Slide 1 / 91 Slide 2 / 91 AP Calculus Differential Equations 2015-11-23 www.njctl.org Slide 3 / 91 Slide 4 / 91 Table of Contents Indefinite Integrals Slope Fields U-Substitution U-Substitution & Definite Integrals Differential Equations (Separable First Order) Indefinite Integrals Integration by Parts Population Growth Return to Table of Contents Slide 5 / 91 Slide 6 / 91 Indefinite Integrals do not have bounds. If the gravity on Planet X is 10 ft/sec 2 and a rock is They will give you an initial value and have you thrown upward from the top of a 20' building. If the rock find C. was thrown with a velocity of 5 ft/sec, when will it hit the ground? Example: 2.562 seconds to hit the ground.
Slide 7 / 91 Slide 8 / 91 Find f(5) if f(0) = 9 and 2 Slide 9 / 91 Slide 10 / 91 If the gravity on Planet Y is 12 ft/sec 2 and a rock is thrown 3 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? Slope Fields Return to Table of Contents Slide 11 / 91 Slide 12 / 91 Now that we have our slope field, we A Slope Field is of graph the slopes of an equation at can find a particular function. specific points. Sketch the curve (0,1) is on Given: , sketch the slope field. y=f(x) Remember that Graph the point given and then is the slope. smoothly flow from dash to dash. So substituting the order pair into the equation will give you It helps if you can integrate the slope at that point. and know what the graph should look like. Each dash has the slope of y at that point.
Slide 13 / 91 Slide 14 / 91 Example: , sketch the slope field and the curve if (0,1) is on y=f(x). Q:Why are there so many dashed lines? A: Because of the unknown constant in an indefinite integral. The slope field shows all of the family of a graph. Slide 15 / 91 Slide 16 / 91 Does the following slope field have a horizontal asymptote? If Does the following slope field have a vertical asymptote? If so, 4 5 so, where? where? A No horizontal asymptote A No vertical asymptote y = 3 x = 0 B B x = 2 x = 2 C C y = -3 y = -3 D D Slide 17 / 91 Slide 18 / 91 If (-1,0) is on y=f(x), which these other points could be on y? The family of graphs shown is for 6 7 A (-4,-1) A circle (1,3) exponential B B (-8,-4) rational C C D (1, -1) D quadratic
Slide 19 / 91 Slide 20 / 91 The concavity of f(x) at (4,2) is 8 positive A B 0 negative U-Substitution C D undefined Return to Table of Contents Slide 21 / 91 Slide 22 / 91 U-substitution is used to find the antiderivative of the chain rule. Example: 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 23 / 91 Slide 24 / 91 Example: Example:
Slide 25 / 91 Slide 26 / 91 Example: Given what should u = ? 9 A I recalled the derivative of tan was B sec 2 , thought this would make u and du easier to find. C D Slide 27 / 91 Slide 28 / 91 Given what should du = ? Given following u-substitutions is 10 11 A A B B C C D D Slide 29 / 91 Slide 30 / 91 Given 12 13 A A B B C C D D
Slide 31 / 91 Slide 32 / 91 Given what should u = ? Given what should du = ? 14 15 A A B B C C D D Slide 33 / 91 Slide 34 / 91 Given what should 8xdx = ? Given following u-substitutions is 16 17 A A B B C C D D Slide 35 / 91 Slide 36 / 91 Given 18 19 A A B B C C D D
Slide 37 / 91 Slide 38 / 91 what should u = ? Given Given what should du = ? 20 21 A A B B C C D D Slide 39 / 91 Slide 40 / 91 Given following u-substitutions is 22 23 A A B B C C D D Slide 41 / 91 Slide 42 / 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. U-Substitution & Definite Integrals Return to Table of Once bounds are converted, x is not Contents used again.
Slide 43 / 91 Slide 44 / 91 Example: what should u = ? 24 Given A B C D Slide 45 / 91 Slide 46 / 91 Given what should du = ? 25 26 The lower bound for becomes what for u-substitution? A B C D Slide 47 / 91 Slide 48 / 91 following u-substitutions is Given 28 27 The upper bound for becomes what for u-substitution? A B C D
Slide 49 / 91 Slide 50 / 91 29 Differential Equations (Separable First Order) Return to Table of Contents Slide 51 / 91 Slide 52 / 91 ? Why does lead to This is why it is called separable. Note: Why? An unknown constant minus another unknown constant is still a constant. Slide 53 / 91 Slide 54 / 91 Separation of variables is used to integrate implicit differentiation. Example: Find the general solution of the differential equation: and find y= Steps 1) Separate variables 2)Integrate both sides 3)Find C as soon as possible. 4)Sub in C if found 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 + 5)If the directions ask for y= or -. solve for y.
