AP * Calculus Review

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1 AP * Clculus Review The Fundmentl Theorems of Clculus Techer Pcket AP* is trdemrk of the College Entrnce Emintion Bord. The College Entrnce Emintion Bord ws not involved in the production of this mteril. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

2 Pge of The Fundmentl Theorems of Clculus I. If f is continuous on [, b], then the function F( ) = f( t) dt hs derivtive t every point in [, b], nd df d = d d f () t dt = f ( ). II. If f is continuous on [, b], nd if F is ny ntiderivtive of f on [, b], then b f () t dt = Fb ( ) F ( ). Note: These two theorems my be presented in reverse order. Prt II is sometimes clled the Integrl Evlution Theorem. Don t overlook the obvious! d. f () t dt d = 0, becuse the definite integrl is constnt b. f '( ) d = f ( b) f( ) Upgrde for prt I, pplying the Chin Rule g( ) If F( ) = f( t) dt, then df d g( ) = () d d f t dt = f ( g ( )) g ( ). 3 d For emple, sin( ) t dt sin( 3 3 = 3 sin 6 d = (( ) )( ) ( ) An importnt lternte form for prt II Fb ( ) F ( ) f( t) dt = + b [Think of this s: ending vlue = strting vlue plus ccumultion.] For emple, given tht Using the lternte formt, f '( ) d= 4 nd (3) f =, find f (). f () = f(3) + f '( ) d = 35 + ( 4) = 3. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

3 Pge of Smple Problems Multiple Choice No Clcultor. d ln tdt d = (A) ln (B) ln (C) (D) (E) ln ln. If π ( ) cos( ) π g= t dt, then g ( ) = (A) sin ( π ) (B) πsin ( π ) (C) π cos( π ) (D) cos( π ) (E) π cos( π ) 3. d d (A) 4 + t dt = sin + sin (B) cos + sin (C) + sin (D) cos sin + (E) + cos 4. If f hs two continuous derivtives on [5, 0], then 0 f "( t) dt = 5 (A) f (0) f (5) (B) f (0) f (5) (C) f (0) f (5) (D) f (0) f (5) (E) f (5) f (0) Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

4 Pge 3 of 5. The grph of f is given, nd g is n ntiderivtive of f. If g (3) = 6, find g (0). (A) (B) (C) 4 (D) 5 (E) 0 6. The grph of f is given. F( ) = f( t) dt 0 Which of the following sttements is true? (A) F decreses on (, ). (B) F hs reltive minimum t = (C) F decreses on (, 4) (D) F hs reltive mimum t =. (E) F hs point of inflection t = 4. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

5 Pge 4 of 7. d tn( t) dt d = (A) tn ( ) tn (B) tn tn ( ) (C) tn tn ( ) (D) ( ) (E) sec ( ) sec tn tn 8. e 5 d = (A) e (B) 9 e (C) e (D) 3 e (E) e Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

6 Pge 5 of Free Response No Clcultor The grph of f is given. It consists of two line segments nd semi-circle. g ( ) = ft ( ) dt () Find g (0), g (), nd g (5). (b) Find g (), g (), nd g (4) or stte tht it does not eist. (c) For wht vlue(s) of does the grph of g hve point of inflection? Justify your nswer. (d) Find the bsolute mimum nd bsolute minimum vlues of g on [0, 5]. Justify your nswer. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

7 Pge 6 of Multiple Choice Clcultor Allowed. If ( ) sin g = tdt, then () 0 g = (A) 0 (B) 0.00 (C) 0.73 (D) 0.87 (E).89. A cr sold for $6,000 nd deprecited t rte of the vlue of the cr 3 yers fter the purchse? e dollrs per yer. Wht is (A) $06.7 (B) $ (C) $3,0.9 (D) $6,06.7 (E) $8, The grph of f is given, nd F() is n ntiderivtive of f. If find F(4) F(0). 4 f ( ) d= 7.5, (A) (B).5 (C) 7.5 (D).5 (E) 8.5 Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

