The Mean Value Theorem for Integrals – Average ValueAverage Value of a FunctionSlide 3Slide 4Finding Where This HappensTry It OutApplicationAssignmentThe Mean Value Theorem for Integrals – Average ValueLesson 5.7Just as mean as beforeJust as mean as beforeAverage Value of a Function•Consider recording a temperature eachhour and taking an average•If we take it more often and take a limit …2411( )24tT f t==�11lim ( )niniT f tn��==�Average Value of a Function•Now apply the concept to a continuous functionf(x)ab11lim ( )ninib ay f xb a n��=-= �-�1( )baf x dxb a=-� is the height of the "box" which is equal to the area under the curve. is the height of the "box" which is equal to the area under the curve. yAverage Value of a Function•Find the average value of these functions( ) on [0,1]2 3xf xx=+2( ) 2 7 on [0,1]f x x x= � +Finding Where This Happens•This height willbe f(c) for somex = c (at least 1)•And we can solve for that value of cf(x)ab1( ) ( )bay f x dx f cb a= =-�c( ) ( ) ( )baf x dx a b f c= - ��Try It Out•Given f(x) = x2 + 4x +1 on [0, 2]•Find a value for c such that2204 1 (2 0) ( )x x dx f c+ + = - ��Application•Suppose a study comes up witha model that says t years fromnow CO2 in the area of a citywill be•What is the average level in the first 3 years?•At what point in time does that average level actually occur20.01( )tL t t e-= �Assignment•Lesson 5.7•Page 332•Exercises 1 – 35
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