Chapter 1 to 4 - Quick Review Problems 1. . State why ... · 5) 1 2 1 '( ) x F x 6) F x 2 5 5 '( ) The function F is an anti-derivative of f on an interval I if F x f x'( ) ( ) for

1

Chapter 1 to 4 - Quick Review Problems

1. Use a graphing calculator to graph 2/3( )f x x . State why Rolle’s Theorem does not apply to f on

the interval 1,1 .

a) f is not continuous on 1,1 b) f is not defined on the entire interval c) ( 1) (1)f f

d) f is not differentiable at 0x e) Rolle’s Theorem does apply

2. The graph of 'f is shown below. Estimate the open intervals in which f is increasing or decreasing.

a) Increasing ,1 and 3, ; decreasing (1, 3)

b) Increasing (0,2); decreasing ,0 and 2,

c) Increasing ,

d) Increasing ,0 and 2, ; decreasing (0,2)

e) Increasing 0.5,0.5 and 2.8, ; decreasing , 0.5 and 0.5,2.8

3. Given that 2( ) 12 28f x x x has a relative maximum at 6x , choose the correct statement.

a) f is negative on the interval ( ,6) b) f is positive on the interval ,

c) f is negative on the interval 6, d) f is positive on the interval 6,

e) None of these

4. Let f x be a polynomial function such that ( 2) 5f , ( 2) 0f , and ( 2) 3f . The point

( 2,5) is a(n)________________________ the graph of f .

a) Relative maximum b) Relative minimum c) Intercept

d) Point of Inflection (e) Absolute minimum

1 2 3–1 x

1

–1

–2

–3

y

2

5. Use a graphing calculator to graph 2

1

( 1)f x

x

. Use the graph to determine the open intervals

where the graph of the function is concave upward or concave downward.

a) Concave downward: , b) Concave downward: , 1 ; Concave upward: ( 1, )

c) Concave downward: , 1 and ( 1, ) d) Concave upward: ( , 1) and ( 1, )

e) Concave upward: , 1 ; Concave downward: ( 1, )

6. Find all extrema, if any, in the interval 0,2 if sinf x x x . Write as ordered pairs.

7. A differentiable function has only one critical number: 3x . Identify the relative extrema of f if

142

f and 2 1f .

8. Find all points of inflection of the function 4 55 2f x x x .

9. State why the Mean Value Theorem does not apply to the function

2

2

1f x

x

on the interval 3,0 . Give your answer as a complete sentence.

3

AP Calculus

Notes 5.1

Anti-derivatives and Indefinite Integration

Exploration:

For each of the following derivatives, find the original function )(xf .

1) '( ) 2F x x 2)

' 2( ) 6F x x 3) '( ) cosF x x

4) 3)(' xexF 5) 21

1)('

xxF

6)

xxF

52

5)('

The function F is an anti-derivative of f on an interval I if '( ) ( )F x f x for all x . F is an anti-derivative

rather than the anti-derivative because any constant C would work.

Explanation:

Find the derivative of 2)( xxf , 5)( 2 xxf , and

exxf 2)( .

Because of this, you can represent all anti-derivatives of ( ) 2f x x by :___________

The constant C is called the constant of integration. The function represented by F is the general anti-

derivative of f , and 2( )F x x C is the general solution of the differential equation

'( ) 2F x x .

Notation for Anti-derivatives:

The operation of finding all solutions of this equation is called anti-differentiation or indefinite integration

and is denoted by the integral sign . The general solution is:

( ) ( )y f x dx F x C

4

The expression ( )f x dx is read as the anti-derivative of f with respect to x. So, dx serves to identify x as the

variable of integration. The term indefinite integral is a synonym for anti-derivative.

Basic Integration Rules

The inverse nature of integration and differentiation can be used to obtain:

'( ) ( )F x dx F x C

If ( ) ( )f x dx F x C then ( ) ( )d

f x dx f xdx

Exploration:

What is the anti-derivative of each of the following? Try to develop the basic power rule for integration:

a) 2)( xxf b)

3)( xxf c) 4)( xxf

So, the Power Rule for Integration is: _______________dxxn

Ex. 1: Integrate each of the following polynomial functions:

a) ( 2)x dx b) dx c) 4 2(3 5 )x x x dx

5

Differentiation Formula Integration Formula

kxdx

d

)(xkfdx

d

)()( xgxfdx

d

nxdx

d

xdx

dsin

xdx

dcos

xdx

dtan

xdx

dsec

xdx

dcsc

xdx

dcot

xedx

d

xadx

d

xdx

dln

6

The most important step in integration is rewriting the integral in a form that fits the basic integration rules.

Ex. 2: Rewrite each of the following before integrating:

Original Integral Rewrite Integrate Simplify

a) 3

1dx

x

b) dxx2

1

c) dtttt )193(2 2

d) 2 2( 1)t dt

e) 3

2

3xdx

x

f) 3 ( 4)x x dx g. 2

sin

cos

xdx

x h.

d22 csc

2sec

7

Initial Conditions and Particular Solutions:

There are several anti-derivatives for a function, depending on C. In many applications of integration, you

are given enough information to determine a particular solution. To do this you need only know the value of

)(xfy for one value of x . (This information is called an initial condition).

