Final Review Guide

# Isare we just want the values of x y the context of

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is(are), we just want the value(s) of ( x, y ). * The context of the problem is usually good enough to indicate what type of constrained extremum you will find. If you get two or more different values, pay attention to the what the question asks for: for example, if it asks for the constrained relative maximum, then you choose the largest of the values you obtained using the method. * Warning: DO NOT ATTEMPT TO USE THE SECOND DERIVATIVE TEST TO SOLVE A CON- STRAINED MAX/MIN PROBLEM. * Exercises: pp.583-584 1-16, 19.20 8.7 Double Integrals * See pp. 592-593 for a development of double integrals. * Let R be the rectangle defined by the inequalities a x b and c y d . Then, Z R Z f ( x, y ) dA = Z d c " Z b a f ( x, y ) dx # dy = Z b a " Z d c f ( x, y ) dy # dx * Suppose g 1 ( x ) and g 2 ( x ) are continuous functions on [ a, b ] and the region r is defined by R = { ( x, y ) | g 1 ( x ) y g 2 ( x ) , Then, Z R Z f ( x, y ) dA = Z b a " Z g 2 ( x ) g 1 ( x ) f ( x, y ) dy # dx * Suppose h 1 ( y ) and h 2 ( y ) are continuous functions on [ c, d ] and the region R is defined by R = { ( x, y ) | h 1 ( y ) x h 2 ( y ) Then, Z R Z f ( x, y ) dA = Z d c " Z h 2 ( y ) h 1 ( y ) f ( x, y ) dx # dy * See pp. 594-597 for examples of these three facts in action. * Exercises: pp.597-598 1-25 8.8 Applications of Double Integrals * Average Value of f ( x, y ) over a rectangle R given by a x b, c y d : R R R f ( x, y ) dA ( b - a )( d - c ) * Exercises: p.605 Find average value of function over given region in 9,11,17 Chapter 9 Differential Equations 9.1 Differential Equations * Terms: differential equaton, general solution, particular solution * Know how to verify that a given expression is a general solution to a given differential equation. 7

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* Know how to verify that a given expression is a particular solution to a given differential equation (possibly with an initial condition). * Exercises: 9.2 Separation of Variables * Terms: order, first-order equation, separable equation, initial value problem * The Method of Separation of Variables: Suppose we are given a first-order separable differential equation in the form dy dx = f ( x ) g ( y ) · Step 1: Write the above equation in the form dy g ( y ) = f ( x ) dx · Step 2: Integrate each side of the equation in step 1 with respect to the appropriate variable. * Don’t forget the arbitrary constant of integration in this process. * With an initial value problem, you use the given initial condition to find a particular value for the arbitrary constant. * Exercises: p.617 1-23 9.3 Applications of Separable Differential Equations * Unrestricted Growth Models have the following differential equation describing them: dQ dt = kQ where Q ( t ) represents the size of a certain population at time r and k is a positive constant. This has solution Q ( t ) = Q 0 e kt where Q 0 is the initial population at time t = 0. * Restricted Growth Models have the following differential equation describing a them: dQ dt = k ( C - Q ) where both k and C are positive constants. This has solution C - Q = Ae - kt . The constant A depends on the initial population.
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