy x 2  2 x
and f(0) = 2

dx
2y
4.
3
2
and f(0) = 2
dy x  2
and f(1) = –3

dx
y
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 605
5.
dy
 x 4 ( y  2) and f(0) = 0
dx
6.
dy y  1
and f(2) = 0

dx
x2
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 606
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 607
The acceleration of a particle moving along the x – axis at time t is given by a(t) = 6t – 2. If the
velocity is 25 when t = 3 and the position is 10 when t = 1, then the position x(t) =
A. 9t2 + 1
B. 3t2 – 2t + 4
C. t3 – t2 + 4t + 6
D. t3 – t2 + 9t – 20
E. 36t3 – 4t2 – 77t + 55
A particle moves along the x-axis so that, at any time t > 0, its acceleration is given by a(t) = 6t + 6. At
time t = 0, the velocity of the particle is –9 and its position is –27.
a. Find v(t), the velocity of the particle at any time t.
b. Find the net distance traveled by the particle over the interval [0, 2].
c. Find the total distance traveled by the particle over the interval [0, 2].
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 608
Name_________________________________________Date____________________Class__________
Day #59 Homework
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 609
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 610
Slope Fields
Graphical Representations of Solutions to Differential Equations
A slope field is a pictorial representation of all of the possible solutions to a given differential
equation.
Remember that a differential equation is the first derivative of a function, f ' ( x) or
dy
. Thus, the
dx
solution to a differential equation is the function, f(x) or y.
There is an infinite number of solutions to the differential equation
dy
 x  3 . Show your work and
dx
explain why.
For the AP Exam, you are expected to be able to do the following four things with slope fields:
1.________________________________________________________________________________
________________________________________________________________________________
2.________________________________________________________________________________
________________________________________________________________________________
3.________________________________________________________________________________
________________________________________________________________________________
4.________________________________________________________________________________
________________________________________________________________________________
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 611
#1
Sketch a slope field for a given differential equation.
Given the differential equation below, compute
the slope for each point
Indicated on the grid to the right.
Then, make a small mark that
approximates the slope
through the point.
dy
 x 1
dx
Given the differential equation below, compute
the slope for each point
Indicated on the grid to the right.
Then, make a small mark that
approximates the slope
through the point.
dy
 x y
dx
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 612
#2
Given a slope field, sketch a solution curve through a given point.
To the right is pictured the slope field that you
developed for the differential equation
on the previous page.
dy
 x 1
dx
Sketch the solution curve through the point
(1, -1).
To do this, you find the point and then follow
the slopes as you connect the lines.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 613
#3
Match a slope field to a differential equation.
Since the slope field represents all of the particular solutions to a differential
equation, and the solution represents the ANTIDERIVATIVE of a differential
equation, then the slope field should take the shape of the antiderivative of dy/dx.
Match the slope fields to the differential equations on the next page.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 614
Separate the variables and find the general solution to each differential equation below to determine what
the slope field should look like for each. Then, match to the graphs of slope fields on the previous page.
1.
dy
 sin x
dx
2.
dy
 2x  4
dx
3.
dy
 ex
dx
4.
dy
2
dx
5.
dy
 x 3  3x
dx
6.
dy
 2 cos x
dx
7.
dy
 4  2x
dx
8.
dy
x
dx
9.
dy
 x2
dx
10.
dy
1

dx
x
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 615
#4
Match a slope field to a solution to a differential equation.
When given a slope field and a solution to a differential equation, then the slope
field should look like the solution, or y.
Match the slope fields below to the solutions on the next page.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 616
1. y  x
2. y  x 2
3. y  e x
4. y 
5. y  x 3
6. y  sin x
7. y  cos x
8. y  x
9. y  1
10. y  tan x
1
x2
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 617
Shown below is a slope field for which of the following differential equations? Explain your reasoning
for each of the choices below.
Consider the differential equation
dy x
 to answer the following questions.
dx y
a. On the axes below, sketch a slope field for the equation.
b. Sketch a solution curve that passes through the point (0, –1) on your slope field.
c. Find the particular solution y = f(x) to the differential equation with the initial condition f(0) = –1.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 618
Name_________________________________________Date____________________Class__________
Day #60 Homework
For the indicated points on each grid, draw the slope field for the given differential equation.
1.
dy
 x y
dx
2.
dy
y

