CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version):

Transcription

1 CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS 6 cos Secon Funamenal Theorem of Calculus: f a 4 a f 6 cos Secon Funamenal Theorem of Calculus (Chain Rule Version): g f a E. Use he Secon Funamenal Theorem o evaluae: (a) 3 (b) an 3 (c) 3 () sin 3

2 E. The graph of a funcion f consiss of a quarer circle an line segmens. Le g be he funcion given by g f. (a) Fin g, g, g, g 5. y Graph of f (b) Fin all values of on he open inerval, 5 a which g has a relaive maimum. (c) Fin he absolue minimum value of g on, 5 an he value of a which i occurs. () Fin he -coorinae of each poin of inflecion of he graph of g on your answer., 5. Jusify

3 CALCULUS WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND FUNCTIONS DEFINED BY INTEGRALS. Fin he erivaives of he funcions efine by he following inegrals: sin cos (a) (b) e (c) () an e (e), (f) cos (g) s s s (h) cos 5 3 cos (i) 7 sin an 4. The graph of a funcion f consiss of a semicircle an wo line segmens as shown. Le g be he funcion given by g f. (a) Fin g, g 3, g,an g 5. (b) Fin all values of on he open inerval,5 a which g has a relaive maimum. Jusify your answers. (c) Fin he absolue minimum value of g on he close inerval [,5] an he value of a which i occurs. () Wrie an equaion for he line angen o he graph of g a = 3. (e) Fin he -coorinae of each poin of inflecion of he graph of g on he open inerval,5. (f) Fin he range of g.

4 3. Le g f, where f is he funcion whose graph is shown. (a) Evaluae g, g, g,ang 6. (b) On wha inervals is g increasing? (c) Where oes g have a maimum value? Wha is he maimum value? () Where oes g have a minimum value? Wha is he minimum value? (e) Skech a rough graph of g on [, 7]. 4. Le g whose graph is shown. (a) Evaluae g 3 an g 3. 3 f, where f is he funcion (b) A wha values of is g increasing? Jusify. (c) A wha values of oes g have a maimum value? Jusify. () A wha values of oes g have a minimum value? Jusify. (e) A wha values of oes g have an inflecion poin? Jusify.

5

6 CALCULUS WORKSHEET ON FUNCTIONS DEFINED BY INTEGRALS. Fin he equaion of he angen line o he curve y F where a he poin on he curve where =. 3 F 7. Suppose ha (a) Wha is f? c f. (b) Fin he value of c. 3. If 4 F 3, for wha values of is F ecreasing? 4. Le H f where f is he coninuous funcion wih omain [, ] shown on he righ. (a) Fin H. y (b) On wha inerval(s) of is H increasing? Graph of f (c) On wha inerval(s) of is H concave up? () Is H posiive or negaive? Eplain. (e) For wha value of oes H achieve is maimum value? Eplain.

7 5. The graph of a funcion f consiss of a semicircle an wo line segmens as shown on he righ. Le g f. (a) Fin g, g 3, g. y (b) On wha inerval(s) of is g ecreasing? Jusify your answer. Graph of f (c) Fin all values of on he open inerval 3, 4 a which g has a relaive minimum. Jusify your answer. () Fin he absolue maimum value of g on he inerval occurs. 3, 4 an he value of a which i (e) On wha inerval(s) of is g concave up? (f) For wha value(s) of oes he graph of g have an inflecion poin? (g) Wrie an equaion for he line angen o he graph of g a. 6. The graph of he funcion f, consising of hree line segmens, is shown on he righ. Le g f. (a) Fin g, g 4, g. y (b) Fin g an g 3. (c) Fin he insananeous rae of change of g wih respec o a =. Graph of f () Fin he absolue maimum value of g on he inerval, 4. (e) The secon erivaive of g is no efine a = an a =. Which of hese values are -coorinaes of poins of inflecion of he graph of g?

8 CALCULUS WORKSHEET 3 ON FUNCTIONS DEFINED BY INTEGRALS Work he following on noebook paper.. The funcion g is efine on he inerval [, 6] by g f where f is he funcion graphe in he figure. (a) For wha values of, < < 6, oes g have a relaive maimum? (b) For wha values of is he graph of g concave own? (c) Wrie an equaion for he angen line o g a he poin where = 3. () Skech a graph of he funcion g. Lis he coorinaes of all criical poin an inflecion poins.. Suppose ha f is a coninuous funcion, ha f 3, an ha f 7. Fin he average value of f over he inerval [, ]. 3. The graph of a iffereniable funcion f on he close inerval [ 4, 4] is shown. Le G f for 4 4. (a) Fin G 4. (b) Fin G 4. 4 (c) On which inerval or inervals is he graph of G ecreasing? () On which inerval or inervals is he graph of G concave own? (e) For wha values of oes G have an inflecion poin?

9 4. The funcion F is efine for all by (a) Fin F. F 8. (b) Fin F. (c) Fin F. () Fin F. 5. If 5 F 6, on wha inervals is F ecreasing? 6. The graph of he velociy v, in f/sec, of a car raveling on a sraigh roa, for 35, is shown in he figure. (a) Fin he average acceleraion of he car, in f / sec, over he inerval 35. (b) Fin an approimaion for he acceleraion of he car, in f / sec, a =. Show your compuaions. (c) Approimae 35 5 v wih a Riemann sum, using he mipoins of hree subinervals of equal lengh. Eplain he meaning of his inegral.

10 7. The funcion F is efine for all by F where f is he funcion graphe in he figure. The graph of f is mae up of sraigh lines an a semicircle. (a) For wha values of is F ecreasing? (b) For wha values of oes F have a local maimum? A local minimum? (c) Evaluae F, F,an F. f, () Wrie an equaion of he line angen o he graph of F a = 4. (e) For wha values of oes F have an inflecion poin?