ap review chapter 4

8/10/2019 AP Review Chapter 4

1/20

Chapter 4 1

1. Define ( ) 3 23 3f x x x x= + .

(a) Estimate the area between the graph of fand the axis on the interval [ ]1,3 using a left-hand sum with four rectangles of equal width.

(b) Is the estimate in part (a) an over-estimate or underestimate of the actual area? Justify your

conclusion.

(c) Use a definite integral to calculate the exact area between the graph of f and the axis on

the interval [ ]1,3 .


2/20

Chapter 4 2

1. Define ( ) 3 23 3f x x x x= + .

(a) Estimate the area between the graph of fand the axis on the

interval [ ]1,3 using a left-hand sum with four rectangles of equal

width.

x ( )f x

1 13 10.5

4x

= =

1.5 1.125

2 2

2.5 4.375

3 9

+1 0.5x =

+1(1) (1.5) (2) (f f f f+ + +

+1 4.25

( ) ( )1 1.125 2 4.375 0.5 4.25LHS = + + + =

(b) Is the estimate in part (a) an over-estimate or underestimate of the

actual area? Justify your conclusion.Observe that

( )

( )

( )

2

2

2

3 6 3

3 2 1

3 1

f x x x

x x

x

= +

= +

=

Since 0f > , for 1x > , the function f is increasing on (1,3].

Furthermore, since (1) 1f = , f is positive and increasing on [ ]1,3 .As a result, each rectangle in the left-hand sum will lie below the

graph of f . Thus the area estimate is an underestimate.

+1 Underestimate

+1 Correct supporting work

(c) Use a definite integral to calculate the exact area between the graph

of fand the x axis on the interval [ ]1,3 .

( ) ( ) ( ) ( ) ( ) ( )

33

3 2 4 3 2

11

4 3 2 4 3 2

1 33 3

4 2

1 3 1 33 3 3 1 1 1

4 2 4 2

6

x x x dx x x x

+ = +

= + +

=

+1 Correctly integrate f

+1 Correct limits of

integration

+1 Apply the FundamentalTheorem of Calculus

+1 Area = 6


3/20

Chapter 4 3

2. Define ( ) 2 4f x x x= + .

(a) Estimate the area between the graph of fand the axis on the interval [ ]1, 4 using a left-hand sum with four rectangles of equal width.

(b) Estimate the area between the graph of fand the axis on the interval [ ]1, 4 using a right-hand sum with four rectangles of equal width.

(c) Use a definite integral to calculate the exact area between the graph of f and the axis on

the interval [ ]1, 4 .


4/20

Chapter 4 4

2. Define ( ) 2 4f x x x= + .

(a) Estimate the area between the graph of fand the axis on

the interval [ ]1, 4 using a left-hand sum with four rectangles

of equal width.

x ( )f x

1 34 10.75

4x

= =

1.75 3.9375

2.5 3.75

3.25 2.4375

4 0

+1 0.75x =

+1 (1) (1.75) (2.5) (3.25)f f f f+ + +

+1 9.844

( ) ( )3 3.9375 3.75 2.4375 0.75 9.844= + + + LHS

(b) Estimate the area between the graph of fand the axis on

the interval [ ]1, 4 using a right-hand sum with fourrectangles of equal width.

x ( )f x

1 34 10.75

4x

= =

1.75 3.9375

2.5 3.75

3.25 2.4375

4 0

+1 0.75x =

+1 (1.75) (2.5) (3.25) (4f f f f+ + +

+1 7.594

( ) ( )3.9375 3.75 2.4375 0 0.75 7.594= + + +

RHS

(c) Use a definite integral to calculate the exact area between the

graph of fand the axis on the interval [ ]1, 4 .

( ) ( ) ( ) ( )

44

2 3 2

11

3 2 3 2

14 2

3

1 14 2 4 1 2 1

3 3

9

+ = +

= + +

=

x x dx x x

+1 Correctly integrate f

+1 Apply the Fundamental Theorem

of Calculus

+1 Correct limits of integration andarea = 9


5/20

Chapter 4 5

3. Define ( ) ( )2

2 4f x x x

= + .

(a) Find the antiderivative of f that goes through ( )0,1 .

(b) Calculate ( )21f x dx

(c) Determine the area between f and the x axis on the interval [ ]1,1 .


6/20

Chapter 4 6

3. Define ( ) ( )2

2 4f x x x

= + .

(a) Find the antiderivative of f that goes through ( )0,1 .

Let 2 4u x= + . Then 2du x dx= and 0.5xdx du= .

( ) ( )

( )

( )

22

2

1

12

4

0.5

0.51

0.5 4

F x x x dx

u du

uC

x C

= +

=

= +

= + +

Since ( )0 1F = ,

( )( )( )

121 0.5 0 4

1 0.5 0.25

1 0.125

1.125

C

C

C

C

= + +

= +

= +

=

The specific antiderivative is ( ) ( )1

20.5 4 1.125F x x

= + + .