dx
x
3.
dy
 x 1
dx
4.
dy
1

dx x  1
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 619
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 620
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 621
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 622
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 623
AP Calculus
Quiz #12
Answer Key & Rubric
Multiple Choice:
*
1. D
C
2. A
E
3. C
A
4. E
A
5. A
B
6. A
E
7. B
A
Free Response Part A – 2 points total
4
____ 1 g (4)   f (t )dt  5
2
1
____ 1 g (2)  
2
1
f (t )dt  
2
1
f (t )dt  6
Free Response Part B – 2 points total
x
____ 1 If g ( x)   f (t )dt , then g ' ( x)  f ( x)  1 . So, g ' (1)  f (1)  4
1
____ 1 If g ' ( x)  f ( x) , then g ' ' ( x)  f ' ( x) . g ' ' (1)  f ' (1) is undefined because the graph of f
has a cusp when x = 1.
Free Response Part C – 3 points total
____ 1 Identifies the only relative maximum of g on (–2, 4) to be x = 3.
3
____ 1 Finds the value of g (3)   f (t )dt  1 (1)(1  4)  1 (1)(1)  3 .
1
2
2
____ 1 On a closed interval, the absolute maximum occurs at an endpoint of the interval or at any
relative maximum on the interval. g(–2) = –6, g(3) = 3, and g(4) = 5 . Thus, since g(3) is the
2
greatest, the point (3, 3) is the absolute maximum of g on the interval [–2, 4].
Free Response Part D – 2 points total
____ 1 The graph of g has a point of inflection when g ' '  f ' changes signs. f ' changes signs when
the graph of f has a relative maximum or minimum.
____ 1 The graph of f changes from increasing to decreasing at x = 1. Thus, g has a point of
inflection at x = 1.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 624
AP CALCULUS
QUIZ #12
Name____________________________________________________Date_____________________
Calculator Permitted
1.
2.
3.
4.
5.
6.
7.
MULTIPLE CHOICE − Graphing Calculator NOT Permitted
The graph of a function, f, which consists of a two line segments and a semi-circle is pictured below. Let
G ( x)  x 2 
2x
 2f (t)dt . Use this information to answer questions 1 and 2.
1. What is the value of G(2)?
A. 4 + 2π
B. –1 + 2π
C. 5 + 2π
D. 3 + 2π
E. 4
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 625
2. Find the value of G ' (2) .
A. 4
B. 12
C. 7
D. 27
E. 8
3. If g(x) =

x2
3t
3
t 1
1
dt , then what is the value of g ' (2) ?
A. –3
B. 8
3
C. 48
65
2
D.
3
E. 12
65
4. If g ( x) 
A.
B.
C.
D.
E.
x
0
t 3e t dt , find g ' ' (1) .
e
2e
e–1
3e
4e
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 626
5.

2x 2
3
x 2
dx 
 
3
B. 1 x 3  2
3
C. 2 x 3  2
3
D. 2x 3  2
E. 3x 3  2
1
A. 4 x 3  2
2
C
2
C
2
C
2
C
1
1
1
1
C
2
 x 3 2x  3dx 
6. If u = 2x – 3, then

C. 1 3 u du
2
E. 1 (2u  3)3 u du
2
4
1
A. 1 u 3  3u 3 du
4
7. If
dy
dx

D. 1 3 u du
4
2
4
B. 1 u 3  3u 3 du
2
2
 x and f(0) = –4, find the particular solution to the differential equation.
y
A. f(x) = 1 x 3  4
3
B. f(x) =  2 x 3  16
3
2 x 3  16
3
C. f(x) =
D. f(x) = 1 x 3
3
E. f(x) =  2 x 3  8
3
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 627
FREE RESPONSE
The graph of a function f, consisting of three line segments, is pictured above. Let g ( x) 
x
1 f (t)dt .
a. Compute the values of g(4) and g(–2).
b. Find g ' (1) and g ' ' (1) . Show or explain your work.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 628
c. Find the coordinates of the absolute maximum of g on the closed interval [–2, 4]. Justify your
answer.
d. The second derivative of g is not defined at x = 1 and x = 2. Which of these two values is/are
coordinates of points of inflection of the graph of g? Justify your answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 629
AP CALCULUS
*QUIZ #12*
Name____________________________________________________Date_______________________
Calculator Permitted
1.
2.
3.
4.
5.
6.
7.
MULTIPLE CHOICE − Graphing Calculator NOT Permitted
The graph of a function, f, which consists of a two line segments and a semi-circle is pictured below. Let
G ( x)  x 2 
2x
 2f (t)dt . Use this information to answer questions 1 and 2.
1. What is the value of G(1)?
A. 2 + π
B. –1 + 2π
C. 1 + 2π
D. 3 + 2π
E. 1 + π
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 630
2. Find the value of G ' (1) .
A. 4
B. 8
C. 7
D. 27
E. 2
3. If g(x) =
x2
1
3t
3
t 1
dt , then what is the value of g ' (1) ?
A. –3
B. 8
3
C. 24
65
3
D.
2
E. 12
65
4. If g ( x) 
A.
B.
C.
D.
E.
x 3 t
t e dt , find g ' ' (1) .
0

4e
2e
e–1
3e
e
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 631
5.