Ex. 3: The function dxxy 13 2 has only one curve passing through the point (2,4) . Find the particular

solution that satisfies this condition.

Ex. 4: Find the general solution of xexF )(' and find the particular solution that satisfies the initial

condition 3)0( F .

Ex. 5: Find the general solution of 2sin)(" xxf and find the particular solution that satisfies the initial

condition 3)0(' f and 3)( f .

8

Ex. 6: A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet

and shown in the figure.

Remember: To go from position to velocity to acceleration –

To go from acceleration to velocity to position –

a) Find the position function giving the height s as a function of time t.

b) What is the speed of the ball when it hits the ground?

c) After how many seconds after launch is the ball back at its initial height?

9

Ex. 7: A particle, starting at the origin, moves along the x-axis and it’s velocity is modeled by the equation

24306)( 2 tttv where t is in seconds and )(tv is meters per second.

a) How is the velocity changing at any time t?

b) What is the particle’s speed at 3 seconds?

c) What is the particle’s position when the acceleration is 2/6 sm ?

d) When is the particle changing directions?

e) When is the particle furthest to the left?

10

Ex. 8: A missile is accelerating at a rate of 2/4 smt from a position at rest in a silo 750m below ground.

How high above the ground will the missile be after 6 seconds.

Ex. 9: The motion of a grizzly bear stalking its prey, walking left and right of a fixed point in ft/s, can be

modeled by the motion of a particle moving left and right along the x-axis, according to the

acceleration equation 1

( ) sin( )3

a t t . If the bear’s velocity is 1 ft/s when 0t …

a) Find the velocity equation.

b) How fast was the bear traveling when 7t ?

c) In what direction is the bear traveling when 5t ?

11

AP Calculus I

5.1 Quiz Review

1. dxxx )576( 2 2. 342 3x x dx 3. 2

6

11dx

x

4. 2 (3 1)x x dx 5. 2(2 5)x dx 6. dxxxx )3(

7. dxxxx )sin4tansec3( 8. 22csc cosx x dx 9.

dxx

xxx

3

59 24

10. Find the original function )(xf given '( ) 4 2f x x and the condition 1)4( f .

12

11. A particle moves along the x-axis with velocity given by 2( ) 3 6v t t t . If the particle is at position

2x at time 0t , what is the position of the particle at time 1t ? [2008 AP MC#7]

12. A cannonball is shot upward from the ground with an initial velocity of sm /30 . The acceleration is 2/8.9 sm .

a) What is the height and velocity function of the cannonball?

b) What is the maximum height of the cannonball?

c) What is the velocity of the cannonball when it hits the ground?

13

AP Calculus I

Notes 5.2

Area Under a Curve

Area under the curve from [0,4]

= __________________________


= __________________________


= __________________________

14

Area under the curve from [-0.5,4.5]

= __________________________


= __________________________

15

16

In this section we will examine the problem of finding the area of a region in a plane.

EX 1: Suppose you have to find the area under the curve 225 xy from x = 0 to x = 4.

Graph:

Method 1: Divide the region into four rectangles, where the left endpoint of each rectangle comes just under

the curve, and find the area.

Is this an over- or under-approximation of the actual?

17

Graph:

Method 2: Divide the region into four rectangles, where the right endpoint of each rectangle comes just under



18

Graph:

Method 3: Divide the region into four rectangles, where the midpoint of each rectangle comes just under the

curve, and find the area.


19

EX 2: Suppose you have to find the area under the curve 13 xy from x = 1.5 to x = 3.

Graph:

Method 1: Divide the region into six rectangles, where the left endpoint of each rectangle comes just under



20

Graph:

Method 2: Divide the region into six rectangles, where the right endpoint of each rectangle comes just under



21

1 2 3 4 5 6 7 8 9 x

1

2

3

4

5

6

7

y

1 2 3 4 5 6 7 8 9 x

1

2

3

4

5

6

7

y

1 2 3 4 5 6 7 8 9 x

1

2

3

4

5

6

7

y

Formulas

Area Using Left Endpoints: 0 1 2 3 1... n

b aAREA y y y y y

n

Area Using Right Endpoints: 1 2 3 4 ... n

b aAREA y y y y y

n

Area Using Midpoints: 1 3 5 2 1

2 2 2 2

... n

b aAREA y y y y

n

As n becomes larger (we add more, smaller rectangles) these become:

0 1 2 3 1lim ... nn

b ay y y y y

n

and so on for each of these.