+1 ( )2 0.5u du +1 Initial condition

+1 ( ) ( )1

20.5 4 1.125F x x

= + +

(b) Calculate ( )2

1f x dx

( ) ( )1

20.5 4F x x C

= + +

( ) ( )

( )

2

1( ) 2 1

0.0625 .1

0.0375

=

=

=

f x dx F F

+1 Fundamental Theorem of

Calculus

+1 Correct limits

+1 0.0375

(c) Determine the area between f and the x axis on the interval

[ ]1,1 .

( ) ( ) ( ) ( )0 1

1 00.125 0.1 0.1 .125

0.05

f x dx f x dx

+ = +

=

+1 Two integrals with correctintegrand

+1 Correct limits of integration

+1 Area = 0.05


7/20

Chapter 4 7

4. The graph of a function f is shown in the figure below. It consists of two lines and a

semicircle. The regions between the graph of f and the -axis are shaded.

Editor: 2

3 3 2

( ) 4 1 2 2

3 2 4

+

= + <


8/20

Chapter 4 8

4. The graph of a function f is shown in the figure below. It consists of two lines and a

semicircle. The regions between the graph of f and the -axis are shaded.

Editor: 2

3 3 2

( ) 4 1 2 2

3 2 4

+

= + <


9/20

Chapter 4 9

5. Define ( ) 34 4f x x x= .

(a) Calculate

1

0

( )f x dx .

(b) What is the average value of f on [ ]0, 2 ?

(c) What is the area of the region(s) bounded by the graph of f and thexaxis ? Show the work

that leads to your conclusion?


10/20

Chapter 4 10

5. Define ( ) 34 4f x x x= .

(a) Calculate

2

0

( )f x dx .

( ) ( )

2 24 2

00

( ) 2

16 8 0 0

8

f x dx x x=

=

=

+1 ( ) 4 22F x x x= +1 Fundamental Theorem of

Calculus

+1 8

(b) What is the average value of f on [ ]0, 2 ?2

0

( )

avg value2 0

82

4

f x dx

=

=

=

+1

( )b

a

f x dx

b a

+1 Correct values for a and b

+1 4

(c) What is the area of the region(s) bounded by the graph of

f and thexaxis ? Show the work that leads to your

conclusion?

Thex-intercepts of f are 1,0,1x = .0 1

1 0

( ) ( )

1 1 2

Area f x dx f x dx

= +

= + =

+1 Two integrals

+1 Correct limits of integration

+1 Area = 2


11/20

Chapter 4 11

6. Define

( )2

2

4 3( )

2 3

xf x

x x

+=

+

.

(a) Use a change of variable to rewrite the integrand of

( )2

2

4 3

2 3

xdx

x x

+

+

as an easily integrable

function.

(b) Use integration by substitution to evaluate

3

1

( )f x dx .

(c) What is the average value of f on [ ]1,3 ?


12/20

Chapter 4 12

6. Define

( )2

2

4 3( )

2 3

xf x

x x

+=

+

.

(a) Use a change of variable to rewrite the integrand of

( )2

24 3

2 3x dx

x x

+

+as an easily integrable function.

Let 22 3u x x= + . Then 4 3du x= + . The integral may be

rewritten as2

1du

u.

+1 22 3u x x= +

+12

1du

u

(b) Use integration by substitution to evaluate

3

1

( )f x dx .

Since 22 3u x x= + , we need to determine the values of u

that correspond with the limits of integration 1x = and 3x = .( ) ( )

22 1 3 1 5u = + =

( ) ( )2

2 3 3 3 27u = + =

2727

2 1

55

1 1 220.163

27 5 135u du u

= = =

+1 Upper limit 27u =

+1 Lower limit 5u=

+1 Use integrand 2u

+122

or 0.163135

(c) What is the average value of f on [ ]1,3 ?3

1

( )

avg value

3 111

135

0.0815

f x dx

=

=

+1

( )

b

a

f x dx

b a

+1 Correct limits of integration+1 0.0815


13/20

Chapter 4 13

7. Define1

( )f xx

= .

(a) Use the trapezoidal rule with 5n = to estimate2

1

( )f x dx .

(b) ( ) lnf x x C= + . Use the Fundamental Theorem of Calculus to calculate the exact value of2

1

( )f x dx .

(c) Explain why the trapezoidal rule cannot be used to estimate1

1

( )f x dx

.


14/20

Chapter 4 14

7. Define1

( )f xx

= .

(a) Use the trapezoidal rule with 5n = to estimate

2

1

( )f x dx .