2x 2
x3  2
dx 
 
3
B. 4 x 3  2
3
C. 2 x 3  2
3
D. 2x 3  2
E. 3x 3  2
A. 1 x 3  2
1
1
2
C
2
C
2
C
1
1
1
C
2
C
2
x
6. If u = 2x – 3, then

 u du
 u  3u
3
2 x  3dx 
2
4
B. 1 u 3  3u 3 du
C. 1 3
D. 1 3
2
2
E. 1
4
7. If

 u du
A. 1 (2u  3)3 u du
dy
dx
4
3
2
4
1
3
du
2
 x and f(0) = –4, find the particular solution to the differential equation.
y
A. f(x) =  2 x 3  16
3
B. f(x) = 1 x 3  4
3
2 x 3  16
3
C. f(x) =
D. f(x) = 1 x 3
3
E. f(x) =  2 x 3  8
3
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 632
FREE RESPONSE
The graph of a function f, consisting of three line segments, is pictured above. Let g ( x) 
x
1 f (t)dt .
a. Compute the values of g(4) and g(–2).
b. Find g ' (1) and g ' ' (1) . Show or explain your work.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 633
c. Find the coordinates of the absolute maximum of g on the closed interval [–2, 4]. Justify your
answer.
d. The second derivative of g is not defined at x = 1 and x = 2. Which of these two values is/are
coordinates of points of inflection of the graph of g? Justify your answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 634
Finding the Area between Two Curves
An Application of Integration
Graph the function f ( x)   12 x 2  2 x  4 and find the value of
0
5 f ( x)dx . Using one color, shade the
region for which this value represents the area.
Graph the function g ( x)  1 x  4 on the same grid above and then find the value of
2
0
5g ( x)dx . Using a
different color, shade the region for which this value represents the area.
What do you suppose you would do to find the area of the region that is located in between the graphs of
f(x) and g(x)? Find this value.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 635
Now, find the value of the definite integral below if f ( x)   12 x 2  2 x  4 and g ( x)  1 x  4 . Show
2
your work.