_____________________________________________________________________________________

Problems

1. Approximate the area under the curve 4)5.0sin(2 xy from 0x to 8x using:

4 left Riemann rectangles 4 right Riemann rectangles 4 midpoint Riemann rectangles

2. Approximate the area under the curve 3y x from 2x to 3x using:

a) five left-endpoint rectangles b) five right-endpoint rectangles

22

HW - LEA, REA, MID Approximations

1. Approximate the area under the curve 3y x from x = 2 to x = 3 using three left-endpoint rectangles.

2. Approximate the area under the curve 3y x from x = 2 to x = 3 using three right-endpoint rectangles.

3. Approximate the area under the curve 3y x from x = 2 to x = 3 using three midpoint rectangles.

4. Approximate the area under the curve 32 xy from x = 4 to x = 9 using five left-endpoint rectangles.

5. Approximate the area under the curve 32 xy from x = 4 to x = 9 using five right-endpoint

rectangles.

6. Approximate the area under the curve 32 xy from x = 4 to x = 9 using five midpoint rectangles.

7. Approximate the area under the curve xy sin2 from x = 0 to x = 2

using eight left-endpoint rectangles.


using eight right-endpoint

rectangles.


using eight midpoint rectangles.

23

PVA – Problems

1) A silver dollar is dropped from the top of a building that is 1362 feet tall (acceleration is 2/32 sft )

a) Determine the position and velocity functions for the coin.

b) Determine the average velocity on the interval 1 2, .

c) How long does it take for the coin to reach the ground?

d) What is the velocity of the coin at impact?

2) A missile is accelerating at a rate of 24 /t km s from a position at rest in a silo 1 km below ground.

How high above the ground will the missile be after 6 seconds.

24

3) A particle, starting at the origin, moves along the x-axis and it’s velocity is modeled by the equation

24306)( 2 tttv where t is in seconds and )(tv is meters per second.

a) How is the velocity changing at any time t?

b) What is the particle’s speed at 3 seconds?

c) What is the particle’s position when the acceleration is 2/6 sm ?

d) When is the particle changing directions?

e) When is the particle furthest to the left?

25

PVA - Homework

1) From the top of a 300m cliff, a ball is thrown straight up at a velocity of 20m/s. Assuming the ball misses

the cliff on the way down, how high is it 5 seconds after it is thrown and how fast is it going? (gravity =

29.8ms

)

2) The motion of a grizzly bear stalking its prey, walking left and right of a fixed point in ft/s, can be

modeled by the motion of a particle moving left and right along the x-axis, according to the acceleration

equation 1

( ) sin( )3

a t t . If the bear’s velocity is 1 ft/s when 0t …

a) Find the velocity equation.

c) How fast was the bear traveling when 7t ?

c) In what direction is the bear traveling when 5t ?

26

AP Calculus I

5.1 – 5.2 Quiz Review

1. Find the area under the curve 22y x x from x = 1 to x = 2 with n = 4 using LEA.

2. Find the area under the curve y x from x = 0 to x = 1 with n = 4 using REA.

3. Find the area under the curve 3 2y x from x = 0 to x = 2 with n = 8 using MPA.

4. Find the area under the curve 21y x from x = 0 to x = 1 with n = 5 using LEA.

5. Evaluate dxxx )576( 2 6. Evaluate dxxxx )3(

7. Evaluate dxxxx )sin4tansec3( 8. Evaluate

dxx

xxx

3

59 24

9. Find the original function )(xf given dxx )24( and the condition 1)4( f .

10. A cannonball is shot upward from the ground with an initial velocity of sm /30 . The acceleration is 2/8.9 sm .

d) What is the height and velocity function of the cannonball?

e) What is the maximum height of the cannonball?

f) What is the velocity of the cannonball when it hits the ground?

27

AP Calculus I

Notes 5.6

The Trapezoidal Rule

In addition to the three approximation techniques, there is a fourth technique that changes the geometric

shape of the approximation. The trapezoidal rule approximates the area using a certain number of trapezoids.

Remember the area of a trapezoid is:

So, to use the trapezoidal rule to approximate the area under the curve of a function…

x1 x2 x3 x4 x5 x6

Area of trapezoids:

28

1 2 3 4 5 6–1–2 x

1

2

3

4

5

6

7

8

y

Trapezoid Rule:

Ex. 1: Use the trapezoidal rule to approximate the area under xxf sin)( from ],0[ with 4n .

Ex. 2: Approximate the area between the x-axis and xxxf 2)( 2 from ]3,1[ using 5 trapezoids.

Ex. 3: Approximate the area between the x-axis and )(xf , found below, from ]5,1[ using 3 trapezoids.

29

Ex. 4: Readings from a car’s speedometer at 10-minute intervals during a 1-hour period are given in

the table: t = minutes, v = speed in miles per hour:

t 0 10 20 30 40 50 60

v 26 40 55 10 60 32 45

a) Draw a graph that could represent the car’s speed during the hour.

b) Find the area under the curve after 40 minutes of driving using the Right Riemann Sum.

30

c) After converting the x-axis into hours, what would be the meaning of the area underneath the

graph?

d) Approximate the distance traveled by the car for the hour using the Trapezoid Rule. (recreating

the graph may be helpful).

