Indefinite Integrals Return to Table of Contents Slide 5 / 91 - - PDF document

Slide 1 / 91

AP Calculus

Differential Equations

2015-11-23 www.njctl.org

Slide 2 / 91

Table of Contents Indefinite Integrals Slope Fields U-Substitution U-Substitution & Definite Integrals Differential Equations (Separable First Order) Integration by Parts Population Growth

Slide 3 / 91

Indefinite Integrals

Return to Table of Contents

Slide 4 / 91

Indefinite Integrals do not have bounds. They will give you an initial value and have you find C. Example:

Slide 5 / 91

If the gravity on Planet X is 10 ft/sec2 and a rock is thrown upward from the top of a 20' building. If the rock was thrown with a velocity of 5 ft/sec, when will it hit the ground? 2.562 seconds to hit the ground.

Slide 6 / 91

Slide 7 / 91

2 Find f(5) if f(0) = 9 and

Slide 8 / 91

3

If the gravity on Planet Y is 12 ft/sec2 and a rock is thrown upward from the top of a 30' building. If the rock was thrown with a velocity of 8 ft/sec, when will it hit the ground?

Slide 9 / 91

Slope Fields

Return to Table of Contents

Slide 10 / 91

A Slope Field is of graph the slopes of an equation at specific points. Given: , sketch the slope field. Remember that is the slope. So substituting the order pair into the equation will give you the slope at that point. Each dash has the slope of y at that point.

Slide 11 / 91

Now that we have our slope field, we can find a particular function. Sketch the curve (0,1) is on y=f(x) Graph the point given and then smoothly flow from dash to dash. It helps if you can integrate and know what the graph should look like.

Slide 12 / 91

Example: , sketch the slope field and the curve if (0,1) is on y=f(x).

Slide 13 / 91

Q:Why are there so many dashed lines? A: Because of the unknown constant in an indefinite integral. The slope field shows all

• f the family of a graph.

Slide 14 / 91

4 A B C D Does the following slope field have a horizontal asymptote? If so, where? No horizontal asymptote y = 3 x = 2 y = -3

Slide 15 / 91

5 A B C D Does the following slope field have a vertical asymptote? If so, where? No vertical asymptote x = 0 x = 2 y = -3

Slide 16 / 91

6 A B C D If (-1,0) is on y=f(x), which these other points could be on y? (-4,-1) (1,3) (-8,-4) (1, -1)

Slide 17 / 91

7 A B C D The family of graphs shown is for circle exponential rational quadratic

Slide 18 / 91

8 A B C D The concavity of f(x) at (4,2) is positive negative undefined

Slide 19 / 91

U-Substitution

Return to Table of Contents

Slide 20 / 91

U-substitution is used to find the antiderivative of the chain rule. Recall the chain rule: We took the derivative of the composite of functions starting with the outer one first. For u-substitution method we're going in reverse. We start with the inner most function and called it u. The find du/dx and make substitutions. The integral should be much easier to find.

Slide 21 / 91

Example:

Slide 22 / 91

Example:

Slide 23 / 91

Example:

Slide 24 / 91

Example: I recalled the derivative of tan was sec2, thought this would make u and du easier to find.

Slide 25 / 91

9 A B C D Given what should u = ?

Slide 26 / 91

10 A B C D Given what should du = ?

Slide 27 / 91

11 A B C D Given following u-substitutions is

Slide 28 / 91

12 A B C D

Slide 29 / 91

13 A B C D Given

Slide 30 / 91

14 A B C D Given what should u = ?

Slide 31 / 91

15 A B C D Given what should du = ?

Slide 32 / 91

16 A B C D Given what should 8xdx = ?

Slide 33 / 91

17 A B C D Given following u-substitutions is

Slide 34 / 91

18 A B C D

Slide 35 / 91

19 A B C D Given

Slide 36 / 91

20 A B C D Given what should u = ?

Slide 37 / 91

21 A B C D Given what should du = ?

Slide 38 / 91

22 A B C D Given following u-substitutions is

Slide 39 / 91

23 A B C D

Slide 40 / 91

U-Substitution & Definite Integrals

Return to Table of Contents

Slide 41 / 91

For the first step in the equation below to be equal to the second the bounds have to be rewritten in terms of u. Use u = x - 4, to convert bounds. Once bounds are converted, x is not used again.

Slide 42 / 91

Example:

Slide 43 / 91

24 A B C D Given what should u = ?

Slide 44 / 91

25 A B C D Given what should du = ?

Slide 45 / 91

26 The lower bound for becomes what for u-substitution?

Slide 46 / 91

27 The upper bound for becomes what for u-substitution?

Slide 47 / 91

28 A B C D Given following u-substitutions is

Slide 48 / 91

29

Slide 49 / 91

Differential Equations (Separable First Order)

Return to Table of Contents

Slide 50 / 91

Why does lead to ?

Slide 51 / 91

This is why it is called separable. Note: Why? An unknown constant minus another unknown constant is still a constant.