0
5
f ( x)  g ( x)dx
What do you notice about this value?
This brings about the general way that we will find the area between two curves.
Find the area of the shaded region, R, that is bounded by y = sin(x) and y = x3 – 4x.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 636
3
2
Pictured to the right is the graph of f ( x)  x  x  x  3 cos x
4
3
2
and a line, l, which is tangent to f(x) at the point (0, 3).
Find the area of Region R.
Find the equation of line l if it is tangent to the graph of f(x) at (0, 3).
At what ordered pair, other than (0, 3), does the graph of line l intersect the graph of f(x)?
Find the area of Region S.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 637
Pictured to the right are regions R and S, which are formed by the
graphs of f ( x)  1  sin(x) and g ( x)  4  x
4
Identify the points of intersection of f(x) and g(x).
Find the area of Region R.
Find the area of Region S.
Find the area of the unshaded region bounded by the graphs of f, g, and the x – axis.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 638
Volumes of Solids of Revolution
A solid of revolution is formed when a flat, two-dimensional shape is rotated around an axis. Consider
the flat region below to the left. When that region is rotated about the x – axis, the solid pictured below to
the right is formed. This objective of this lesson is to learn to find the volume of such a solid.
Now, imagine slicing the solid into individual discs of height 1 unit. The volume of one of those discs is
V = πr2h, or V = πr2.
b
Sum of all the discs =
a   f ( x)  Axis of Rotation  dx
Volume of the Solid = 
2
b
a
 f ( x)  Axis of
Rotation 2 dx
Notice that the axis of rotation is the x – axis and the bottom function of the region is also the x – axis.
Imagine for a moment what the solid would look like if the axis of rotation were still the x – axis but the
bottom function of the region was the line y = c. The solid would look similar except for the fact that
there would be a cylinder that is cut out of the center.
To find the volume of this solid, we would find the volume of the
whole solid that we found previously and then subtract out the solid
in the form of a cylinder.
In order to do this, we use the formula below to find the volume of
such a solid.
Volume    OuterFunct ion  axis 2  InnerFunct ion  axis 2 dx
b
a
The “outer function” is defined to be the function that is farther from
the axis of rotation. The “inner function” is defined to be the function that is closer to the axis of rotation.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 639
If the axis of rotation is the x-axis or is parallel to the x-axis, the integrand needs to be in terms of x and
the limits of integration need to be the x-values of the points of intersection of the curves that form the
region being rotated.
Consider the region pictured to the right that is bounded by the graphs of y = x2 and y = x + 2.
Find the volume of the solid formed when the
region is rotated about the x – axis.
Find the volume when the region is rotated about
the line y = 4.
Find the volume when the region is rotated about
the line y = –2.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 640
If the axis of rotation is the y-axis or is parallel to the y-axis, the integrand needs to be in terms of y and
the limits of integration need to be the y-values of the points of intersection of the curves that form the
region being rotated.
Consider the region pictured to the right that is bounded by the graphs y =  x and y = x – 2.
Find the volume when the region is rotated about
the y – axis.
Find the volume when the region is rotated about
the line x = 4.
Find the volume when the region is rotated about
the line x = 7.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 641
Name_________________________________________Date____________________Class__________
Day #62 Homework
Let R be the region bounded by the graphs of y = ln x and the line y = x – 2 as shown below. Though you
may use a calculator, show the integral that you found to arrive at your answer.
1. Find the coordinates of the points at which the two graphs
intersect each other. Then, find the area of R.
2. Find the volume of the solid generated when R is rotated about the horizontal line y = –3.
3. Write and evaluate an integral expression that can be used to find the volume of the solid
generated when R is rotated about the y-axis.
Let f and g be the functions given by f ( x)  1  sin(x) and g ( x)  4  x . Let R be the region in the first
4
quadrant enclosed by the y-axis and the graphs of f and g, and let S be the region in the first quadrant
enclosed by the graphs of f and g shown to the right. Though you may use a calculator, show the integral
that you found to arrive at your answer.
4. Find the points of intersection of f and g.
5. Find the area of the region bounded by the graphs of f and g and
the x – axis.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 642
4. Find the volume of the solid generated when R is revolved about the horizontal line y = 8.
5. Find the volume of the solid generated when S is revolved about the horizontal line y = –1.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 643
CALCULATOR NOT PERMITTED
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 644
Volumes of Solids with Known Cross Sections
Calculus cannot only be used to find the volume of solids created by revolving two-dimensional shapes
around an axis but also the volume of solids formed by cross sections that are geometric shapes. In this
lesson, you will derive the formulas for finding volumes of solids given that their cross sections are
squares, isosceles right triangles, equilateral triangles, and semi circles.
Cross Sections that are Squares
Find the area of the square above in terms of f(x)
and g(x).
If all of the cross sectional areas were added up,
the total would be the volume of the solid
pictured. How do we write this in calculus using
an integral?
Cross Sections that are Isosceles Right
Triangles
Find the area of the triangle above in terms of f(x)
and g(x).
If all of the cross sectional areas were added up,
the total would be the volume of the solid
pictured. How do we write this in calculus using
an integral?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 645
Cross Sections that are Equilateral Triangles
Find the area of the triangle above in terms of f(x)
and g(x).
If all of the cross sectional areas were added up,
the total would be the volume of the solid
pictured. How do we write this in calculus using
an integral?
Cross Sections that are Semicircles
Find the area of the semicircle above in terms of
f(x) and g(x).
If all of the cross sectional areas were added up,
the total would be the volume of the solid
pictured. How do we write this in calculus using
an integral?
What do you notice about the integral-defined formulas for finding the volume of solids with certain cross
sections?
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 646
Region R is bounded by y = sin(x) and y = x3 – 4x.
Find the volume of the solids formed whose cross sections are the shapes indicated below. The cross
sections are perpendicular to the x – axis.
a. Cross sections are equilateral triangles
b. Cross sections are semi-circles
c. Cross sections are isosceles right triangles
d. Cross sections are squares.
e. Cross sections are rectangles whose height is
twice the length of the base.
f. Cross sections are rectangles whose height
is one-third the length of the base.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 647
Name_________________________________________Date____________________Class__________
Day #63 Homework
Let R be the region bounded by the graphs of y = ln x and the line y = x – 2 as shown below. Though you
may use a calculator, show the integral that you found to arrive at your answer.
1. Find the volume of the solid whose base is region R that is formed by cross
sections that are semi-circles that are perpendicular to the x – axis.
2. Find the volume of the solid whose base is region R that is formed by cross sections that are squares
that are perpendicular to the x – axis.
Let f and g be the functions given by f ( x)  1  sin(x) and g ( x)  4  x . Let R be the region in the first
4
quadrant enclosed by the y-axis and the graphs of f and g, and let S be the region in the first quadrant
enclosed by the graphs of f and g shown to the right. Though you may use a calculator, show the integral
that you found to arrive at your answer.
3. Find the volume of the solid whose base is the cross section area of
region S and is formed by squares that are perpendicular to the x-axis.
4. Find the volume of the solid whose base is the cross section area of region S and is formed by
equilateral triangles that are perpendicular to the x – axis.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 648
Let f and g be the functions given by f ( x)  2 x(1  x) and g ( x)  3( x  1) x for 0 < x < 1. The graphs of
f and g are shown in the figure to the right. Though you may use a calculator, show the integral that you
found to arrive at your answer.
5. Find the volume of the solid whose base is the cross section of the
region bounded by the graphs of f and g and is formed by squares that
are perpendicular to the x-axis.
6. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f
and g and is formed by semi-circles that are perpendicular to the x-axis.
7. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f
and g and is formed by equilateral triangles that are perpendicular to the x-axis.
8. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f
and g and is formed by right isosceles triangles that are perpendicular to the x-axis.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 649
CALCULATOR PERMITTED
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 650
CALCULATOR PERMITTED
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 651
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 652
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 653
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 654
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 655
AP Calculus
Quiz #13
Answer Key & Rubric
Multiple Choice:
*
1. C
E
2. A
C
3. D
B
4. D
A
5. D
B
6. C
E
7. E
B
Free Response Part A – 4 points total
____ 1 Correct integrand and limits for [0, 1]
____ 1 Correct integrand and limits for [1,
3]
____ 1 Correct answer
Free Response Part B – 3 points total
____ 1 Correct integrand
____ 1 Correct limits and constant
____ 1 Correct answer
Free Response Part B – 3 points total
____ 1 Correct constant and limits
____ 1 Correct integrand