31

Uneven Interval Widths

Ex. 5:: The table below shows the rate at which water is coming out of a faucet in (mL/sec.) over different

periods of time. t = seconds. R = rate of volume of water in mL/sec.

t 0 3 4 8 10

R 8 12 15 11 5

a) Draw a graph that could represent the rate of volume of water during the 10 seconds.

b) Approximate the area under the curve after 10 seconds using the Left Riemann Sum.

32

c) What would be the meaning of the area underneath the graph?

d) Approximate the volume of water that has come out of the faucet after 10 seconds using the

Trapezoid Rule. (recreating the graph may be helpful).

33

Accumulation Problems: (Approximate area under a curve – Trapezoid, LEA, REA)

Ex. 1

Oil is flowing down a pipeline but there is a leak from where oil is dripping. The hole is getting bigger so the

rate in which the oil is dripping changes as shown in the table below:

Time (min) 1 2 3 4 5 6 7

Rate (L/min_) 6 10 15 24 20 16 2

a.) Calculate the amount of oil spilled using the trapezoidal rule with 6 sub intervals from

1 ≤ x ≤ 7. Be sure to include units to calculate the total oil spilled.

b.) Use the LEA rule with the same number of sub intervals over the same period of time and

compare the results.

34

Ex. 2

People enter a concert at the following rates, E(t) in people per hour over t hours.

Time (hours) 0 1 2 3 4 5 6

People entering 120 156 176 126 150 80 0

c. Estimate the total number of people who have entered using all three (LEA, REA and

Trapezoidal rules.) Use 6 sub intervals from 0 ≤ x ≤ 6.

Function Attribute vs Over / Under Estimate

Function Attribute

Type of

Approx.

Increasing Decreasing Concave

UP

Concave

DOWN

LEA Under Over ---- ----

REA Over Under ---- ----

TRAP ---- ---- Over Under

35

AP Calculus Driving Project

You are to go on a fifteen-minute drive with someone else driving. At the beginning of the 15-minute

time interval, record the exact odometer reading. Each minute of the drive (on the minute), ask the driver for

the speedometer reading and record these. Between speedometer readings you need to make notes on

significant observations about the about the speed of the car (stop and go traffic, steady traffic flow, traffic

lights, etc.). At the close of the fifteen minutes, record the exact odometer reading.

Use the trapezoid rule and the speedometer readings to approximate the distance traveled, and compare your

approximation with the difference between the odometer readings from the car. In addition to the trapezoidal

rule, you will complete a left-end, right-end Riemann Sum and a midpoint approximation. Next, write up

your results – include your purpose, data, comments, a nice graph, calculations and results, analysis, and

conclusions.

This project will be worth 20 points scored as follows.

Scoring Rubric

1. Graph (correct scale and accurate) (2 points) __________

2. Trapezoidal Calculation (5 points) __________

3. Riemann Sums (LEA, REA, MPA) (3 points) __________

4. Accuracy of computations (show all work!) (4 points) __________

5. Write-up and analysis (6 points) __________

Due date: ____________________________

36

5.1 – 5.2, 5.6 Quiz Review

Find the area and decide whether it is an over/underestimate

1. 22y x x from ]2,1[ with n = 4 using LEA. 2. y x from ]8,0[ with n = 4 using REA.

3. 3 2y x from ]10,2[ with n = 4 using MPA. 4. 216 xy from ]4,2[ with n = 5 using TZA.

5. dxexx x )576( 2 6. dxxxx )3( 7. dxxxx )sin4tansec3(

8.

dxx

xxx

3

59 24

9.

dxx

xx

tan

csc3sin 10.

dxx

xx

4 5

23

54

11. Find the original function )(xf given 24

)(' x

xf and the condition 1)4( f .

37

12. A cannonball is shot up from the ground with a velocity of sm /30 . The acceleration is 2/8.9 sm .

a) What is the height and velocity function of the cannonball?

b) What is the maximum height of the cannonball?

c) What is the velocity of the cannonball when it hits the ground?

13. A region’s beverage consumption )(tC , in L/month, over various months, t , where 0t is the

beginning of the first month can be modeled by the following table:

t 0 3 6 8 12

C(t) 15 25 30 25 10

a) Approximate the area under the curve of )(tC from ]12,0[ using the Trapezoidal Rule with 4

subintervals. Describe the meaning of this answer.

b) Assuming )(tC is a function that is concave down everywhere, will the answer from a) be an over

or under estimate?

38

AP Calculus I Notes 5.3

Definite Integrals

Definite Integrals

Theorem – The Definite Integral as the Area of a Region

If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by

the graph of f, the x-axis, and the vertical lines x = a and x = b is given by: Area = ( )

b

a

f x dx

** A definite integral is a number, whereas an indefinite integral is a family of functions.

Ex. 1: Sketch the region corresponding to each definite integral. Then evaluate each integral using

a geometric formula:

a)

3

1

4 dx b)

4 - x2 dx-2

2

ò

Negative Area – What happens when the curve is below the x-axis?

