First Semester Review Calculus BC


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1 First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y ? The grph of piecewiseliner function f, for 4, is shown below. y Wht is the vlue of 4 f d? d 7 4 ln
2 4. If f is continuous for < < b nd differentible for < < b, which of the following could be flse? f ' c f b f b for some c such tht < < b f ' c 0 for some c such tht < < b f hs minimum vlue on < < b f hs mimum vlue on < < b b f d eists sin t dt sin cos cos cos cos 6. If y 0, then when =, dy d
3 7. e d = e e e e e e e e 3 8. Let f nd g be differentible functions with the following properties: i) g () > 0 for ll ii) f (0) = If h () = f () g () nd h' f g', then f f ' g () e 0 9. Wht is the instntneous rte of chnge t = of the function f given by f? 6 6
4 0. If f is liner function nd 0 < < b, then f '' 0 b d b b b. If f ( ) = ln for 0 < ln for < 4, then lim ( ) f = ln ln 8 ln 6 4 noneistent. The grph of the function f shown in the figure below hs verticl tngent t the point (, 0) nd horizontl tngents t the points (, ) nd (3, ). y For wht vlues of, < < 4, is f not differentible? 0 only 0 nd only nd 3 only 0,, nd 3 only 0,,, nd 3
5 3. prticle moves long the is so tht its position t time t is given by t t 6t 5. For wht vlue of t is the velocity of the prticle zero? If 3 F t dt, then F ' If f sine, then f ' cose cose cose e e e cose e cose
6 6. The grph of twicedifferentible function f is shown in the figure below. Which of the following is true? y f f ' f '' f f '' f ' f ' f f '' f '' f f ' f '' f ' f 7. n eqution of the line tngent to the grph of y cos t the point (0, ) is y = + y = + y = y = y = 0 8. If f '', then the grph of f hs inflection points when = only only nd 0 only nd only, 0, nd only
7 9. Wht re ll vlues of k for which d 0? nd 3 3, 0, nd 3 k The function f is given by f. On which of the following intervls is f incresing?,, 0,,0,. The grph of f is shown below. b Which of the following could be the grph of the derivtive of f? b b b b b
8 . The mimum ccelertion ttined on the intervl 0 < t < 3 by the prticle whose velocity is given by 3 vt t3t t 4 is vlute: e lim + N 0 47 nd hlf 4. If f tn, then f ' 5 3 undefined 3 5
9 First Semester Review lculus Multiple hoice: lcultor llowed 5. The grph of function f is shown below. y Which of the following sttements bout f is flse? f is continuous t =. f hs reltive mimum t =. = is in the domin of f. lim f is equl to lim f lim f eists 6. Let f be the function given by f 3e 3 nd let g be the function given by g 6. t wht vlue of do the grph of f nd g hve prllel tngent lines? The rdius of circle is decresing t constnt rte of 0. centimeter per second. In terms of the circumference, wht is the rte of chnge of the re of the circle in squre centimeters per second?
10 8. The grphs of the derivtives of the functions f, g, nd h re shown below. y y y b b b Which of the functions f, g, or h hve reltive mimum on the open intervl < < b? f only g only h only f nd g only f, g, nd h cos 9. The first derivtive of the function f is given by f '. How mny criticl vlues does f hve on the 5 open intervl (0, 0)? One Three Four Five Seven 30. Let f be the function given by f. Which of the following sttements bout f re true? I. f is continuous t = 0. II. f is differentible t = 0. III. f hs n bsolute minimum t = 0. I only II only III only I nd III only II nd III only
11 3. If 0, then lim 4 4 = 6 0 noneistent 3. The function f is continuous on the closed intervl [, 8] nd hs vlues tht re given in the tble below f () Using the subintervls [, 5], [5, 7], nd [7, 8], wht is the trpezoidl pproimtion of 8 f d? Which of the following is n eqution of the line tngent to the grph of f t the point where f '? y = 8 5 y = + 7 y = y = 0. y =.46
12 34. Let F () be n ntiderivtive of ln , If F () = 0, the F (9) = 35. If g is differentible function such tht g () < 0 for ll rel numbers nd if f ' 4g following is true? f hs reltive mimum t = nd reltive minimum t =. f hs reltive minimum t = nd reltive mimum t =. f hs reltive minim t = nd t =. f hs reltive mim t = nd t =., which of the It cnnot be determined if f hs ny reltive etrem. 36. If the bse b of tringle is incresing t rte of 3 inches per minute while its height h is decresing t rte of 3 inches per minute, which of the following must be true bout the re of the tringle? is lwys incresing is lwys decresing is decresing only when b < h is decresing only when b > h remins constnt 37. pproimte cos( 0.0 ) using lineriztion
