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# Class Note for MATH 1314 with Professor Marks at UH

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This 6 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 21 views.

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Date Created: 02/06/15
Math 1314 Lesson 20 Evaluating De nite Integrals We will sometimes need these properties when computing de nite integrals Properties of De nite Integrals Suppose fand g are integrable functions Then 1 fxdx 0 2 j fxdx fxdx 3 fcfxdxc fxdx 4 flfxigx1dx jf xwxi I gmdx 5 f7 fxdx I fxdx f7 fxdx where a lt c lt b We will need to use substitution to evaluate some problems Example 1 Evaluate 1034xx2 35 dx Example 2 Evaluate xzexzdx 2 x dx Example 3 Evaluate J73 3 6 x Applications Example 4 A company purchases a new machine for which the rate of depreciation is given by 10000t 6 How much value is lost over the rst three years that the machine is in use Example 5 Suppose you are driving a car and that your velocity can be approximated by vt 2N 25 t2 where tis measured in seconds and v is measured in feet per second How far will you travel in the 5 seconds from t 0 to t 5 Example 6 The marginal daily pro t function associated with production and sales of a Video game is estimated to be P39x 00003x2 004x 17 where x is the number of units produced and sold daily and P39x is measured in dollars per unit Find the additional daily pro t realizable if production and sales is increased from 200 units per day to 300 units per day The Average Value ofa Function We can use the de nite integral to nd the average value of a function Suppose f is an integrable function on the interval a b Then the average value of f over 1 17 61 the interval is r f xdx This is what average value represents Example 7 Find the average value of f x J over the interval 1 16 Example 8 Find the average value of fx x2 3x 5 0n 2 5 Example 9 The sales of ABC Company in the rst I years of its operation is approximated by the function S t txl02t2 4 where S t is measured in millions of dollars What were the company s average annual sales over its rst ve years of operation From this lesson you should be able to Use the FTOC to compute de nite integrals including problems that require substitution De ne average value Use the FTOC to nd average value Solve word problems using FTOC

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