Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION


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1 lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer.. Wht is the coordinte of the point of inflection on the grph of y ? If f is continuous for < < nd differentile for < <, which of the following could e flse? f ' c f f for some c such tht < < f ' c 0 for some c such tht < < f hs minimum vlue on < < f hs mimum vlue on < < 3. If y 0, then when =, dy d
2 4. Let f nd g e differentile 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 5. Wht is the instntneous rte of chnge t = of the function f given y f? 6 6 f ln for 0 < = ln for < 4 6. If ( ), then lim f ( ) = ln ln 8 ln 6 4 noneistent
3 7. The grph of the function f shown in the figure elow hs verticl tngent t the point (, 0) nd horizontl tngents t the points (, ) nd (3, ). y For wht vlues of, < < 4, is f not differentile? 0 only 0 nd only nd 3 only 0,, nd 3 only 0,,, nd 3 8. prticle moves long the is so tht its position t time t is given y t t 6t 5. For wht vlue of t is the velocity of the prticle zero? 3 4 5, then f ' 9. If f sine cose cose cose e e e cose e cose
4 0. The grph of twicedifferentile function f is shown in the figure elow. Which of the following is true? y f f ' f '' f f '' f ' f ' f f '' f '' f f ' f '' f ' f. n eqution of the line tngent to the grph of y cos t the point (0, ) is y = + y = + y = y = y = 0. If f '', then the grph of f hs inflection points when = only only nd 0 only nd only, 0, nd only
5 4 3. The function f is given y f. On which of the following intervls is f incresing?,, 0,,0, 4. The grph of f is shown elow. Which of the following could e the grph of the derivtive of f?
6 5. The mimum ccelertion ttined on the intervl 0 < t < 3 y the prticle whose velocity is given y 3 vt t3t t 4 is The function f is continuous on the closed intervl [0, ] nd hs vlues tht re given in the tle elow. 0 f () k The eqution f ( ) = must hve t lest two solutions in the intervl [0, ] if k = If f tn, then f '
7 lculus Section I Prt LULTOR IS RQUIR FOR SOM QUSTIONS ON THIS PRT OF TH XMINTION In this test: ) The ect numericl vlue of the correct nswer does not lwys pper mong the choices given. When this hppens, select from mong the choices the numer tht est pproimtes the ect numericl vlue. ) Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer. 76. The grph of function f is shown elow. y Which of the following sttements out 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 77. Let f e the function given y f 3e 3 nd let g e the function given y g 6. t wht vlue of do the grph of f nd g hve prllel tngent lines?
8 78. The grphs of the derivtives of the functions f, g, nd h re shown elow. y y y f ' y g' y y h' Which of the functions f, g, or h hve reltive mimum on the open intervl < <? f only g only h only f nd g only f, g, nd h cos 79. The first derivtive of the function f is given y f '. How mny criticl vlues does f hve on the 5 open intervl (0, 0)? One Three Four Five Seven 80. Let f e the function given y f. Which of the following sttements out f re true? I. f is continuous t = 0. II. f is differentile t = 0. III. f hs n solute minimum t = 0. I only II only III only I nd III only II nd III only
9 8. If 0, then lim 4 4 = 6 0 noneistent 4 8. 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 = If g is differentile function such tht g () < 0 for ll rel numers 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 e determined if f hs ny reltive etrem.
10 84. Let f e function tht is differentile on the open intervl (, 0). If f 5, f 5 5, nd f 9 5, which of the following must e 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 Solutions: Prt : Prt :
11 lculus Section II PRT GRPHING LULTOR IS RQUIR FOR SOM PROLMS OR PRTS OF PROLMS. n isosceles tringle, whose se 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 y (0, 0) (c, 0) Justify your nswer:
12 . Given the following tle 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 ) 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
13 for 3. Let f e function defined y f. k p for ) For wht vlues of k nd p will f e continuous nd differentile t =? ) 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.
14 4. onsider the curve defined y y y 7. ) Write n epression for the slope of the curve t ny point (, y). ) 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.
15 5. The figure elow 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 y f ' 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. ) For wht vlues of is the grph of f concve up? Justify your nswer. c) Use the informtion found in prts nd nd the fct tht f ( 3) = 0 to sketch possile grph of f on the es provided elow.
16 3 6. Find the vlue of c tht stisfies the Men Vlue Theorem for f 5 over [ 4, ] n oject moves long the is with velocity vt t t 5t where t 0,. ) When is the oject stopped? Justify your nswer. ) When is the oject moving right? Justify your nswer. c) Wht is the ccelertion t time t =.3? Show ll your work. d) When is the oject speeding up? Justify your nswer.
17 8. Find the derivtive of the given function. ) g( ) = log ( ) 3 4 ) f tn = d) ( ) = ( 5 6) c) y cot ( ln(5 ) ) 3 cos h e 9. vlute ech limit. If limit N, eplin why. Show ll work ) lim 4 sin(3 ) 0 ) lim e + c) lim 4 + d) lim For the limits in question 9, prts c nd d, drw conclusion out the ehvior of the function s pproches the given vlue. Justify, with clculus, your conclusions.
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