AP Calculus Notes 5.1 Anti-Derivatives and Indefinite Integration

Chapter 1 to 4 - Quick Review Problems

1. Use a graphing calculator to graph f() x x2/3 . State why Rolle’s Theorem does not apply to f on the interval 1,1.

a) f is not continuous on b) is not defined on the entire interval c) ff( 1) (1) d) is not differentiable at x  0 e) Rolle’s Theorem does apply

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

–1 1 2 3 x a) Increasing ,1and 3,  ; decreasing (1, 3) –1

b) Increasing (0,2); decreasing ,0 and 2, –2

c) Increasing ,  –3 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

2 3. Given that f( x )  x  12 x  28 has a relative maximum at x  6 , 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 fx  be a polynomial function such that f ( 2) 5 , f ( 2) 0 , and f ( 2) 3 . 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

1 5. Use a graphing calculator to graph fx   . Use the graph to determine the open intervals (x  1)2 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 e) Concave upward: ; Concave downward:

6. Find all extrema, if any, in the interval 0,2  if f x sin x x . Write as ordered pairs.

7. A differentiable function has only one critical number: x 3. Identify the relative extrema of f if f  4 1 and f  21   .   2  

8. Find all points of inflection of the function f x 52 x45 x .

2 9. State why the Mean Value Theorem does not apply to the function fx   x 12 on the interval 3,0 . Give your answer as a complete sentence.

AP Calculus Notes 5.1 Anti-derivatives and Indefinite Integration

Exploration: For each of the following derivatives, find the original function f (x) .

1) F' ( x ) 2 x 2) F'2( x ) 6 x 3) F' ( x ) cos x

1  5 4) F'(x)  ex  3 5) F'(x)  6) F'(x)  1 x2 2  5x

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 f (x)  x2 , f (x)  x2  5, and f (x)  x2  e .

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

The constant C is called the constant of integration. The function represented by F is the general anti- derivative of f , and F() x x2 C is the general solution of the differential equation F' ( x ) 2 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

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

d If f()() x dx F x C then f()() x dx f x  dx 

Exploration:

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

a) f (x)  x2 b) f (x)  x3 c) f (x)  x4

So, the Power Rule for Integration is:  xndx  ______

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

a) (x 2) dx b)  dx c) (3x42 5 x x ) dx

Differentiation Formula Integration Formula d kx  dx d kf(x)  dx d  f (x)  g(x)  dx d xn   dx d sin x  dx d cos x  dx d tan x  dx d sec x  dx d csc x  dx d cot x  dx d ex   dx d a x   dx d ln x  dx

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

1 a) dx  x3

1 b) dx  2 x

c) 2t(3t 2  9t 1)dt

d) (t22 1) dt

x3  3 e) dx  x2

sin x  2  f) 3 x( x 4) dx g. dx h. sec2    csc2  d   cos2 x   

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 y  f (x) for one value of x . (This information is called an initial condition).

Ex. 3: The function y  3x2 1dx 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 F'(x)  ex and find the particular solution that satisfies the initial condition F(0)  3.

Ex. 5: Find the general solution of f "(x)  sin x  2 and find the particular solution that satisfies the initial condition f '(0)  3 and f ()  3.

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?

Ex. 7: A particle, starting at the origin, moves along the x-axis and it’s velocity is modeled by the equation v(t)  6t 2  30t  24 where t is in seconds and v(t) 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 6m/ s2 ?

d) When is the particle changing directions?

e) When is the particle furthest to the left?

Ex. 8: A missile is accelerating at a rate of 4t m/ s 2 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 1 acceleration equation a( t )  sin( t )  . If the bear’s velocity is 1 ft/s when t  0 … 3

a) Find the velocity equation.

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

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

AP Calculus I 5.1 Quiz Review

6 1. (6x2  7x  5)dx 2. 23x 4 x3 dx 3. dx    11x2

4. 2x (3 x 1) dx 5. (2x 5)2 dx 6.  x(x  3 x)dx

9x4  x2  5x 7. (3sec xtan x  4sin x)dx 8. 2csc2 x cos x dx 9. dx    3x

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

11. A particle moves along the x-axis with velocity given by v( t ) 3 t2 6 t . If the particle is at position x  2 at time t  0, what is the position of the particle at time t 1? [2008 AP MC#7]

12. A cannonball is shot upward from the ground with an initial velocity of 30m/ s . The acceleration is  9.8m/ s2 .

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?