2
1 1
Volume   3  x 2  2 x dx
8 a
 2.436
____ 1 Correct answer
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 656
AP CALCULUS
QUIZ #13
Name____________________________________________________Date________________________
Calculator Permitted
1.
2.
3.
4.
5.
6.
7.
MULTIPLE CHOICE − Graphing Calculator Permitted
1. Find the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x and
the y – axis.
A. 0.127
B. 0.385
C. 0.400
D. 0.600
E. 0.947
2. Find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x – x2
and y = 0 about the x – axis.
A. 107.233
B. 34.133
C. 33.510
D. 10.667
E. 129.322
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 657
3. A solid is generated when the region in the first quadrant enclosed by the graph of y = (x2 + 1)3, the
line x = 1, the x – axis, and the y – axis is revolved about the x – axis. Its volume is found by
evaluating which of the following integrals?
3
2


x

1
dx
1
8
6
B.   x 2  1 dx
1
1 2
3
C.   x  1 dx
0
1
6
D.   x 2  1 dx
0
1
6
E. 2  x 2  1 dx
0
A. 
8
4. The region bounded by the graph of y = 2x – x2 and the x – axis is the base of a solid. For this solid,
each cross section perpendicular to the x – axis is an equilateral triangle. What is the volume of
this solid?
A. 1.333
B. 1.067
C. 0.577
D. 0.462
E. 0.267
2
5. The base of a loud speaker is determined by the two curves y = x and
10
y
x2
10
for 1 < x < 4 as shown in the figures to the right. For this loud
speaker, the cross sections perpendicular to the x – axis are squares. What is
the volume of this speaker, in cubic units?
A. 2.046
B. 4.092
C. 4.200
D. 8.184
E. 25.711
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 658
6. The slope field pictured below represents all general solutions to which of the following differential
equations?
A.
dy
 2x
dx
B.
dy
 2 x
dx
C.
dy
 y
dx
D.
dy
y
dx
E.
dy
 x y
dx
7. The graph of a function f, which consists of two line segments
and a quarter circle, is pictured to the right. If H ( x) 
x
 2f (t)dt ,
which of the following statements is true?
A. H(4) < H ' (2) < H ' ' (3)
B. H(4) < H ' ' (3) < H ' (2)
C. H ' (2) < H(4) < H ' ' (3)
D. H ' ' (3) < H(4) < H ' (2)
E. H ' ' (3) < H ' (2) < H(4)
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 659
FREE RESPONSE
Let R and S in the figure to the right be defined as follows:
R is the region in the first and second quadrants bounded
by the graphs of y = 3 – x2 and y = 2x.
S is the shaded region in the first quadrant bounded by
the two graphs, the x – axis, and the y – axis.
a. Find the area of region S.
b. Find the volume of the solid generated when R is rotated about the horizontal line y = –1.
c. The region R is the base of a solid. For this solid, each cross section perpendicular to the x – axis is
a semi-circle whose diameter lies on the base of the solid. Find the volume of this solid.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 660
AP CALCULUS
*QUIZ #13*
Name____________________________________________________Date_____________________
Calculator Permitted
1.
2.
3.
4.
5.
6.
7.
MULTIPLE CHOICE − Graphing Calculator Permitted
1. Find the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x and
the y – axis.
A. 0.127
B. 0.385
C. 0.947
D. 0.600
E. 0.400
2. Find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x – x2
and y = 0 about the x – axis.
A. 33.510
B. 34.133
C. 107.233
D. 10.667
E. 129.322
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 661
3. A solid is generated when the region in the first quadrant enclosed by the graph of y = (x2 + 1)3, the
line x = 1, the x – axis, and the y – axis is revolved about the x – axis. Its volume is found by
evaluating which of the following integrals?
3
2