Considering the width (_______) remains positive and the height (_________________) is now negative,

the area is going to be _________________.

39

Ex. 2: Find the area of the following curves over the given intervals:

a)

4

1

)4( dxx b)

6

0

24 dxx

Definition of Two Special Definite Integrals

1. If f is defined at x = a, then ( ) 0

a

a

f x dx

2. If f is integrable on [a, b], then ( ) ( )

a b

b a

f x dx f x dx

Theorem – Continuity Implies Integrability

If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].

Theorem - Additive Interval Property

If f is integrable on the closed intervals determined by a, b, and c, then

b c b

a a c

f x dx f x dx f x dx

Ex. 3: Given

1

1

( ) 3f x dx

and

1

0

( ) 5f x dx , find

0

1

( )f x dx

40

1 2 3 4 5 6 7 8 9 10–1–2–3–4 x

1

2

3

4

5

6

–1

–2

y

Worksheet on Definite Integrals

The graph of f(x) shown at the

right is formed by lines and a

semi-circle.

Use the graph to evaluate the

definite integrals below.

1. ∫ 𝑓(𝑡)𝑑𝑡4

4 2. ∫ 𝑓(𝑡)𝑑𝑡

1

0 3. ∫ 𝑓(𝑡)𝑑𝑡

3

1

4. ∫ 𝑓(𝑡)𝑑𝑡3

0 5. ∫ 𝑓(𝑡)𝑑𝑡

6

3 6. ∫ 𝑓(𝑡)𝑑𝑡

3

6

7. ∫ 𝑓(𝑡)𝑑𝑡6

0 8. ∫ 𝑓(𝑡)𝑑𝑡

10

6 9. ∫ 𝑓(𝑡)𝑑𝑡

6

10

10. ∫ 𝑓(𝑡)𝑑𝑡10

0 11. ∫ 𝑓(𝑡)𝑑𝑡

0

10 12. ∫ 𝑓(𝑡)𝑑𝑡

0

−1

13. ∫ 𝑓(𝑡)𝑑𝑡0

−3 14. ∫ 𝑓(𝑡)𝑑𝑡

−3

0 15. ∫ 𝑓(𝑡)𝑑𝑡

−3

−4

16. ∫ 𝑓(𝑡)𝑑𝑡0

−4 17. ∫ 𝑓(𝑡)𝑑𝑡

10

−4 18. |∫ 𝑓(𝑡)𝑑𝑡

10

0|

19. ∫ |𝑓(𝑡)|10

0𝑑𝑡 20. |∫ 𝑓(𝑡)𝑑𝑡

10

−4| 21. ∫ |2𝑓(𝑡)|

−4

10𝑑𝑡

Suppose: ∫ 𝑓(𝑥)𝑑𝑥 = 18

5

−2, ∫ 𝑔(𝑥)𝑑𝑥 = 5

5

−2, ∫ ℎ(𝑥)𝑑𝑥 = −11

5

−2, and ∫ 𝑓(𝑥)𝑑𝑥 = 0

8

−2 find

22. ∫ (𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥5

−2 23. ∫ (𝑓(𝑥) + 𝑔(𝑥) − ℎ(𝑥))𝑑𝑥

5

−2

24. ∫ 4𝑔(𝑥)𝑑𝑥−2

5 25. ∫ (𝑔(𝑥) + 2)𝑑𝑥

5

−2

26. ∫ (𝑓(𝑥) − 6)𝑑𝑥5

−2 26. ∫ ℎ(𝑥 − 2)𝑑𝑥

7

0

41

Notes 5.4

Fundamental Theorem of Calculus

The two branches of Calculus: differential calculus and integral calculus seem unrelated. However,

the connection between the two was discovered independently by Sir Isaac Newton and

Gottfried Leibniz. This connection is stated in a theorem appropriately named The

Fundamental Theorem of Calculus.

Theorem – The Fundamental Theorem of Calculus

If a function f is continuous on the closed interval ],[ ba and F is the anti-derivative of f on ],[ ba ,

then:

)()()( aFbFdxxf

b

a

Guidelines for Using the Fundamental Theorem of Calculus

Provided you can find an anti-derivative, you now have a way to evaluate a definite integral without

having to use the limit of a sum or geometric means.

When using the FTC, you use the following steps with the given notation:

)()()()( aFbFxFdxxfb

a

b

a

1) Find the anti-derivative of f

2) Evaluate the anti-derivative function at the two bounds

3) Find the difference of the upper bound and the lower bound

It is not necessary to use a constant of integration C in this process

Reason:

42

Ex. 1: Evaluate the definite integral using the Fundamental Theorem of Calculus and then verify by finding

the area under the curve.

a) 3

0

4xdx

b)

2

1

2dx

Ex. 2: Evaluate the following definite integrals:

a)

2

1

3 )3( dxx

b)

4

1

2dt

tet

c)

4

0

2 cossec

d

d)

4

1

2

2

3

24 23dx

x

xxx

43

Definition of the Average Value of a Function on an Interval

If f is integrable on the closed interval ],[ ba , then:

The Average Value of f is:

b

a

dxxfab

)(1

Ex. 3: Find the average value of xxxf 23)( 2 over the interval ]4,1[ :

Ex. 4: A ball is fired from a cannon and its motion can be modeled by the function, 104816)( 2 ttts .

