5.4 Fundamental Theorem of Calculus Calculus. Do you remember the Fundamental Theorem of Algebra? Just thought I'd ask
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1 5.4 FUNDAMENTAL THEOREM OF CALCULUS Do you remember the Funamental Theorem of Algebra? Just thought I' ask The Funamental Theorem of Calculus has two parts. These two parts tie together the concept of integration an ifferentiation an is regare by some to by the most important computational iscovery in the history of mathematics! Be honest! Deep own, you're not only really impresse, but you're fille with anticipation! I know it's har, but try to contain your ecitement! In 5.1, we saw how the area uner a velocity curve can be use to etermine istance travele. Using this same iea, complete the following eamples: Eample: A car is traveling so that its spee is never ecreasing uring a 1 secon interval. The spee at various moments in time is liste in the table below. Time (sec) Spee (ft/sec) a) Use the table to help eplain why the best lower estimate for the istance travele in the first secons is 6 feet. b) Use the table to help eplain why the best upper estimate for the istance travele in the first secons is 7 feet. c) Use the table to give the best lower estimate for the istance travele in the first 1 secons. : An answer of feet (which ignores some of the ata) is not correct. ) Use the table to give the best upper estimate for the istance travele in the first 1 secons. These sums of proucts that you have foun in c an are calle. e) If you choose the lower estimate for your approimation of how far the car travels, what is the maimum amount your approimation coul iffer from the eact istance? f) Choose spees to correspon with t = 1,, 5, 7, an 9 secons. Keep the nonecreasing nature of the above table an o not select the average of the consecutive spees. Fin new best upper an lower estimates for the istance travele for these 1 secons. Time (sec) Spee (ft/sec) g) Write an epression giving the ACTUAL istance this car travele in 1 secons, if it's velocity was v (t). 15
2 If you felt like we haven't one anything new yet, then goo! If you felt like this was the first time you've one anything like what we just i, then I hope you unerstan it. Eample: Suppose you have a car whose position is given by st ( ) = t + t+ where t is time in secons, an t 1. a) What is the position of the car at t = secons? b) What is the position of the car at t = 1 secons? c) What is the change in position of the car from time t = to time t = 1 secons? ) How oes this question relate to the previous page? The Funamental Theorem of Calculus [The Evaluation Part] If f is continuous at every point of [a, b], where F() is an antierivative of f (). b f ( ) = F( b) F( a), a ( that last phrase is actually the toughest part!) Eample: π Eample: ( 1+ cos) π Eample: 1 16
3 Using the evaluation part, we are going to evelop the concept of the other part of the Funamental Theorem of Calculus. You're book calls this Part 1, because it proves them in the opposite orer. Our goal here isn't really to prove the Funamental Theorem of Calculus, Part 1, but to unerstan how it works. First, a quick overview we are going to create a function that is efine as an integral then, using this function we are going to fin the erivative of this function thus tying the two concepts of calculus together forever!!! Keep in min that if we can efine a function as an integral an take a erivative, then we can answer all the same types of questions about increasing, ecreasing, concave up, concave own, an inflection points that we i earlier in the year. So, to see how it is possible to efine a function using an integral, consier the eamples below. Eample: The graph of f (t) given below has o symmetry aroun the point (, ). On the interval [, ], the graph is 1 4 symmetric with respect to the line t = 1. Also, f () t t=. f (t) 1 t OK now to efine the function as an integral. Eample: Let F ( ) = f ( t) t. a) Complete the following table: 1 4 F () b) Sketch your best estimate of the graph of F on the gri below. F () 17
4 Eample: Suppose g( ) = sin t t. What is g '( )? a) Evaluate the right sie of the equation. b) Take the erivative. Eample: Suppose g( ) = sin t t. What is g '( )? a) First how is this problem ifferent than the first one? b) Evaluate the right sie of the equation. c) Take the erivative. ) Di changing the lower limit from to matter? Eplain. Eample: Suppose g( ) = sin t t, where a represents any constant. What is g '( )? a a Eample: Suppose g( ) = sin t t, where a represents any constant. What is g '( )? 18
5 Eample: Let g( ) = 1+ t t. What is g '( )? 1 a) Why is this eample ifferent from the previous eamples? b) Suppose the antierivative of 1+ t is ht (). This means h' () t =. c) Using h (t), evaluate the right sie of the equation. ) Take the erivative. Eample: Try again with g( ) = 1+ t t. Fin g '( ). Eample: Fin sin 1 t t
6 Eample: Fin f () t t Eample: Fin f () t t sin Eample: Fin sin e t t Notice any patterns? 1
7 The Funamental Theorem of Calculus [Part #1 Simple] f () t t f ( ) = a In other wors, the Integral an the Derivative are just. Eample: ( ) t t t The Funamental Theorem of Calculus [Part 1 Etene] u ( ) f () t t f ( u( ) ) u' ( ) f ( v( ) ) v' ( ) = v ( ) 4 Eample: ( ) t t t Notecars from Section 5.4: Funamental Theorem of Calculus Part 1 (Etene); Funamental Theorem of Calculus Part (Evaluation) 11
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