x

1
dx
1
1
6
B.   x 2  1 dx
0
1 2
3
C.   x  1 dx
0
8
6
D.   x 2  1 dx
1
1
6
E. 2  x 2  1 dx
0
A. 
8
4. The region bounded by the graph of y = 2x – x2 and the x – axis is the base of a solid. For this solid,
each cross section perpendicular to the x – axis is an equilateral triangle. What is the volume of
this solid?
A. 0.462
B. 1.067
C. 0.577
D. 1.333
E. 0.267
2
5. The base of a loud speaker is determined by the two curves y = x and
10
y
x2
10
for 1 < x < 4 as shown in the figures to the right. For this loud
speaker, the cross sections perpendicular to the x – axis are squares. What is
the volume of this speaker, in cubic units?
A. 2.046
B. 8.184
C. 4.200
D. 4.092
E. 25.711
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 662
6. The slope field pictured below represents all general solutions to which of the following differential
equations?
A.
dy
 2x
dx
B.
dy
 2 x
dx
C.
dy
 x y
dx
D.
dy
y
dx
E.
dy
 y
dx
7. The graph of a function f, which consists of two line segments
and a quarter circle, is pictured to the right. If H ( x) 
x
 2f (t)dt ,
which of the following statements is true?
A. H(4) < H ' (1) < H ' ' (5)
B. H ' (1) < H ' ' (5) < H(4)
C. H ' ' (5) < H ' (1) < H(4)
D. H ' (1) < H(4) < H ' ' (5)
E. H(4) < H ' ' (5) < H ' (1)
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 663
FREE RESPONSE
Let R and S in the figure to the right be defined as follows:
R is the region in the first and second quadrants bounded
by the graphs of y = 3 – x2 and y = 2x.
S is the shaded region in the first quadrant bounded by
the two graphs, the x – axis, and the y – axis.
a. Find the area of region S.
b. Find the volume of the solid generated when R is rotated about the horizontal line y = –1.
c. The region R is the base of a solid. For this solid, each cross section perpendicular to the x – axis is
a semi-circle whose diameter lies on the base of the solid. Find the volume of this solid.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 664
AP Calculus
Test #8
Answer Key & Rubrics
Raw Score to Percentage Conversion
Multiple Choice
Calculator
1.
2.
3.
4.
5.
6.
7.
A
D
C
D
B
E
C
Calculator NOT Permitted Free Response Part A – 3 points total
____ 1 g (3) 
3
0 f (t )dt  14  (2)
2
 1 (1)(1)    1
2
2
____ 1 g ' (3) = f(3) = –1
____ 1 g ' ' (3) = undefined because g ' ( x)  f ( x) is not differentiable at x = 3.
Calculator NOT Permitted Free Response Part B – 3 points total
____ 1 g(x) has a point of inflection when g ' ' ( x) changes signs.
____ 1 g ' ' ( x) changes signs when the graph of g ' ( x)  f ( x) has a relative maximum or minimum
____ 1 Thus, g has a point of inflection when x = 0 and x = 3.
Calculator NOT Permitted Free Response Part C – 3 points total
____ 1 Correctly finds the values of g(−2) and g(5), as x = −2 and x = 5 are the endpoints of the interval.