Find the average velocity of the projectile over the interval ]3,1[ using the following methods:

a) The average velocity from the position function:

b) The average velocity from the velocity function:

44

Definite Integrals Worksheet Name:

Evaluate each of the following definite integrals- without using calculators!

1) 3

2

0(3 4 1)x x dx 2)

62

3( 2 )x x dx

3) 2

2

21

1xdx

x

4)

4

1(2 )x x dx

5) 4

4

32

w wdw

w

6)

43 2

0( 1)x x dx

45

7) 3

2

1(3 5 1)x x dx

8)

2

1( 1)(2 3)x x dx

9) 2

0sin x dx

10) 0

(2sin 3cos 1)x x dx

The graph of f is shown below. The function g is defined as

x

dttfxg3

)()( . Evaluate the following:

11) )1(g

12) )3(g

13) )6(g

14) )3(g

46

The Second Fundamental Theorem of Calculus Investigation

1. Let 32)( ttf .

a) Find 1

(2 3)

x

t dt b) Find 1

(2 3)

xd

t dtdx

2. Let 563)( 2 tttg

a) Find 3

( )

x

g t dt b) Find 3

( )

xd

g t dtdx

Do you see a pattern? Use it to find ( )

x

a

dj t dt

dx

(where a is a constant)

3. Let ttk cos)(

a) Find

22

6

( )

x

k t dt b) Find

22

6

( )

xd

k t dtdx

4. Let ( ) tl t e

a) Find cos

3

( )

x

l t dt b) Find cos

3

( )

xd

l t dtdx

Do you see another pattern? Use it to find ( )

( )

g x

a

df t dt

dx

(where a is a constant

47

Theorem – The Second Fundamental Theorem of Calculus

If f is continuous on an open interval, then for every x in the interval , ')()( uufdttfdx

du

a

Ex. 5: Evaluate

23

3

2

x

dttdx

d

Ex. 6: Find the derivative of

3

2

cos)(

x

tdtxF

Ex. 7:

x

dttxF

2

2

2 52)(

a) (1)F = b) (1)F c) (1)F

48

Theorem – Net Change Theorem

The definite integral of the rate of change of a quantity )(' xF gives the total, or net change, in that quantity

on the interval ],[ ba .

)()()(' aFbFdxxF

b

a

Ex. 8: A chemical flows into a storage tank at a rate of t3180 liters per minute at time t (in min), where

600 t . Find the amount of the chemical that flows into the tank during the first 20 minutes.

Ex. 9: Given that )(xf is the anti-derivative of )(xF , 3)2( f and 353arctan2)( xxxF , find )5(f .

Ex. 10: Mr. Gough is baking cookies for his favorite Calculus class at a temperature of 350 degrees

Fahrenheit. He then takes out the cookies and turns off the oven ( 0t minutes). The temperature

of the oven is changing at a rate of te 4.0110 degrees Fahrenheit per minute. To the nearest degree,

what is the temperature of the oven at time 5t minutes?

49

Calculator problem – AP BC 2002 #2

Ex. 11: The rate at which people enter an amusement park on a given day is modeled by the function ( )E t and the

rate at which people leave the park on the same day is modeled by the function ( )L t shown below.

2

15600( )

( 24 160)E t

t t

2

9890( )

( 38 370)L t

t t

Both ( )E t and ( )L t are measured in people per hour and t is in hours after midnight. These functions are

valid for 9 23t , the hours in which the park is open. At 9t , there are no people in the park.

a) How many people have entered the park by 5:00 P.M. ( 17t )? Round your answer to the nearest whole

number.

b) The price of admission to the park is $15 until 5:00 P.M. ( 17t ). After 5:00 P.M., the price of admission to

the park is $11. How many dollars are collected from admissions to the park on the given day? Round

your answer to the nearest whole number.

c) Let

9( ) ( ( ) ( ))

tH t E x L x dx for 9 23t . The value of (17)H to the nearest whole number is 3725.

Find the value of (17)H and explain the meaning of (17)H and (17)H in the context of the park.

d) At what time t , for 9 23t , does the model predict that the number of people in the park is a maximum?

50

graph of f

graph of f

Functions Defined by Integrals

Ex. 1: Let 1

( ) ( ) , 1 4

x

F x f t dt x

, where f is the function graphed.

a) Complete the table of values for F.

x -1 0 1 2 3 4

F(x)

b) Sketch a graph of F.

c) Where is F increasing? Why?