( )

2 10.1

2 2 5

b a

n

= =

2

1

1 1 1 1 1 1( ) 0.1 2 2 2 2

1 1.2 1.4 1.6 1.8 2

0.696

f x dx

+ + + + +

+1 0.12

b a

n

=

+2 Proper use of trapezoidal rule

+1 0.696

(b) ( ) lnf x x C= + . Use the Fundamental Theorem of

Calculus to calculate the exact value of

2

1

( )f x dx

2

1

( ) ln 2 ln 1

ln 2

f x dx =

=

+1 Use Fundamental Theorem ofCalculus

+1 ln 2 or 0.693

+1 Exact value

(c) Explain why the trapezoidal rule cannot be used to

estimate1

1

( )f x dx

.

The function1

( )f x = is discontinuous at 0x = . Since the

function f is not continuous on

[ ]1,1 , the trapezoidal rule

may not be used.

+1 f discontinuous at 0x =

+1 continuity required for

trapezoidal rule


15/20

Chapter 4 15

8. A continuous, integrable function f has exactly twox-intercepts, ( )1,0x and ( )2 , 0x .

(a) What is the difference in meaning between2

1

( )

x

x

f x dx and ( )f x dx ?

(b) Write an integral for the area bounded by f and thex-axis.

(c) Given that ( ) ( )F x f x dx= ,2

1

( ) 2

x

x

f x dx = , and ( )1 5F x = , determine ( )2F x .


16/20

Chapter 4 16

8. A continuous, integrable function f has exactly twox-intercepts, ( )1,0x and ( )2 , 0x .

(a) What is the difference in meaning between2

1

( )

x

x

f x dx and

( )f x dx ?2

1

( )

x

x

f x dx is the signed area of the region bounded by f and

thex-axis.

( )f x dx is the family of functions with derivative f

+22

1

( )

x

x

f x dx explanation (1 if

state areanotsigned area)

+2 ( )f x dx explanation

(b) Write an integral for the area bounded by f and thex-

axis.

Since the function f is continuous and has exactly two

xintercepts, the area of the region bounded by f and thex-

axis is given by2

1

( )

x

x

f x dx

+1 limits of integration

+1 integrand f or f

+1 integrand f

(c) Given that ( ) ( )F x f x dx= ,2

1

( ) 2

x

x

f x dx = , and

( )1 5F x = , determine ( )2F x .

( ) ( )

( )

2

1

2 1 ( )

5 2

3

x

xF x F x f x dx

= +

= +

=

+1 ( ) ( )2

1

2 1 ( )

x

x

F x F x f x dx= +

+1 ( )2 3F x =


17/20

Chapter 4 17

9. Let graph of a continuous, twice-differentiable function f is shown in the figure below. The

three regions between the graph of f and the x -axis are markedA, B, and Cand have areas

5.5, 8, and 15.5, respectively.

The function Fis an antiderivative of f with the property that ( )1 9F = .

(a) Which value is larger (0)F or (4)F ? Justify your answer.

(b) How many times does Fequal 5 on the interval [ ]0,4 ? Show the work that leads to yourconclusion.

(c) On what interval(s) is Fincreasing? Justify your answer.


18/20

Chapter 4 18

9. The graph of a continuous, twice-differentiable function f is shown in the figure below. The

three regions between the graph of f and the x -axis are markedA, B, and Cand have areas

5.5, 8, and 15.5, respectively.

The function Fis an antiderivative of f with the property that ( )1 9F = .


( ) ( ) ( )

( ) ( ) ( )

( )

1

0

1

0

1 0

0 1

9 5.5

14.5

F F f x dx

F F f x dx

= +

=

=

=

( ) ( ) ( )

( )

4

14 1

9 8 15.5

9 7.5

16.5

F F f x dx= +

= + +

= +

=

( )4F is larger.

+1 Use ( ) ( ) ( )b

a

F b F a f x dx= +

+1 Use appropriate limits of

integration in each integral

+1 Conclusion with correct

justification

(b) How many times does Fequal 5 on the interval [ ]0, 4 ?Show the work that leads to your conclusion.

We know ( )0 14.5F = , ( )1 9F = , and ( )4 16.5F = . We

calculate ( )3

1

(3) (1) ( ) 9 8 1F F f x dx= + = + = . Since

( )1 5 (3)F F> > , ( ) 5F c = for some c in ( )1,3 . Since

( )3 5 (4)F F< < , ( ) 5F c = for some c in ( )3,4 . So the

function Fequals 5 two times on the interval [ ]0, 4 .

+1 (3) 1F =

+1 Correct supporting work

+1 Two times


When 0f > , Fis increasing. Since 0f > on (3,4], F

is increasing on (3,4] .

+1 When 0f > , Fis increasing.

+1 0f > on (3,4]

+1 Fis increasing on (3,4].


19/20

Chapter 4 19

10. The graph of a continuous, twice-differentiable function f is shown in the figure below.

The three regions between the graph of f and the x -axis are markedA, B, and Cand have

areas 4, 4, and 6.25, respectively.

The function Fis an antiderivative of f with the property that (0) 6F = .


(b) How many times does Fequal 4 on the interval [ ]0,5 ? Show the work that leads to yourconclusion.



20/20

ap review chapter 4

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