2
g (2) 
f (t )dt  
f (t )dt   1  (2) 2  
4
2
0
5
g (5) 

0
f (t )dt  1  (2) 2  1 (1)(1)    1
0
4
2
2
____ 1 Correctly finds the value of g(4), as x = 4 is the only relative minimum of g on the interval
4
g (4) 
 f (t)dt 
0
1  (2) 2
4
 1 (2)(1)    1
2
____ 1 According to the Extreme Value Theorem, the absolute minimum value of g on −2 < x < 5 is –π.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 665
Multiple Choice
Non Calculator
8.
9.
10.
11.
12.
13.
14.
C
D
B
E
C
A
C
Free Response – Non Calculator
Part A – 2 points total
_____ 1 Correctly drawn zero slopes
_____ 1 Correctly drawn non-zero slopes
Part B – 6 points total
_____ 1 Correct separation of variables

_____ 1 Correct anti-differentiation of y
_____ 1 Correct anti-differentiation of x
_____ 1 Includes constant of integration
ln y  1   1  c
_____ 1 Uses the initial condition
ln 0  1   1  c  ln 1   1  c  0   1  c  c  1
1 dy
y 1

1
x2
dx
x
2
y 1  e
_____ 1 Correctly solves for y

2
2
2
 1x  12
y 1  e
 1x  12
y  1 e
 1x  12
or
y  1  e
 1x  12
Part C – 1 point total
_____ 1 Since
dy
dx

y 1
x2
represents the slope of the tangent line, then it is only positive for points such
that y – 1 > 0, or such that y > 1, provided that x ≠ 0.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 666
AP CALCULUS AB
TEST #8
Unit #7 – Advanced Integration and Applications
Name______________________________________________________Date_____________________
A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS
IN THIS PART OF THE EXAMINATION.
(1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices the number that best approximates the
exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers
x for which f(x) is a real number.
MULTIPLE CHOICE
1. Let R be the region in the first quadrant bounded by the graphs of y  2  sin x , y  e x  3 , and the
y – axis as shown in the figure above. Find the volume of the solid generated when R is rotated around
the line y = 4.
A. 115.380
B. 36.727
D. 23.052
E. 7.338
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
C. 67.036
Page 667
2. The graph of the piecewise linear function f is shown in the figure above. If g ( x) 
x
 2f (t)dt , which
of the following values is the greatest?
A. g(–3)
B. g(–2)
C. g(0)
D. g(1)
E. g(2)
3. The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis and the graph of
the line x + 2y = 8, as shown in the figure below. If the cross sections of the solid perpendicular to the
x-axis are semicircles, what is the volume of the solid?
A. 12.566
B. 14.661
C. 16.755
D. 67.021
E. 134.041
4. The region bounded by the graph of y = 2x – x2 and the x – axis is the base of a solid. For this solid,
each cross section perpendicular to the x – axis is an equilateral triangle. What is the volume of this
solid?
A. 1.333
B. 1.067
C. 0.577
D. 0.462
E. 0.267
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 668
5. The graph of a function f is shown in the figure below and has a horizontal tangent at x = 4 and
x = 8. If g ( x)  x 2 
2x
 f (t) dt , what is the value of
g ' (3) ?
0
A. –2.5
B. 10
C. –4
D. 13
E. 2
6. Which of the following is the solution to the differential equation
condition y( )  1.
A.
dy
 2 sin x with the initial
dx
y  2 cos x  3
B. y  2 cos x  1
C. y  2 cos x  3
D. y  2 cos x  1
E. y  2 cos x  1
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 669
7. The graph of the function f shown above has horizontal tangents at x = 2 and x = 5. Let g be the
function defined by g ( x) 
x
0 f (t)dt . For what values of x does the graph of g have a point of
inflection?
A. 2 only
B. 4 only
C. 2 and 5 only
D. 2, 4, and 5
E. 0, 4, and 6
FREE RESPONSE
The graph of a function f, pictured above, consists of a semicircle and two line segments as shown to the
x
right. Let g be the function given by g ( x)   f (t )dt .
0
a. Find the values of g(3), g ' (3) , and g ' ' (3) , if they exist. Show the computations that lead to your
answers or give a reason for your answers.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 670
b. Find the x-coordinate of each point of inflection of the graph of g on the open interval –2 < x < 5.
Justify your answer.
c. On the closed interval –2 < x < 5, what is the absolute minimum value of g. Show your work and
justify your answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 671
AP CALCULUS AB
TEST #8
Test #8: Unit #7 – Advanced Integration and Applications
Name______________________________________________________Date_____________________
A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM.
(1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices the number that best approximates the
exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers
x for which f(x) is a real number.
MULTIPLE CHOICE
8. Using the substitution u  x ,
16
4e
1
x
x
dx is equal to which of the following?
4
A. 2 e u du
B. 2 e u du
2
2
D. 1  e u du
1
C. 2 e u du
1
E.
4 u
1 e
1
2 1
du
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 672
1
9.