Slide 52 / 91

Separation of variables is used to integrate implicit differentiation. Steps 1) Separate variables 2)Integrate both sides 3)Find C as soon as possible. 4)Sub in C if found 5)If the directions ask for y= solve for y. and find y=

Slide 53 / 91

Example: Find the general solution of the differential equation: NOTE: Since there was not an initial value given we leave ±, had one been given C would have been found in line 3 and subbing inital value back in again line 6, would have decided +

• r -.

Slide 54 / 91

30 A B C D and y(4)=0, separate the variables.

Slide 55 / 91

31 A B C D and y(4)=0, the antiderivative is

Slide 56 / 91

32 and y(4)=0, C=

Slide 57 / 91

33 A B C D and y(0)=3, separate the variables.

Slide 58 / 91

34 A B C D and y(0)=3, y=

Slide 59 / 91

35 and y(0)=3, C=

Slide 60 / 91

Example: y(0)=3, find y. Since an initial value is given, it can be determined whether + or - is used. But which?

Slide 61 / 91

Example: y(0)=3, find y. Since an initial value is given, it can be determined whether + or - is used. But which?

• r

Sub in initial value and see which equation is true.

Slide 62 / 91 Slide 63 / 91

Integration by Parts

Return to Table of Contents

Slide 64 / 91

Integration by Parts is used when you have the product

• f 2 functions that you want to integrate.

The time to use it is when u-substitution doesn't work because the one function doesn't derive to the other.

Slide 65 / 91 Slide 66 / 91

Example: Make u the function that reduces when it is derived.

Slide 67 / 91

Example: Whenever there is ln(x) in an integration, that is the u.

Slide 68 / 91

36 Consider the following integration by parts problem: what should u= ? A B C D

Slide 69 / 91

37 Consider the following integration by parts problem: what should du= ? A B C D

Slide 70 / 91

38 Consider the following integration by parts problem: what should dv= ? A B C D

Slide 71 / 91

39 Consider the following integration by parts problem: what should v= ? A B C D

Slide 72 / 91

Slide 73 / 91

41 Consider the following integration by parts problem: A B C D

Slide 74 / 91 Slide 75 / 91

Slide 76 / 91

u dv x3 e3x 3x2 (1/3)e3x 6x (1/9)e3x 6 (1/27)e3x (1/81)e3x

Slide 77 / 91

In this case neither ex nor cos x will derive to zero. We will use the trig function as u and derive twice. We've now done integration by parts twice and we've gotten back to the same integral we started with. Now use algebra.

Slide 78 / 91

Slide 79 / 91

Population Growth

Return to Table of Contents

Slide 80 / 91

There are 4 types of Population Growth. 1) Linear Growth This is direct variation so y= kt + C As opposed to indirectly: y= k/t +C

Slide 81 / 91

There are 4 types of Population Growth. 2) Sinusoidal Growth Think of it as the population of a college town. Crests during the fall and is at a low over the summer.

Slide 82 / 91 Slide 83 / 91

There are 4 types of Population Growth. 4) Exponential Growth The amount of growth depends on population present.

Slide 84 / 91

If you recognize the model you can go to the equation and skip the integration to get there. Example: A bacteria population grows at y'=ky, where k is constant and y is current population. P(0)=1000 and population tripled in the first 5 days. a) write an expression for y at any time t. Recognizing y'=ky as exponential use

Slide 85 / 91

Example: A bacteria population grows at y'=ky, where k is constant and y is current population. P(0)=1000 and population tripled in the first 5 days.

• b. By what factor did the population increase in the first 10 days?

Using the equation from part a Population increased by a scale factor

• f 9 in the first 10 days

Slide 86 / 91

Example: A bacteria population grows at y'=ky, where k is constant and y is current population. P(0)=1000 and population tripled in the first 5 days. c.How long will it take for the population to reach 6000? It will take 8.155 days for population to reach 6000.

Slide 87 / 91

If you don't recognize the growth model from the rate, seperate the variables and integrate. Example: A sphere's volume increases at a rate proportional with the reciprocal of its radius. At t=0, r=1 and at t=15, r=2

• a. Find r in terms of t

x

Slide 88 / 91

Example: A sphere's volume increases at a rate proportional with the reciprocal of its radius. At t=0, r=1 and at t=15, r=2 b.Find when the Volume is 27 times its initial volume. initial volume: What is the radius when V is 27 times greater? At what time does r=3? It takes 80 seconds for the volume to be 27 times the initial volume.

Slide 89 / 91

42 A wolf population grows at a rate of increase that is directly proportional to 800-P(t), where k is the constant

• f proportion. If p(2)=700, find k.

HINT k=-.549

Slide 90 / 91

43 A wolf population grows at a rate of increase that is directly proportional to 800-P(t), where k is the constnat

• f proportion. If p(2)=700, find the limit of the population

as t goes to # .

Slide 91 / 91