AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt computtionl iscovery in the history of mthemtics. Emple. In section 6. we foun n estimte for the istnce trvele by fining the re between velocity function n the -is. If v(t) is the velocity function (tht is bove the -is) n the time is from to secons, how might we use clculus to efine the istnce trvele? Emple. Suppose cr s position is given by t. A. Wht is the position of the cr t t = secons? s t = t + t + 5 where t is time in secons, n B. Wht is the position of the cr t t = secons? C. Wht is the chnge in position of the cr from t = to time t = secons? D. How oes this question relte to the previous emple? The Funmentl Theorem of Clculus Prt, The Evlution Prt b If f is continuous t every point of [, b], then = f F b F f(). This lst phrse is the toughest prt to unerstn., where F() is n ntierivtive of 6.4 Pge of 6
The rules below is list of ntierivtives you shoul become fmilir with. These rules re stte in the net chpter, but wht you know bout erivtives, you shoul be ble to mke these connections. Integrl Formuls. Power Rule for. Rule for. n when n : n when n = : e k k k e = + C 4. sin 5. cos cos( k) k = + C k sin ( k) k = + C k n+ n = + C n + = ln + C 6. sec tn = + C 7. csc cot = + C 8. sec tn sec = + C 9. csc cot csc. = + C = + C ln Emple. π = Emple 4. ( + ) π cos = Emple 5. 9 = Emple 6. f = 4 Emple 6 will be n etremely importnt concept throughout the rest of the semester. 6.4 Pge of 6
Using the Evlution Prt of the Funmentl Theorem of Clculus, we re going to evelop the concept of the Antierivtive Prt of the Funmentl Theorem of Clculus. Our gol here isn't relly to prove the Funmentl Theorem of Clculus, the Antierivtive Prt, but to unerstn how it works. Here is quick overview:. We re going to crete function tht is efine s n integrl, then,. Using this function, we re going to fin the erivtive of this function; thus, tying the two concepts of clculus together forever. Keep in min tht if we cn efine function s n integrl n tke erivtive, then we cn nswer ll the sme types of questions bout incresing, ecresing, concve up, concve own, n inflection points tht we i erlier in the yer; you hven t forgotten ll those resons, hve you? Step #: So, to see how it is possible to efine function using n integrl, consier the emples below. The grph of f(t) given below hs o symmetry n is perioic (with perio = ). Also = f t t. Emple 7. Let F Complete the following tble. y 4 f t t =. F() - Emple 8. Let g A. Fin g(). = w t t, where the Grph of w(t) is given below. w(t) B. Fin g(). C. Fin g( ). 6.4 Pge of 6
Now tht function is efine s n integrl, let us see how to fin erivtive of such function. While interpreting function efine s n integrl is vli skill in its own right, our gol here is to simply iscover ptterns foun when tking the erivtive. 9 In Emple 6, f ( ) = f ( 9) f ( 4), where f() is the ntierivtive of f 4 Emple 9. Fin h t t, where is constnt.. Emple. Now, fin h t t. The Funmentl Theorem of Clculus, Antierivtive Prt (Simple) Emple. If g = w t t, then g =? = f t t f Emple. ( 5 6 + ) t t t = 6.4 Pge 4 of 6
The net emple hs limits on the integrl tht re functions of, s oppose to simply limit of n n upper limit of constnt. u Emple. v h t t The Funmentl Theorem of Clculus, Antierivtive Prt (Etene) Emple 4. Fin ( ) ( ) u = v f t t f t t f u u f v v Emple 5. Let g t t. Fin g 5 = +. 6.4 Pge 5 of 6
Putting It All Together. = f t t. Emple. Suppose the functions is the grph of f(t) n g A. Complete the tble: g() f(t) t B. Wht re the intervls on which g is incresing or ecresing? Justify ech response. C. Wht re the intervls on which g is concve up or concve own? Justify ech response. D. For wht vlue of oes g hve reltive mimum? Justify your response. E. For wht vlue of oes g hve n inflection point? Justify your response. F. Grph g(). g() 6.4 Pge 6 of 6