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]

= ______

Area under the curve from [-0.5,4.5]

= ______

Area under the curve from [1,5]

= ______

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 y  25  x2 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?

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?

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?

EX 2: Suppose you have to find the area under the curve y  x3 1 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? 19

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?

Formulas ba Area Using Left Endpoints: AREA y0  y 1  y 2  y 3 ...  yn 1  n ba Area Using Right Endpoints: AREA y1  y 2  y 3  y 4 ...  yn  n

ba AREA y  y  y ...  y 1 3 5 2n 1 Area Using Midpoints: n 2 2 2 2

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

ba lim y0 y 1  y 2  y 3  ...  yn 1  and so on for each of these. n n ______

Problems

1. Approximate the area under the curve y  2sin(0.5x)  4 from x  0 to x  8 using:

4 3 2 1

1 2 3 4 5 6 7 8 9 x 4 left Riemann rectangles 4 right Riemann rectangles 4 midpoint Riemann rectangles

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

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

HW - LEA, REA, MID Approximations

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

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

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

4. Approximate the area under the curve y  2 x  3 from x = 4 to x = 9 using five left-endpoint rectangles.

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

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

 7. Approximate the area under the curve y  2sin x from x = 0 to x = using eight left-endpoint rectangles. 2

 8. Approximate the area under the curve y  2sin x from x = 0 to x = using eight right-endpoint 2 rectangles.

9. Approximate the area under the curve from x = 0 to x = using eight midpoint rectangles.

PVA – Problems

1) A silver dollar is dropped from the top of a building that is 1362 feet tall (acceleration is  32 ft / s2 )

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

b) Determine the average velocity on the interval 12, .

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 4/t km s2 from a position at rest in a silo 1 km below ground. How high above the ground will the missile be after 6 seconds.

3) A particle, starting at the origin, moves along the x-axis and it’s velocity is modeled by the equation v(t)  6t 2  30t  24 where t is in seconds and v(t) 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 6m/ s2 ?

d) When is the particle changing directions?

e) When is the particle furthest to the left?

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 = 9.8m ) s2

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 1 equation a( t )  sin( t )  . If the bear’s velocity is 1 ft/s when t  0 … 3

a) Find the velocity equation.

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

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

AP Calculus I

5.1 – 5.2 Quiz Review

1. Find the area under the curve y2 x x2 from x = 1 to x = 2 with n = 4 using LEA.

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

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

4. Find the area under the curve yx1 2 from x = 0 to x = 1 with n = 5 using LEA.

5. Evaluate (6x2  7x  5)dx 6. Evaluate  x(x  3 x)dx

9x4  x2  5x 7. Evaluate (3sec xtan x  4sin x)dx 8. Evaluate dx   3x

9. Find the original function f (x) given (4 x  2)dx and the condition f (4)  1.

10. A cannonball is shot upward from the ground with an initial velocity of 30m/ s . The acceleration is  9.8m/ s2 .

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?

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:

Trapezoid Rule:

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

Ex. 2: Approximate the area between the x-axis and f (x)  x2  2x from [1,3] using 5 trapezoids.

Ex. 3: Approximate the area between the x-axis and f (x) , found below, from [1,5] using 3 trapezoids. y 8

4 3 2 1

–2 –1 1 2 3 4 5 6 x

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.

29 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).

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.

31 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).

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.

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

Increasing Decreasing Concave Concave Type of UP DOWN Approx. LEA Under Over ------

REA Over Under ------TRAP ------Over Under

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: ______

5.1 – 5.2, 5.6 Quiz Review

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

1. y2 x x2 from [1,2] with n = 4 using LEA. 2. yx from [0,8] with n = 4 using REA.

3. yx3 2 from [2,10] with n = 4 using MPA. 4. y  16  x2 from [2,4] with n = 5 using TZA.

