AP Calculus. Fundamental Theorem of Calculus
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1 AP Clculus Fundmentl Theorem of Clculus Student Hndout EDITION Click on the following link or scn the QR code to complete the evlution for the Study Session Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t
2 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t
3 Fundmentl Theorem of Clculus Students should be ble to: Use the fundmentl theorem to evlute definite integrls. Use vrious forms of the fundmentl theorem in ppliction situtions. Clculte the verge vlue of function over prticulr intervl. Use the other fundmentl theorem.
4 Multiple Choice 1. (clcultor not llowed) 4 3 t 4 t dt (A) 4 (B) 1 3 (C) 4 3 (D) 4 8. (clcultor not llowed) Wht is the verge vlue of y for the prt of the curve qudrnt? y 3 which is in the first (A) 6 (B) 3 (C) (D) (E) (clcultor not llowed) The grph of, the derivtive of, is the line shown in the figure bove. If f () 5, then f (1) (A) (B) 3 (C) 6 (D) 8 (E) 11 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 1
5 4. (clcultor not llowed) d sin( 3 t ) dt d (A) (B) (C) (D) (E) 6 cos( ) 3 sin( ) 6 sin( ) 3 sin( ) 6 sin( ) 5. (clcultor not llowed) 3 t Let f ( ) e dt. At wht vlue of is f ( ) minimum? (A) For no vlue of 1 (B) 3 (C) (D) (E) 3 6. (clcultor not llowed) cos d 1 sin (A) 1 (B) (C) (D) 1 (E) 1 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t
6 7. (clcultor not llowed) The grph of the function f shown in the figure bove hs horizontl tngents t 3 nd 6. If g( ) f ( t) dt, wht is the vlue of g 3? (A) (B) 1 (C) (D) 3 (E) 6 8. (clcultor llowed) Let h be the function defined by h ( ) g() 3, wht is the vlue of g (4)? 1. If g is n ntiderivtive of h nd 5 1 (A). (B).15 (C) 3.31 (D) (clcultor not llowed) Which of the following is n eqution of the line tngent to the grph of t the point where 1? (A) y (B) y ( 1) e (C) y e( 1) (D) y ( e 1) t y e dt 1 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 3
7 1. (clcultor not llowed) 3 1 If f ( ) dtfor 1, then f '() 1 1 lnt 1 (A) 1 ln 1 (B) 1 ln 1 (C) 1 ln8 1 (D) 1 ln8 11. (clcultor llowed) If f '( ) ln( ) nd f (5) 8, then f (3) (A).497 (B) (C) 5.5 (D) (clcultor llowed) Let g be the function given by intervls is g decresing? ( ) sin( ) g t dtfor 1 3. On which of the following (A) 1 (B) 1.77 (C) (D) (E) (clcultor llowed) If 4, of the following, which is the gretest vlue of such tht ( t t) dt t dt? (A) 1.35 (B) 1.38 (C) 1.41 (D) 1.48 (E) 1.59 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 4
8 14. (clcultor llowed) Let f ( ) h( t) dt, where h hs the grph shown bove. Which of the following could be the grph of f? (A) (B) (C) (D) (E) 15. (clcultor llowed) A pizz, heted to temperture of 35 degrees Fhrenheit ( o F), is tken out of n oven nd plced in 75 o F room t time t minutes. The temperture of the pizz is.4t chnging t rte of 11e degrees Fhrenheit per minute. To the nerest degree, wht is the temperture of the pizz t time t 5 minutes? (A) 11 F (B) 119 F (C) 147 F (D) 38 F (E) 335 F Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 5
9 16. (clcultor llowed) For ll vlues of, the continuous function f is positive nd decresing. Let g be the function given by g ( ) ftdt ( ). Which of the following could be tble of vlues for g? (A) g ( ) (B) g ( ) (C) g ( ) (D) g ( ) (E) g ( ) (clcultor llowed) The rte of chnge of the ltitude of hot ir blloon is given by 3 r t t 4t 6 for t 8. Which of the following epressions gives the chnge in ltitude of the blloon during the time the ltitude is decresing? (A) (B) (C) (D) (E) r t dt r t dt r t dt r t dt r t dt Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 6
