Slide 1 / 91 Slide 2 / 91 AP Calculus Differential Equations 2015-11-23 www.njctl.org Slide 3 / 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 4 / 91 Indefinite Integrals Return to Table of Contents Slide 5 / 91 Indefinite Integrals do not have bounds. They will give you an initial value and have you find C. Example: Slide 6 / 91 If the gravity on Planet X is 10 ft/sec 2 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 7 / 91 Slide 8 / 91 Find f(5) if f(0) = 9 and 2 Slide 9 / 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?
Slide 10 / 91 Slope Fields Return to Table of Contents Slide 11 / 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 12 / 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 13 / 91 Example: , sketch the slope field and the curve if (0,1) is on y=f(x). Slide 14 / 91 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 Does the following slope field have a horizontal asymptote? If 4 so, where? A No horizontal asymptote y = 3 B x = 2 C y = -3 D
Slide 16 / 91 Does the following slope field have a vertical asymptote? If so, 5 where? A No vertical asymptote x = 0 B x = 2 C y = -3 D Slide 17 / 91 If (-1,0) is on y=f(x), which these other points could be on y? 6 A (-4,-1) (1,3) B (-8,-4) C D (1, -1) Slide 18 / 91 The family of graphs shown is for 7 A circle exponential B rational C quadratic D
Slide 19 / 91 The concavity of f(x) at (4,2) is 8 A positive B 0 negative C D undefined Slide 20 / 91 U-Substitution Return to Table of Contents Slide 21 / 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 22 / 91 Example: Slide 23 / 91 Example: Slide 24 / 91 Example:
Slide 25 / 91 Example: I recalled the derivative of tan was sec 2 , thought this would make u and du easier to find. Slide 26 / 91 Given what should u = ? 9 A B C D Slide 27 / 91 Given what should du = ? 10 A B C D
Slide 28 / 91 Given following u-substitutions is 11 A B C D Slide 29 / 91 12 A B C D Slide 30 / 91 Given 13 A B C D
Slide 31 / 91 Given what should u = ? 14 A B C D Slide 32 / 91 Given what should du = ? 15 A B C D Slide 33 / 91 Given what should 8xdx = ? 16 A B C D
Slide 34 / 91 Given following u-substitutions is 17 A B C D Slide 35 / 91 18 A B C D Slide 36 / 91 Given 19 A B C D
Slide 37 / 91 what should u = ? Given 20 A B C D Slide 38 / 91 Given what should du = ? 21 A B C D Slide 39 / 91 Given following u-substitutions is 22 A B C D
Slide 40 / 91 23 A B C D Slide 41 / 91 U-Substitution & Definite Integrals Return to Table of Contents 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. Once bounds are converted, x is not used again.
Slide 43 / 91 Example: Slide 44 / 91 what should u = ? 24 Given A B C D Slide 45 / 91 Given what should du = ? 25 A B C D
Slide 46 / 91 26 The lower bound for becomes what for u-substitution? Slide 47 / 91 27 The upper bound for becomes what for u-substitution? Slide 48 / 91 Given following u-substitutions is 28 A B C D
Slide 49 / 91 29 Slide 50 / 91 Differential Equations (Separable First Order) Return to Table of Contents Slide 51 / 91 ? Why does lead to
Slide 52 / 91 This is why it is called separable. Note: Why? An unknown constant minus another unknown constant is still a constant. Slide 53 / 91 Separation of variables is used to integrate implicit differentiation. and find y= 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. Slide 54 / 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 + or -.
Slide 55 / 91 and y(4)=0, separate the variables. 30 A B C D Slide 56 / 91 and y(4)=0, the antiderivative is 31 A B C D Slide 57 / 91 32 and y(4)=0, C=
Slide 58 / 91 and y(0)=3, separate the variables. 33 A B C D Slide 59 / 91 and y(0)=3, y= 34 A B C D Slide 60 / 91 35 and y(0)=3, C=
Slide 61 / 91 y(0)=3, find y. Example: Since an initial value is given, it can be determined whether + or - is used. But which? Slide 62 / 91 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. Slide 63 / 91
Slide 64 / 91 Integration by Parts Return to Table of Contents Slide 65 / 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 66 / 91
Slide 67 / 91 Example: Make u the function that reduces when it is derived. Slide 68 / 91 Example: Whenever there is ln(x) in an integration, that is the u. Slide 69 / 91 36 Consider the following integration by parts problem: what should u= ? A B C D
Slide 70 / 91 37 Consider the following integration by parts problem: what should du= ? A B C D Slide 71 / 91 38 Consider the following integration by parts problem: what should dv= ? A B C D Slide 72 / 91 39 Consider the following integration by parts problem: what should v= ? A B C 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 u dv x 3 e 3x 3x 2 (1/3)e 3x 6x (1/9)e 3x 6 (1/27)e 3x 0 (1/81)e 3x 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. We've now done integration by parts twice and we've gotten back to the same integral we started with. Now use algebra.
Slide 79 / 91 Slide 80 / 91 Population Growth Return to Table of Contents Slide 81 / 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 82 / 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 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 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 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. 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 of 9 in the first 10 days Slide 87 / 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.
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