5. (6x2  7x  5ex )dx 6.  x(x  3 x)dx 7. (3sec xtan x  4sin x)dx

9x4  x2  5x sin x  3csc x  5  8. dx 9. dx 10. 4x  34 x5 dx  3x  tan x   x2 

4 11. Find the original function f (x) given f '(x)   2 and the condition f (4)  1. x

12. A cannonball is shot up from the ground with a velocity of 30m/ s . The acceleration is  9.8m/ s2 .

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 C(t) , in L/month, over various months, t , where t  0 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 from [0,12] using the Trapezoidal Rule with 4 subintervals. Describe the meaning of this answer.

b) Assuming is a function that is concave down everywhere, will the answer from a) be an over or under estimate?

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

3 2  4dx ò 4 - x2 dx a) 1 b) -2

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 ______.

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

4 6 a)  (x  4)dx b)   x  4  2dx 1 0

Definition of Two Special Definite Integrals

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

ab 2. If f is integrable on [a, b], then f()() x dx f x dx ba

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 fxdx    fxdx   fxdx  a a c

1 1 0 Ex. 3: Given  f( x ) dx  3 and  f( x ) dx  5, find  f() x dx 1 0 1

Worksheet on Definite Integrals

y 6 The graph of f(x) shown at the 5 right is formed by lines and a

4 semi-circle.

2 Use the graph to evaluate the

1 definite integrals below.

–4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10 x –1 –2

4 ( ) 1 ( ) 3 ( ) 1. ∫4 푓 푡 푑푡 2. ∫0 푓 푡 푑푡 3. ∫1 푓 푡 푑푡

3 ( ) 6 ( ) 3 ( ) 4. ∫0 푓 푡 푑푡 5. ∫3 푓 푡 푑푡 6. ∫6 푓 푡 푑푡

6 ( ) 10 ( ) 6 ( ) 7. ∫0 푓 푡 푑푡 8. ∫6 푓 푡 푑푡 9. ∫10 푓 푡 푑푡

10 ( ) 0 ( ) 0 ( ) 10. ∫0 푓 푡 푑푡 11. ∫10 푓 푡 푑푡 12. ∫−1 푓 푡 푑푡

0 ( ) −3 ( ) −3 ( ) 13. ∫−3 푓 푡 푑푡 14. ∫0 푓 푡 푑푡 15. ∫−4 푓 푡 푑푡

0 ( ) 10 ( ) 10 ( ) 16. ∫−4 푓 푡 푑푡 17. ∫−4 푓 푡 푑푡 18. |∫0 푓 푡 푑푡|

5 5 5 8 푓(푥)푑푥 = 18 푔(푥)푑푥 = 5 ℎ(푥)푑푥 = −11 푓(푥)푑푥 = 0 Suppose: ∫−2 , ∫−2 , ∫−2 , and ∫−2 find

5 ( ) ( ) 5 ( ( ) ( ) ) 22. ∫−2(푓 푥 + 푔 푥 )푑푥 23. ∫−2 푓 푥 + 푔 푥 − ℎ(푥) 푑푥

−2 ( ) 5 ( ) 24. ∫5 4푔 푥 푑푥 25. ∫−2(푔 푥 + 2)푑푥

5 ( ) 7 ( ) 26. ∫−2(푓 푥 − 6)푑푥 26. ∫0 ℎ 푥 − 2 푑푥 40

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 [a,b] and F is the anti-derivative of on , then:

b  f (x)dx  F(b)  F(a) 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:

b b f (x)dx  F(x)  F(b)  F(a)  a a

1) Find the anti-derivative of 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:

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

3 4xdx  2 a) 0 b)  2dx 1

Ex. 2: Evaluate the following definite integrals:

2 4  2  (x3  3)dx et  dt   t a) 1 b) 1  

 4 2 3 sec   cos d 4     3x4  2x2  x 2  c) 0 d) dx  2  1  x   

Definition of the Average Value of a Function on an Interval

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

1 b The Average Value of is:  f (x)dx b  a a

Ex. 3: Find the average value of f (x)  3x2  2x over the interval [1,4] :

Ex. 4: A ball is fired from a cannon and its motion can be modeled by the function, s(t)  16t2  48t 10 . Find the average velocity of the projectile over the interval [1,3] using the following methods:

a) The average velocity from the position function:

b) The average velocity from the velocity function:

Definite Integrals Worksheet Name:

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

3 6 1) (3x2  4 x 1) dx 2) (x2  2 x ) dx 0 3

2 2 x 1 4 3) dx 4) x(2 x ) dx 1 x2 1

4 4 ww 4 5) dw 6) (x32 x 1) dx 2 w3 0

3 2 7) (3x2  5 x 1) dx 8) (x 1)(2 x 3) dx 1 1

  9) 2 sin x dx 10) (2sinx 3cos x 1) dx 0 0

x The graph of f is shown below. The function g is defined as g(x)   f (t)dt . Evaluate the following: 3 11) g(1)

12) g(3)

13) g(6)

14) g(3)

The Second Fundamental Theorem of Calculus Investigation

1. Let f (t)  2t  3. x d x a) Find (2t 3) dt b) Find (2t 3) dt  dx  1 1

2. Let g(t)  3t 2  6t  5 x d x a) Find g() t dt b) Find g() t dt  dx  3 3

d x Do you see a pattern? Use it to find j() t dt (where a is a constant) dx  a

3. Let k(t)  cost 2x2 2 d 2x a) Find k() t dt b) Find k() t dt  dx  6 6

4. Let l() t et cos x d cos x a) Find l() t dt b) Find l() t dt  dx  3 3

gx() d  Do you see another pattern? Use it to find f() t dt (where a is a constant dx  a

Theorem – The Second Fundamental Theorem of Calculus

d u  If f is continuous on an open interval, then for every x in the interval ,  f (t)dt  f (u) u' dx  a 

3x2 d  2  Ex. 5: Evaluate  t dt dx  3 

x3 Ex. 6: Find the derivative of F(x)  costdt  2

2x Ex. 7: F(x)  2t 2  5dt 2 a) F(1) = b) F(1)  c) F(1)

Theorem – Net Change Theorem

The definite integral of the rate of change of a quantity F'(x) gives the total, or net change, in that quantity on the interval [a,b]. b  F'(x)dx  F(b)  F(a) a

Ex. 8: A chemical flows into a storage tank at a rate of 180  3t liters per minute at time t (in min), where 0  t  60 . Find the amount of the chemical that flows into the tank during the first 20 minutes.

Ex. 9: Given that f (x) is the anti-derivative of F(x) , f (2)  3 and F(x)  2 arctan3x  5x3 , find f (5) .

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 ( t  0 minutes). The temperature of the oven is changing at a rate of 110e0.4t degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the oven at time t  5 minutes?

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 Et() and the rate at which people leave the park on the same day is modeled by the function Lt() shown below.

15600 9890 Et() Lt() (tt2  24 160) (tt2  38 370)

Both and are measured in people per hour and t is in hours after midnight. These functions are valid for 9t 23, the hours in which the park is open. At t  9 , there are no people in the park.

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

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

t c) Let H( t ) ( E ( x ) L ( x )) dx for 9t 23. The value of H(17) to the nearest whole number is 3725.  9 Find the value of H(17) and explain the meaning of H(17) and in the context of the park.

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

Functions Defined by Integrals

x Ex. 1: Let F( x ) f ( t ) dt ,  1  x  4 , where f is the function graphed. graph of f 1  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?

x graph of f Ex. 2: Let A( x ) f ( t ) dt ,  3  x  4 , where f is graphed.  3  a) Which is larger, A(1) or A(1) ? Justify. 

       

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

c) Where is A increasing? Justify.

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

Particle Motion

Ex. 3: A particle is moving along a line such that its velocity can be modeled by v(t)  4t3  6t 2  7et .

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

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

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

d) Find the average velocity from the time interval .

Functions Defined by Integrals / Particle Motion - Classwork / Homework

x 1. Let G( x ) g ( t ) dt ,  3  x  3, where g is the function graphed. 3 graph of g  a) Put the following in increasing order before completing part b):  G(3) , G(1) , G(1) , G(3) 



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?

x graph of f 2. Let B( x ) f ( t ) dt ,  2  x  10 0

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 x  10 . Justify your answer.

t 3. A particle moves along a coordinate axis. Its position at time t (sec) is s()() t  f x dx feet, where the 3 graph of f is shown below as line segments and a semicircle.

y a) What is the particle’s position at t  0? 5 4

b) What is the particle’s position at t  3? –1 1 2 3 4 5 6 7 x –1

–2 –3

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

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.