Ex. 2: Let 3

( ) ( ) , 3 4

x

A x f t dt x

, where f is graphed.

a) Which is larger, )1(A or )1(A ? Justify.

b) Which is larger, )2(A or )4(A ? Justify.

c) Where is A increasing? Justify.

d) Does A have a relative minimum, relative maximum or neither at x = 1. Justify your answer.

51

Particle Motion

Ex. 3: A particle is moving along a line such that its velocity can be modeled by tetttv 764)( 23 .

a) Find the position of the particle at time 3t given the particle is at 13)0( s

b) Find the total distance travelled by the particle over the first 3 seconds.

c) Find the average acceleration from the time interval ]5,3[ .

d) Find the average velocity from the time interval ]5,3[ .

52

graph of g

graph of f

Functions Defined by Integrals / Particle Motion - Classwork / Homework

1. Let 3

( ) ( ) , 3 3

x

G x g t dt x

, where g is the function graphed.

a) Put the following in increasing order before completing part b):

)3(,)1(,)1(,)3( GGGG

b) Complete the table of values for G.

x -3 -2 -1 0 1 2 3

G(x)

c) Sketch a graph of G.

d) Where is G increasing? Why?

2. Let 0

( ) ( ) , 2 10

x

B x f t dt x

a) Which is larger, B(1) or B(5)? Justify your

answer.

b) Where is B decreasing?

c) Determine where the absolute extrema of B occur on 2 10x . Justify your answer.

53

3. A particle moves along a coordinate axis. Its position at time t (sec) is 3

( ) ( )

t

s t f x dx feet, where the

graph of f is shown below as line segments and a semicircle.

a) What is the particle’s position at 0t ?

b) What is the particle’s position at 3t ?

c) What is the particle’s speed at 4t ?

d) Approximately when is the acceleration of the particle positive? Justify your answer.

e) At what time during the 1st 7 seconds does s have its smallest value? Justify your answer.

1 2 3 4 5 6 7–1 x

1

2

3

4

5

–1

–2

–3

y

54

4.

Three trains, A, B, and C each travel on a straight track for 0 16t hours. The graphs above, which consist

of line segments, show the velocities, in kilometers per hour, of trains A and B. The velocity of C is given by

the function 2( ) 8 0.25v t t t . Be sure to indicate units of measure for all answers.

a) Find the velocities of A and C at time t = 6 hours.

b) Find the accelerations of B and C at time t = 6 hours.

c) Find the positive difference between the total distance that A traveled and the total distance that B

traveled in 16 hours.

d) Find the total distance that C traveled in 16 hours.

55

AP Calculus I

Notes 5.5

Integration by Substitution

In this section, we look at how to integrate composite functions. The major technique involved is called

u-substitution. The objective is to know the few rules from before and rewrite the integrand to fit those

rules. The role of substitution is comparable to the role of The Chain Rule in differentiation.

Theorem – U-Substitution Integration

Let g and f be a function that is continuous and differentiable on an interval I . If F is an anti-derivative

of f on I , then,

CxgFdxxgxgf ))(()('))((

If )(xgu , then dxxgdu )(' and CuFduuf )()(

Ex. 1: The integrand in each of the following integrals fits the pattern )('))(( xgxgf . Identify the

pattern and use the result to evaluate the integral. Then check through differentiation.

a) dxxx 42 )1(2 b) dxxx )3(tansec2

c) dxe x33 d) dxx)5cos(5

56

For Example 1, the integrands fit the )('))(( xgxgf pattern exactly – you only had to recognize the pattern.

You can extend this technique (if it doesn’t fit perfectly) with the Constant Multiple Rule:

dxxfkdxxkf )()(

Ex. 2: Evaluate the following indefinite integrals:

a) dxxx 22 )1( b) d)4tan()4sec(7

c) dxx

x

22)21(

5 d) dxexe xx )1)((

57

e) xdxx tansec2 f) xdxxsincos3 2

g)

dxxx5 ln6

4 h) xdxx 3cos3sin2

58

Ex. 3: Evaluate dxx 12

Sometimes you have to get a little creative and do some rewriting.

Ex. 4: Evaluate dxe xe x

Ex. 5: Evaluate dxxx 12

59

Definite Integrals

When it comes to definite integrals, you can still integrate the function using any technique and then just use

the Fundamental Theorem of Calculus to evaluate the bounds. Another method is to rewrite the definite

integral limits when you do the u-substitution.

Ex. 6: Evaluate the following definite integrals:

a)

5

1 110x

dx b)

3

1

2

3

dyy

e y

60

c) 3

1

6

1

tansec6 dtttt d)

0

2 )sin( dxxx

61

Ex. 7: A particle is moving along the x-axis for all 0t and has an acceleration modeled by the function 3( ) 8(2 1)a t t . The particle has an initial velocity of (0) 3v and starts at the origin.

a) Find the velocity function ( )v t for any time 0t .

b) Find the position function ( )x t for any time 0t .

c) Find the average velocity of the particle from [0,1] .

d) Find the total distance travelled by the particle from [0,2] .