e  4 x dx =
0
A.  e
4
4
B.  4e 4
C. e 4  1
4
D. 1  e
4
4
E. 4  4e 4
10. Region R is the region in the first quadrant bounded by the graphs of f ( x)  x , g(x) = 6 – x and
the x – axis. Find the area of R.
A. 4
B. 22
3
C. 14
3
D. 13
3
E. 6
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 673
11. If f is the function given by f ( x) 

2x
t 2  t dt , then f ' (2) 
4
A. 0
B.
7
2 12
C.
2
D.
12
E. 2 12
12.
Shown above is a slope field for which of the following differential equations?
A.
dy x

dx y
dy x 3
B.

dx
y
D.
dy x 2

dx y 2
E.
dy x 3
C.

dx y 2
dy x 2

dx
y
x
13. The graph of a differentiable function f is shown at right. If h( x)   f (t )dt , which of the following
0
is true?
A. h(6)  h' (6)  h' ' (6)
B. h(6)  h' ' (6)  h ' (6)
C. h ' (6)  h(6)  h' ' (6)
D. h' ' (6)  h(6)  h ' (6)
E. h' ' (6)  h ' (6)  h(6)
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 674
14.
x
x
2
A.
B.
C.
4
dx 
1
2
4x 2  4
1

2 x2  4
C
 C
1
ln x 2  4  C
2
D. 2 ln x 2  4  C
E.
2
x2  4
C
FREE RESPONSE
Consider the differential equation
dy y  1

, where x ≠ 0.
dx
x2
a. On the axes provided, sketch a slope field for the given differential equation at the nine points
indicated.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 675
b. Find the particular solution y = f(x) to the differential equation with the initial condition f(2) = 0.
c. Describe all points in the x – y coordinate plane for which the slope of the tangent line would be
positive. Give a reason for your answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 676
Let F(x) =
x
3
f (t )dt , where the graph of f(t) is shown to the right. Answer the following questions.
1. Complete the following table for values of F(x).
x
2
3
5
6
9
F(x)
2. On what interval(s) is f(t) positive?
3. On what interval(s) is f(t) negative?
4. On what interval(s) is F(x) increasing?
Justify your answer.
5. On what interval(s) is F(x) decreasing?
Justify your answer.
6. On what interval(s) is F(x) concave up? Justify your answer.
7. On what interval(s) is F(x) concave down? Justify your answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 677
Pictured to the right is the graph of f, which consists of two semi-circles and one line segment on the
interval [0, 17]. Let g ( x) 
 f (t )dt .
x
0
45. Find the values of g(8), g ' (8) and g ' ' (8) .
46. On what interval(s) is the graph of g(x) concave down? Justify your answer.
47. On what interval(s) is the graph of g(x) increasing? Justify your answer.
48. Find all values on the open interval (0, 17) at which g has a relative minimum. Justify your answer.
49. What are the x – coordinates of each point of inflection of g(x)? Justify your answer.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 678
Test #8 Additional Free Response – Calculator NOT Permitted
Consider the differential equation
dy
2x
 .
dx
y
a. On the axes provided, sketch a slope field for the given differential equation at the twelve points
indicated.
b. Write an equation of the tangent line to the graph of f at (1, –1) and use it to approximate f(1.1).
Explain why the tangent line gives a good approximation of f(1.1).
c. Find the particular solution y = f(x) to the given differential equation with the initial condition
f(1) = –1.
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 679
Calculator NOT Permitted Free Response Part A – 2 points total
____ 1 Accurately drawn slope segments are provided
for all six points above the x – axis as pictured
____ 1 Accurately drawn slope segments are provided
for all six points below the x – axis as pictured.
Calculator NOT Permitted Free Response Part B – 3 points total
____ 1 Equation of the tangent line is y = 2x – 3 or equivalent form.
____ 1 f(1.1)  –0.8 or  4
5
____ 1 At x = 1, the value of the function and the value of the tangent line are equivalent
because they intersect each other. Thus, at x = 1.1, the tangent line would give a
very close approximation of the function because the graph of the tangent line
would be a very close under or over approximation depending upon the concavity
of the function at x = 1.
Calculator NOT Permitted Free Response Part C – 4 points total
____ 1 Separation of variables
____ 1 Correct anti-differentiation of y variable expression: ½y2
____ 1 Correct anti-differentiation of x variable expression: –x2 + c
____ 1 Correct function y = f(x) =   2 x 2  3
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course
Mark Sparks 2012
Page 680