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

Info icon This preview shows pages 7–9. Sign up to view the full content.

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
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

* 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.
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern