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# Class Note for MATH 1314 at UH

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
Math 13 14 Antiderivatives So far in this course we have been interested in nding derivatives and in the applications of derivatives In this chapter we will look at the reverse process Here we will be given the answer and we ll have to nd the problem In other words if we are given a function and told that it is the derivative we ll want to nd the original function Antiderivatives Definition A function F is an antiderivative of f on interval I if F 39x f x for all x in I The process of nding an antiderivative is called antidifferentiation or nding an indefinite integral Example 1 Determine if F is an antiderivative of f if F x x3 x2 4x 1 and fx 3x2 2x4 Example 2 Suppose Fx x3 2 Gx x3 5 Hx x3 10 and Kx x3 27 If f x 3x2 show that each of F G H and K is an antiderivative and draw a conclusion Notation We will use the integral sign I to indicate integration antidifferentiation Problems will be written in the form I f x dx F x C This indicates that the inde nite integral of f x with respect to the variable x is F x C where F x is an antiderivative off Basic Rules Rule 1 The Inde nite Integral of a Constant jkdxkxc Example 3 J 9 dx Rule 2 The Power Rule n1 x Ixquotdx Cn 1 n1 Example 4 If dx Example 5 Example 6 I dx x3 Rule 3 The Inde nite Integral of a Constant Multiple of a Function Icfxdx 0quot fxdx Example 7 15x4 dx 3 Example 8 I4xzdx Example 9 J 58dx x Rule 4 The Sum Difference Rule Hfoc i gxdx j fxdx i jgmdx Example 10 J4x2 7x 3dx Rule 5 The Indefinite Integral of the Exponential Function Ie dx e C Example 11 Ze 3x5dx Rule 6 The Inde nite Integral of the Function f x l x 1 j dx1nxc x 0 x Example 12 I 6x 2 izjdx x x Applying the Rules 2 3 Example 13 Jde x 6 Example 14 dx V Example 15 Ixz 3 3 xx2 x3 Example 16 IN x7 26 dx x Differential Equations A differential equation is an equation that involves the derivative or differential of some function So if we write f 39x 3x 5 we have a differential equation We will be interested in solving these A solution of a differential equation is any function that satisfies the differential equation So for the example above fx gxz 5x 3 is a solution ofthe differential equation since f x 3x 5 The general solution of a differential equation is one which gives all of the solutions so the general solution for the example above will be fx gxz 5x C If we are given a point that lies on the function we can nd a particular solution that is we can nd C If we know that f 2 1 we can substitute this information into our general solution and solve for C f 2 1 is called an initial condition Initial Value Problems An initial value problem is a differential equation together with one or more initial conditions If we are given this information we can nd the function f by rst nding the general solution and then nding the value of C that satis es the initial condition Example 17 Solve the initial value problem f x 3x1 f3 2 Example 18 Solve the initial value problem f39x 6x2 9x 1 f 3 0 Example 19 Solve the initial value problem f39x 36 4x no 3 From this section you should be able to Explain what we mean by an antiderivative inde nite integral a differential equation and an initial value problem Determine if one function is an antiderivative of another function Use the basic rules to nd antiderivatives Simplify if necessary before applying the basic rules Solve initial value problems

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