chapter 1 to 4 - quick review problems 1. . state why ... · 5) 1 2 1 '( ) x f x 6) f x 2 5 5...
TRANSCRIPT
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]
= __________________________
Area under the curve from [1,3]
= __________________________
Area under the curve from [0,6]
= __________________________
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Area under the curve from [-0.5,4.5]
= __________________________
Area under the curve from [1,5]
= __________________________
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
the curve, and find the area.
Is this an over- or under-approximation of the actual?
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.
Is this an over- or under-approximation of the actual?
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
the curve, and find the area.
Is this an over- or under-approximation of the actual?
20
Graph:
Method 2: Divide the region into six rectangles, where the right endpoint of each rectangle comes just under
the curve, and find the area.
Is this an over- or under-approximation of the actual?
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
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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.
8. Approximate the area under the curve xy sin2 from x = 0 to x = 2
using eight right-endpoint
rectangles.
9. Approximate the area under the curve xy sin2 from x = 0 to x = 2
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.
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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:
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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.
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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