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# Class Note for MATH 1314 at UH

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This 9 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 13 views.

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Date Created: 02/06/15

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 Jde Rule 2 The Power Rule n 1 x Ixquotdx Cn l n1 Example 4 Ix dx Example 5 dx Example 6 dx x2 Rule 3 The Inde nite Integral of a Constant Multiple of a Function Icfxdx 0J fxdx Example 7 I4x3dx Example 8 dx x Rule 4 The Sum Difference Rule 1 fx i gxdx j fxdxi goodx Example 9 J2x2 5x 1dx Rule 5 The Indefinite Integral of the Exponential Function Ie dx 6x C Example 10 156 4x3 dx Rule 6 The Indefinite Integral of the Function f x l x jldx1nxc x 0 x Example 11 I 3x 2 dx x x Applying the Rules 2 3 Example 12 dx x N Example 13 1x2 l isjdx x x x Example 14 10973 7er Babc x Differential Equations A differential equation is an equation that involves the derivative or differential of some function So if we write f x 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 f x 5x2 5x 3 1s a solutlon of the d1fferent1al equatlon s1nce the der1vat1ve off is 3x5 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 find a particular solution that is we can find C Ifwe know that f 2 1 we can substitute this information into our general solution and solve for C f 2 l 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 find the function f by first finding the general solution and then finding the value of C that satisfies the initial condition Example 15 Solve the initial value problem f x 2x 5 f23 Example 16 Solve the initial value problem f x 3e 2x f 0 7 Example 17 A cable television provider estimates that the number of its subscribers will grow at the 3 rate of 100 210t2new subscribers per month tmonths from the start date of the service Suppose 5000 subscribers signed up for the service before the start date How many subscribers will there be 16 months after the start date From this section you should be able to Explain what we mean by an antiderivative indefinite integral a differential equation and an initial value problem Determine if one function is an antiderivative of another function Use the basic rules to find antiderivatives Simplify if necessary before applying the basic rules Solve initial value problems Math 1314 Lesson 16 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 This process is generally called integration We can use integration to solve a variety of problems Antiderivatives De nition A function F is an antiderivative of f on interval I if F x 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 g gxz 2x 5 and fxx2 3x2 Example 2 Suppose Hx x3 10 and Kx x3 27 If fx 3x2 show that each ofH andK is an antiderivative off and draw a conclusion

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