8 Pge 7 of 4. The ccelertion of n object in motion is defined by t = 6 is. Find the velocity t t =. + t. The velocity t (A).44 (B) (C) (D).03 (E) h ( ) = gt () dt nd t + u gt () = du. Find h (.5). 0 u (A).03 (B).077 (C).54 (D) (E) Find f ( ) d if f( ) = sin, 0 <, 0 (A) 0 (B) (C) (D) 6.6 (E) Let g ( ) be n ntiderivtive of 3. If g () = 3, find (6) ln g. (A) 0.55 (B) 3.55 (C) 08. (D). (E) cost ( ) = ( ) 0 h e dton (3, 6). On which intervl(s) is h decresing? (A) (3.97, 5.498) (B) (5.498, 6) (C) (3, 4.7) (D) Alwys decresing on (3, 6) (E) Never decresing on (3, 6) Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

9 Pge 8 of Free Response Clcultor Active Let g ( ) = (5 8 ln t) dtfor >. Let h ( ) = (5 8 ln t) dtfor >. () Write n eqution of the tngent to g t = 3. (b) Wht is h ( )? (c) On which open intervl(s) is g decresing? Justify your nswer? (d) Find ll vlues for which h hs reltive etrem. Lbel them s mimum or minimum nd justify your nswer. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

10 Pge 9 of Key No Clcultor. A. E 3. B 4. C 5. B 6. C 7. D 8. A Clcultor Allowed. D. C 3. D 4. B 5. D 6. D 7. E 8. A Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

11 Pge 0 of Free Response No Clcultor The grph of f is given. It consists of two line segments nd semi-circle. g( ) = f( t) dt () Find g (0), g (), nd g (5). (b) Find g (), g (), nd g (4) or stte tht it does not eist. (c) For wht vlue(s) of does the grph of g hve point of inflection? Justify your nswer. (d) Find the bsolute mimum nd bsolute minimum vlues of g on [0, 5]. Justify your nswer. 0 () g(0) = f( t) dt = pts: pt g (0) g() = f( t) dt = 0 pt g () nd g (5) 5 g(5) = f( t) dt = π 3 (b) g () = f() = pts: pt g () g () = f () = DNE g (4) = f (4) = 0 pt g () nd g (4) (c) g hs point of inflection t pts: pt = 4 = 4 becuse g = f chnges from pt justifiction incresing to decresing. (d) Cndidtes re = 0, 3, 5, the endpoints 3 pts: pt for cndidtes of the intervl nd the criticl number. pt evluting cndidtes pt for nswers g ( ) π The bsolute minimum vlue is 3. The bsolute mimum vlue is. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

12 Pge of Free Response Clcultor Allowed Let g( ) = (5 8 ln t) dt for >. Let h ( ) = (5 8 ln t) dtfor >. () Write n eqution of the tngent to g t = 3. (b) Wht is h ( )? (c) On which open intervl(s) is g decresing? Justify your nswer? (d) Find ll vlues for which h hs reltive etrem. Lbel them s mimum or minimum nd justify your nswer. () y+.354 = 3.385( 3) 3 pts: pt g (3) =.354 pt g (3) = pt eqution = pts for h ( ) (b) h( ) ( 5 8 ln ) (c) g is decresing where g ( ) < 0 pts: pt correct intervl g ( ) = 5 8 lnt pt justifiction (.4779, ) (d) h hs reltive mimum t pts: pt correct reltive mimum =.56 becuse h pt justifiction chnges sign from positive to negtive. Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

13 Pge of AP Clculus Em Connections The list below identifies free response questions tht hve been previously sked on the topic of the Fundmentl Theorems of Clculus. These questions re vilble from the CollegeBord nd cn be downloded free of chrge from AP Centrl. Free Response Questions 004 AB Question Form B AB Question AB Question 3 Copyright 008 Lying the Foundtion, Inc., Dlls, TX. All rights reserved. Visit:

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