Slide 55 / 91 Slide 56 / 91 and y(4)=0, separate the variables. and y(4)=0, the antiderivative is 30 31 A A B B C C D D Slide 57 / 91 Slide 58 / 91 and y(0)=3, separate the variables. 32 and y(4)=0, C= 33 A B C D Slide 59 / 91 Slide 60 / 91 and y(0)=3, y= 34 35 and y(0)=3, C= A B C D
Slide 61 / 91 Slide 62 / 91 Example: y(0)=3, find y. Example: y(0)=3, find y. Since an initial value is given, it can be determined whether + or - is used. But which? Sub in initial or value and see which equation is true. Since an initial value is given, it can be determined whether + or - is used. But which? Slide 63 / 91 Slide 64 / 91 Integration by Parts Return to Table of Contents Slide 65 / 91 Slide 66 / 91 Integration by Parts is used when you have the product of 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 67 / 91 Slide 68 / 91 Example: Example: Whenever there is ln(x) in an integration, that is the u. Make u the function that reduces when it is derived. Slide 69 / 91 Slide 70 / 91 36 Consider the following integration by parts problem: 37 Consider the following integration by parts problem: what should u= ? what should du= ? A A B B C C D D Slide 71 / 91 Slide 72 / 91 38 Consider the following integration by parts problem: 39 Consider the following integration by parts problem: what should dv= ? what should v= ? A A B B C C D D
Slide 73 / 91 Slide 74 / 91 41 Consider the following integration by parts problem: A B C D Slide 75 / 91 Slide 76 / 91 Slide 77 / 91 Slide 78 / 91 In this case neither e x nor cos x will derive to zero. We will use the trig function as u and derive twice. u dv x 3 e 3x 3x 2 (1/3)e 3x 6x (1/9)e 3x We've now done integration by parts twice and we've gotten 6 (1/27)e 3x back to the same integral we started with. 0 (1/81)e 3x Now use algebra.
Slide 79 / 91 Slide 80 / 91 Population Growth Return to Table of Contents Slide 81 / 91 Slide 82 / 91 There are 4 types of Population Growth. There are 4 types of Population Growth. 1) Linear 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. This is direct variation so y= kt + C As opposed to indirectly: y= k/t +C Slide 83 / 91 Slide 84 / 91 There are 4 types of Population Growth. 4) Exponential Growth The amount of growth depends on population present.
Slide 85 / 91 Slide 86 / 91 If you recognize the model you can go to the Example: A bacteria population grows at y'=ky, where k is constant equation and skip the integration to get there. and y is current population. P(0)=1000 and population tripled in the first 5 days. Example: A bacteria population grows at y'=ky, where k is constant b. By what factor did the population increase in the first 10 days? and y is current population. P(0)=1000 and population tripled in the Using the equation from part a first 5 days. a) write an expression for y at any time t. Recognizing y'=ky as exponential use Population increased by a scale factor of 9 in the first 10 days Slide 87 / 91 Slide 88 / 91 If you don't recognize the growth model from the rate, seperate the Example: A bacteria population grows at y'=ky, where k is constant variables and integrate. and y is current population. P(0)=1000 and population tripled in the first 5 days. 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 c.How long will it take for the population to reach 6000? a. Find r in terms of t It will take 8.155 days for population to reach 6000. x Slide 89 / 91 Slide 90 / 91 Example: A sphere's volume increases at a rate proportional with the 42 A wolf population grows at a rate of increase that is reciprocal of its radius. At t=0, r=1 and at t=15, r=2 directly proportional to 800-P(t), where k is the constant of proportion. If p(2)=700, find k. b.Find when the Volume is 27 times its initial volume. initial volume: HINT What is the radius when V is 27 times greater? At what time does r=3? k=-.549 It takes 80 seconds for the volume to be 27 times the initial volume.
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