62

AP Calculus

Notes 5.7

The Natural Logarithmic Function and Integration

The differentiation rules

1) 1

ln d

xdx x

2) '

ln d u

udx u

produces the following integration rules:

Theorem – Log Rule for Integration

Let u be a differentiable function of x .

1) dxx

1 2)

u

dudu

u

1

Ex. 1: Evaluate the following integrals:

a) 2 dx

x b)

1

4 1 dxx

Ex. 2: Find the area bounded by the graph of 2 1

xy

x, the x-axis and the lines 0x and 3x .

63

Ex. 3: Recognizing the Quotient Forms of the Log Rule

a) 2

1

2

x

dxx x

b) xxdx

dy

ln

1

c) 2sec

tanx

dxx

d)

1

01

dxe

ex

x

64

As we continue our study of integration, we will devote much time to integration techniques. To master these

techniques, you must recognize the “form-fitting” nature of integration. In this sense, integration is not nearly

as straightforward as differentiation.

Guidelines for Integration

1. Memorize a basic list of integration formulas (by the end of section 5.8, you will have 20 rules).

2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a

choice of u that will make the integrand conform to the formula.

3. If you cannot find a u -substitution that works, try altering the integrand. You might try a

trigonometric identity, multiplication and division by the same quantity or addition and subtraction of

the same quantity. Be creative, but PATIENT most of all.

Ex. 4: Evaluate xdxtan

Integrals of the Trigonometric Functions

udusin uducos

udutan uducot

udusec uducsc

udu2sec udu2csc

uduu tansec uduucotcsc

65

Ex. 5: Evaluate:

a) xdx3csc5 b) dxxxxx 2tansec2tansec

Integrals to which the Log Rule can be applied often appear in disguised form. For instance, if a rational

function has a numerator of degree greater than or equal to that of the denominator, divide!

Ex. 6: Evaluate 2

2

1

1

x x

dxx

Ex. 7: A population of bacteria is changing at a rate of tdt

dP

25.01

3000

where t is the time in days. The

initial population is 1000. Find the population of bacteria after 3 days.

66

AP Calculus I

Notes 5.8

Inverse Trigonometric Functions and Integration

The derivatives of the six inverse trigonometric functions fall into three pairs. In each pair, the derivative of

one function is the opposite of the other.

When listing the anti-derivative that corresponds to each of the inverse trigonometric functions, you need to

use only one member from each pair.

Theorem – Integrals Involving Inverse Trigonometric Functions

Let u be a differentiable function of x .

Ca

u

ua

du

arcsin

22

Ca

u

aua

du

arctan1

22

Ca

uarc

aauu

du

sec

1

22

Ex. 1: Integrate each of the following:

a) 24 x

dx b) 292 x

dx c) dx

xx

94

1

2

67

Unfortunately, integration is not usually straightforward. The inverse trigonometric integration formulas can

be disguised in many ways. Remember that REWRITING to a formula is the key to integration!

Ex. 2: Evaluate x

x

e

dxe

21

2

Ex. 3: Evaluate

1

024

4dx

x

x

68

Ex. 4: Evaluate 742 xx

dx

Ex. 5: Find the total distance travelled by a particle whose velocity is modeled by 2

2

3 4 12( )

4

t tv t

t

from

20 t .

69

Guidelines for Integration

1. Memorize a basic list of integration formulas (you now have 20 rules).

2. Find an integration formula that resembles all or part of the integrand, and, by trial and error,

find a choice of u that will make the integrand conform to the formula.

3. If you cannot find a u -substitution that works, try altering the integrand. You might try a

trigonometric identity, multiplication and division by the same quantity or addition and

subtraction of the same quantity. Be creative, but PATIENT most of all.

Ex. 6: Evaluate as many of the following integrals as you can using the formulas and techniques we have

studied so far. (Hint: 2 of the following cannot be integrated as of yet)

a) 12xx

dx b)

12x

xdx c)

12x

dx

d) xx

dx

ln e) dx

x

xln f) xdxln

70

AP Calculus 5.7 – 5.8 Review - No Calculators !

1) 1

20

1

4dx

x

2)

2

1

4 9dx

x

3) 21

x

x

edx

e

4)

2

2

sin

1 cos

xdx

x

5) 4

2

1

1

xdx

x

6) 2

arcsin

1

xdx

x

7) 2

3

3 1xdx

x x

8) 2

33

xdx

x

71

9) 2

3

6 7

xdx

x x

10) 4

0

5

3 1dx

x

11)

2

1

1 lne xdx

x

12)

2 1

xdx

x

13) sec tan

sec 1

x xdx

x 14) sec2

xdx

15) 2

1

1 cos

sind

16) 1

0

1

1

xdx

x

Chapter 1 to 4 - Quick Review Problems 1. . State why ... · 5) 1 2 1 '( ) x F x 6) F x 2 5 5 '( ) The function F is an anti-derivative of f on an interval I if F x f x'( ) ( ) for

Documents