Chapter 6: Messy Integrals
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1 Chapter 6: Messy Integrals Review: Solve the following integrals x 4 sec x tan x 0 0 Find the average value of 3 1 x 3 3 Evaluate ( x 1), then find the area of ( x 1) 4 Section 6.1: Slope Fields and Euler s Method Objectives: Solving differential equations, Using slope fields, and Euler s Method! Differential Equations Definition: An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation. What is a Differential Equation? Really? This is a differential Equation. y + y = 3y It has lots going on, and it MANY equations that could satisfy it. *Verify that y = e x and y = e -3x are TWO of the many solutions to this diff- e.q.
2 Solutions to Differential Equations: Differential Equations vs. Classic Algebraic Equation Ex. 1: Find all functions y that satisfy dy/ = sec x + x + 5 Ex. : Find the particular solution to the equation dy/ = e x 6x whose graph passes through the point (1, 0). Ex. 3: Find the particular solution to the equation dy/ = x sec x whose graph passes through the point (0, 3). Ex. 4: Find the solution to the differential equation x f '( x) e for which f(7) = 3.
3 Ex. 5: Graph the family of functions that solve the differential equation dy/ = cos x. Slope Fields Ex. 6: Construct a slope field for the differential equation dy/ = cos x. Ex. 7: Use a calculator to construct a slope field for the differential equation dy/ = x + y and sketch a graph of the particular solution that passes through the point (, 0). Hints: 1. The slopes are zero along the line:. The slopes are -1 along the line: 3. The slopes get steeper as increases.
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5 Euler s Method Euler s Method for Graphing a Solution to an Initial Value Problem 1. Begin at the point (x, y) specified by the initial condition. This point will be on the graph, as required.. Use the differential equation to find the slope (dy/) at the point. 3. Increase x by a small amount (). Increase y by a small amount (dy), where dy = (dy/). This defines a new point at (x +, y + dy) that lies along the linearization. 4. Using this new point, return to step. Repeating the process constructs the graph to the right of the initial point. 5. To construct the graph moving to the left from the initial point, repeat the process using negative values for. NOTE: Euler s Method does a better job of approximating the curve when the curve is nearly straight, as should be expected! Ex. 8: Let f be the function that satisfies the initial value problem in Ex 7 (dy/ = x + y with x = ). Use Euler s Method and increments of = 0. to approximate f(3). (x, y) dy/ = x + y dy (x +, y + dy) Ex 9: If dy/ = x y and if y = 3 when x =, use Euler s Method with five equal steps to approximate y when x = 1.5. (x, y) dy/ = x y dy (x +, y + dy) Euler s Method is ONE example of a numerical method for solving differential equations. The table of values is the numerical solution. Quick Review Section 6.
6 1. Evaluate the definite integrals: a. 4 5 x b. x Find dy/. a. y = x 3 t dt b. y = x 3 t dt 0 c. y = (x 3 x + 3) 4 d. y = sin (4x 5) e. y = ln (cos x) f. y = ln (sin x) g. y = ln (sec x + tan x) h. y = ln (csc x + cot x) Section 6.: Antidifferentiation by Substitution
7 Objectives: Indefinite Integrals, Leibniz notation and antiderivatives, Substitution in indefinite integrals, substitution in definite integrals. Indefinite Integrals Definition: The family of ALL antiderivatives of a function is the indefinite integral of f with respect to x and is denoted by f ( x). Definition: If F is any function such that F' ( x) f ( x), then f ( x) F( x) C, where C is an arbitrary constant, called the constant of integration. Ex. 1: Evaluate ( x sin x) Properties of Indefinite Integrals kf ( x) k f ( x) for any constant k. f ( x) g( x) f ( x) g( x) Power Formulas n1 n u u du C when n is not equal to -1. n u du du ln u C u Trigonometric Formulas cos udu sin udu sec udu csc udu sec u tanudu csc u cot udu Exponential and Logarithmic Formulas u u e du e C u u a a du C ln a ln udu u ln u u C log a udu u ln u u C ln a Leibniz Notation and Antiderivatives
8 Ex. : Let f(x) = x and let u = x. Find each of the following antiderivatives in terms of x: (a) f ( x) (b) f ( u) du (c) f ( u) Substitution in Indefinite Integrals The Substitution Method: 1. Substitute u = g(x) and du = g (x) to obtain the integral f ( u) du. Integrate with respect to u. 3. Replace u with g(x) in the result. Ex. 3: Evaluate sin xe cos x Ex. 4: 3 Evaluate x 5 x Ex. 5:
9 Evaluate cot 7x Ex. 6: Evaluate cos x Ex. 7: Evaluate cos 3 x Substitution in Definite Integrals Ex. 8: Evaluate 3 tan x sec 0 x
10 Let s Recap. Slope Fields, Section 6.1 Construct a slope field for the differential equation dy/ = x y. Euler s Method, Section 6.1 If dy/ = e x x and if y = when x = 1, use Euler s Method with five equal steps to approximate y when x = 1.5. (x, y) dy/ = x y dy (x +, y + dy) Section 6.: Antidifferentiation by Substitution a) Evaluate cos xe sin x (ln x) b) Evaluate 4 x 6 c) Evaluate the integral ( e x ). 1 x d) Solve the initial value problem: dy/ = x (x + ) when x = 0 and y =.
11 Section 6.4: Separable Differential Equations with Exponential Change Objectives: Separable Differential Equations, Using the Law of Exponential Change, Evaluating Compounded Interest, Investigating Radioactivity and Newton s Law of Cooling Separable Differential Equations Definition: A differential equation of the form dy/ = f(y)g(x) is called separable. We separate the 1 variables by writing it in the form: dy g( x). The solution is found by antidifferentiating f ( y) each side with respect to its thusly isolated variable. Ex. 1: Solve for y if dy/ = (xy) and y = 1 when x = 1. Ex. : Solve for y if dy/ = e x-y.
12 Law of Exponential Change Differential equation: dy/dt = ky Initial condition: y y0 when t = 0. Separate the variables and integrate: Law of Exponential Change: If y changes at a rate proportional to the amount present and y y0 kt when t = 0, then y y0e, where k > 0 represents growth and k < 0 represents decay. The number k is the rate constant of the equation. Ex. 3: Use the fact that the world population was 560 million people in 1950 and 3040 million in 1960 to model the population of the world in the second half of the 0th century. (Assume that the growth rate is proportional to the population size.) What is the relative growth rate k? Use the model to estimate the world population in 1993 and to predict the population in the year 00. Ex. 4: At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.0, what will be the population after 5 hours? How long will it take for the population to double?
13 Section 6.5: Partial Fractions Objectives: Fitting data to exponential models, Partial Fractions, Using the Logistic Growth Model, Fitting data to a logistic regression line Partial Fractions P( x) Definition: If f(x) =, where P and Q are polynomials with the degree of P less than the degree of Q( x) Q, and if Q(x) can be written as the product of distinct linear factors, then f(x) can be written as a sum of rational functions with distinct linear denominators. 1. x 1 x 4x A x B x 4. x 16 x x 6 A B x 3 x Ex. 1: Write the function x 13 f ( x) as a sum of rational functions with linear denominators. x 7x 3
14 Ex. : 4 3x 1 Find x 1 Ex. 3: This example will be our most laborious problem. Find the general solution to dy 6x 8x 4. ( x 4)( x 1)
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