Three trains, A, B, and C each travel on a straight track for 0t 16 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 v( t ) 8 t 0.25 t 2 . 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.

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 on , then,

f (g(x))g'(x)dx  F(g(x))  C 

If u  g(x), then du  g'(x)dx and  f (u)du  F(u)  C

Ex. 1: The integrand in each of the following integrals fits the pattern f (g(x))g'(x) . Identify the pattern and use the result to evaluate the integral. Then check through differentiation.

2x(x2 1)4 dx sec2 x(tan x  3)dx a)  b) 

c) 3e3xdx d) 5cos(5x)dx

For Example 1, the integrands fit the f (g(x))g'(x) 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:

kf(x)dx  k f (x)dx

Ex. 2: Evaluate the following indefinite integrals:

a)  x(x2 1)2 dx b) 7sec(4)tan(4)d

 5x c) dx d) (ex  x)(ex 1)dx  (1 2x2 )2 

56 e) sec2 x tan xdx f) 3cos2 xsin xdx

4 g) dx h) sin2 3xcos3xdx  x5 6  ln x 

Ex. 3: Evaluate  2x 1dx

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

x Ex. 4: Evaluate ee  xdx

Ex. 5: Evaluate  x 2x 1dx

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:

3 5 dx 3 e y a) b) dy   2 1 10x 1 1 y

1 3  c) 6t  sect tantdt d)  xsin(x2   )dx 1 0 6

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

a) Find the velocity function vt() for any time .

b) Find the position function xt() for any time .

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

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

AP Calculus Notes 5.7 The Natural Logarithmic Function and Integration

The differentiation rules

d 1 du' 1) ln x   2) ln u   dx x dx u produces the following integration rules:

Theorem – Log Rule for Integration

Let u be a differentiable function of x .

1 1 du 1) dx  2) du    x  u  u

Ex. 1: Evaluate the following integrals:

2 1 a) dx b) dx  x  41x 

x Ex. 2: Find the area bounded by the graph of y  , the x-axis and the lines x  0 and x  3. x2 1

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

x 1 dy 1 a) dx b)   xx2  2 dx xln x

sec2 x 1 ex c) dx d) dx   x tan x 0 1 e

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 -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  tan xdx

Integrals of the Trigonometric Functions

sinudu  cosudu 

 tanudu  cot udu 

secudu  cscudu 

sec2 udu  csc2 udu 

secu tanudu  cscucot udu 

Ex. 5: Evaluate:

a)  5csc3xdx b)  sec x tan x  2sec x  tan 2xdx

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!

xx2 1 Ex. 6: Evaluate dx  x2 1

dP 3000 Ex. 7: A population of bacteria is changing at a rate of  where t is the time in days. The dt 1 0.25t initial population is 1000. Find the population of bacteria after 3 days.

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 .

du u  arcsin  C  2 2 a  u a

du 1 u  arctan  C  a2  u2 a a

du 1 u  arcsec  C  2 2 u u  a a a

Ex. 1: Integrate each of the following:

dx dx 1 a) b) c) dx  2 2  2 4  x  2  9x x 4x  9

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! 2exdx Ex. 2: Evaluate  2x 1 e

1 x  4 Ex. 3: Evaluate dx  2 0 4  x

dx Ex. 4: Evaluate  x2  4x  7

3tt2  4 12 Ex. 5: Find the total distance travelled by a particle whose velocity is modeled by vt() from t 2  4 0  t  2.

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.

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)

dx xdx dx a) b) c)  2  2  2 x x 1 x 1 x 1

dx ln x d) e) dx f) ln xdx  xln x  x 

AP Calculus 5.7 – 5.8 Review - No Calculators !

1 1 1 1) dx  2) dx   2  2 0 4  x 49 x

ex  sin x 3) dx  4) dx   2x  2 1 e  1 cos x 2

x4 1 arcsin x 5) dx  6) dx   2  2 x 1 1 x

31x2  x2 7) dx 8) dx  xx3   3  x3

x  3 4 5 9) dx 10) dx  2  xx67 0 31x 

2 e 1 ln x x 11) dx 12) dx   2 1 x x 1

secxx tan x 13) dx 14) secdx  secx  1  2

2 1 cos 1 x 1 15)  d 16)  dx 1  sin 0 x 1