13 38. Let f be function tht is differentible on the open intervl (, 0). If f 5, f 5 5, nd f 9 5, which of the following must be true? I. f hs t lest zeros. II. The grph of f hs t lest one horizontl tngent. III. For some c, < c < 5, f (c) = 3. None I only I nd II only I nd III only I, II, nd III 39. The function f is continuous on the closed intervl [0, ] nd hs vlues tht re given in the tble below. 0 f () k The eqution f ( ) = must hve t lest two solutions in the intervl [0, ] if k = 0 3 Solutions: Prt : Prt :
14 First Semester Review lculus Free Response: lc llowed. n isosceles tringle, whose bse is the intervl from (0, 0) to (c, 0), hs its verte on the grph of f. For wht vlue of c does the tringle hve mimum re? y (0, 0) (c, 0) Justify your nswer:
15 . Given the following tble of vlues t = nd =, find the indicted derivtives in prts l. f f ' g g' d f g d ) 3 d g) g f d d b) f g d d h) g g d c) d f d g d i) f g 46 d d) d g d f d j) g 3 d d e) f g d d k) f d d f) f g d l) d f d
16 for 3. Let f be function defined by f. k p for ) For wht vlues of k nd p will f be continuous nd differentible t =? b) For the vlues of k nd p found in prt, on wht intervl or intervls is f incresing? c) Using the vlues of k nd p found in prt, find ll points of inflection of the grph of f. Support your conclusion.
17 4. onsider the curve defined by y y 7. ) Write n epression for the slope of the curve t ny point (, y). b) etermine whether the lines tngent to the curve t the intercepts of the curve re prllel. Show the nlysis tht leds to your conclusion. c) Find the points on the curve where the lines tngent to the curve re verticl.
18 5. The figure below shows the grph of f ', the derivtive of function f. The domin of the function is the set of ll such tht 3 3. y Note: This is the grph of the derivtive of f, not the grph of f. ) For wht vlues of, 3 < < 3, does f hve reltive minimum? reltive mimum? Justify your nswer. b) For wht vlues of is the grph of f concve up? Justify your nswer. c) Use the informtion found in prts nd b nd the fct tht f ( 3) = 0 to sketch possible grph of f on the es provided below. y
19 3 6. Find the vlue of c tht stisfies the Men Vlue Theorem for f 5 over [ 4, ] n object moves long the is with velocity vt t t 5t where t 0,. ) When is the object stopped? Justify your nswer. b) When is the object moving right? Justify your nswer. c) Wht is the ccelertion t time t =.3? Show ll your work. d) When is the object speeding up? Justify your nswer. e) The object ws t position of +4 when t = 0. Where will it be t t =? Show ll your work.
20 8. Given: the grph of f consisting of two line segments nd semicircle f () s shown, nd tht g f d. omin: [0, 4] ) Find g 0, g, nd 4 g. 3 4 b) When does g hve minimum? Justify your response. c) When does g hve point of inflection? Justify your response. d) Sketch g. Lbel the verticl is. g () 3 4
21 9. [00 P lculus Free Response Question #5] continer hs the shpe of n open right circulr cone, s shown below. 0 cm h r 0 cm The height of the continer is 0 cm nd the dimeter of the opening is 0 cm. Wter in the continer is evporting so tht 3 its depth h is chnging t the constnt rte of cm/hr. 0 (Note: The volume of cone of height h nd rdius r is given by V rh.) 3 ) Find the volume V of wter in the continer when h = 5 cm. Indicte units of mesure. b) Find the rte of chnge of the volume of wter in the continer, with respect to time, when h = 5 cm. Indicte units of mesure. c) Show tht the rte of chnge of the volume of wter in the continer due to evportion is directly proportionl to the eposed surfce re of the wter. Wht is the constnt of proportionlity?
22 0. [008 P lculus Free Response Question #, prts nd b] t (hours) L () ( ) oncert tickets went on sle t noon (t = 0) nd were sold out within 9 hours. The number of people witing in line to purchse tickets t time t is modeled by twicedifferentible function of L for 0 t 9. Vlues of L(t) t vrious times t re shown in the tble bove. () Use the dt in the tble to estimte the rte t which the number of people witing in line ws chnging t 5:30pm (t = 5.5). Show the computtions tht led to your nswer. Indicte units of mesure. (b) Use trpezoidl sum with three subintervls to estimte the verge number of people witing in line during the first 4 hours tht tickets were on sle.. Find the lineriztion of f ( ) = 3+ e d t =
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