10 Free Response 18. (clcultor not llowed) Let f be function tht is twice differentible for ll rel numbers. The tble bove gives vlues of f for selected points in the closed intervl 13. (b) Evlute f d. Show the work tht leds to your nswer. 19. (clcultor not llowed) The function g is defined nd differentible on the closed intervl 7, 5 nd stisfies g ( ) 5. The grph of y g, the derivtive of g, consists of semicircle nd three line segments, s shown in the figure bove. () Find g (3) nd g (). Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 7
11 . (clcultor llowed) As pot of te cools, the temperture of the te is modeled by differentible function H for t 1, where time t is mesured in minutes nd temperture H () t is mesured in degrees Celsius. Vlues of H () t t selected vlues of time t re shown in the tble. t (minutes) H () t (degrees Celsius) (c) Evlute 1 H() t dt. Using correct units, eplin the mening of the epression in the contet of this problem (d) At time t, biscuits with temperture 1 C were removed from n oven. The temperture of the biscuits t time t is modeled by differentible function B for.173t which it is known tht B( t) 13.84e. Using the given models, t time t 1, how much cooler re the biscuits thn the te? Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 8
12 1. (clcultor llowed) At certin height, tree trunk hs circulr cross section. The rdius R (t) of tht cross section grows t rte modeled by the function dr 1 3 sin( t ) centimeters per yer dt 16 for t 3, where time t is mesured in yers. At time t, the rdius is 6 centimeters. The re of the cross section t time t is denoted by A (t). () Write n epression, involving n integrl, for the rdius R (t) for t 3. Use your epression to find R (3). 3 t (c) Evlute A dt. Using pproprite units, interpret the mening of tht integrl in terms of cross-sectionl re. Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 9
13 . (clcultor not llowed) Let g be the piecewise-liner defined function on, 4 whose grph is given bove, nd let f g cos. () Find 4 f d. Show the computtions tht led to your nswer. (c) Let h 3 g t dt. Find h. 3 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 1
14 3. (clcultor not llowed) The function f is defined on the closed intervl The grph of f consists of three line segments nd is shown in the figure bove. Let g be the function defined by g( ) = ò f ( t) dt. -3 (b) On wht open intervls contined in concve down? Give reson for your nswer. is the grph of g both incresing nd Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 11
15 4. (clcultor not llowed) The rte t which rinwter flows into drinpipe is modeled by the function R, where t Rt () sin cubic feet per hour, t is mesured in hours, nd t 8. The pipe is 35 prtilly blocked, llowing wter to drin out the other end of the pipe t rte modeled by 3 Dt ( ).4t.4t.96t cubic feet per hour, for t 8. There re 3 cubic feet of wter in the pipe t time t. () How mny cubic feet of rinwter flow into the pipe during the 8-hour time intervl t 8? (c) At wht time t 8, is the mount of the pipe t minimum? Justify your nswer. Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 1
16 Fundmentl Theorem of Clculus Reference Pge Fundmentl Theorem of Clculus b f ( d ) Fb ( ) F ( ) The Fundmentl Theorem of Clculus cn be written in vrious wys. b f ( d ) f ( b ) f ( ) f () b f() f() d f ( ) f( b) f( ) d b f ''( d ) f '( b ) f '( ) b b Averge vlue of function over prticulr intervl f vg b f( ) d F( b) F( ) b b Other Fundmentl Theorem of Clculus d d d d g( ) f () tdt f( ) f () tdt f( g ( )) g( ) Fundmentl Theorem of Clculus in contet Amount ending time begining time Rte dt time time Current Amount Initil Amount " rte in " dt " rte out " dt time1 time1 Copyright 16 Ntionl Mth + Science Inititive, Dlls, Tes. All rights reserved. Visit us online t 13
The Fundamental Theorem of Calculus, Particle Motion, and Average Value
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