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Unit 5 Progress Check: MCQ Part A – A set of calculus multiple-choice questions covering the Mean Value Theorem, intervals of increasing/decreasing, finding local extrema from derivatives, and applying continuity/differentiability on given intervals.

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    Unit 5 Progress Check: MCQ Part A 1- Let / be the function given by f( z ) c* ' In Lr 1 ) 3- The derivative of f is given by / ’ i r ) = ■ 5 cos f y i sin ( y ) 4- - - • - . What 'LTaiue of r satisfies the conclusion of the Mean Value Theorem applied to/ on the interval 1, L ? (A) 2.132 because / ( 2. 132} — 1 1 , 1 / ' (B) 2.749 because/' (2. 749 1 — - 3 - (C) 3.042 because / ' ( 3 .012) 0 (D) 3.252 because/' (3.252) ' . The derivative of the function / is given by / r ( r i r J 2 - 3.T j . On which of the following intervals it’- 4 , 3 ] i s / decreasing? (A) 1, 3.441], [ 1.806, 11.(if>11Land 1 . 5 0 9 . 3 ] (B) 4 , 2.805] and l.227,IM>37| (C) 3.414, 1.806 and 0.660,1.509] (D) : 2.805, 1.227] and 10.637, 3] 3. The temperature inside a vehicle is modeled by the function / . where / ( / } is measured in degrees Fahrenheit and - is measured in minutes. The first derivative of / is given by / ' ( f } t~ 3 t -b cos t - At what times t . for 0 . t 4 5 does the temperature attain a local minimum? (A) 0.354 only (B) 1.962 (C) 3.299 only (D) 0.354 and 3.299 4, Let / be the function given by on the closed interval 7. 7] - Of the following intervals, on which can the Mean Value Theorem be applied to / ? I. 1 , 3 ] because / is continuous on 1 , 3 ] and differentiable on i 1 . 3 . II. •>. 7 because is continuous on 5 . 7 and differentiable on; j , “ |- III. 1 . 5 because ■ is continuous on I . 5 and differentiable on ; I 5 | . (A) None (B) I only (C) I and II only (D) LIL and IH 5. Let / be a differentiable function with f (0) 4 and / I ■ l 1 I . Which of the following must be true for somer in the interval 0.1(1 ?

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    Unit 5 Progress Check: MCQ Part A (A) f ' ( c ) — IK since the Extreme Value Theorem applies. (B) / ' ( c ) = — ■ — — , since the Mean Value Theorem applies. (C) f ' i c i — 3 i ] | ? D *i 3 since the Mean Value Theorem applies. (D) / ' ( <. ■ i — 1 .5, since the Intermediate Value Theorem applies. Let j be the function given by / ] . < —— r — * 1 * —jy ™ the clo sed interval —5,5]. On which of the following closed intervals is the function / guaranteed by the Extreme Value Theorem to have an absolute maximum and an absolute minimum? 6. (A) [ - 5 , 5 ] (E) [-3,1] (C) [ - 2 , 0 (D) [0,5] Let jf be the function defined by / ( z ) = z sin z with domain 0, oc ) ■ The function f has no absolute minimum and no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem? (A) The domain of j is not an open interval. (B) The domain of / is not a closed and bounded interval. (C) The function f is not continuous on its domain. (D) The function f is not differentiable on its domain. z 2 3 4 5 /(*) 1 14 2'1 31 Selected values of a continuous function / are given ui the table above. Which of the following statements could be false? . , , By the Intermediate Value Theorem applied to f on the interval 2 5 ■ there is a value r such that 1 J /(<0 = 1 0 _______________________________________________ (B) By the Mean Value Theorem applied to / on the interval 2 , 5 , there is a value t: such that f‘ - c I — 1(1. By the Extreme Value Theorem applied to / on the interval 2 , 5 . , there is a value e such that 1 J /(c) '1 f(?) forallxin[2 t 5 | . , By the Extreme Value Theorem applied to J on the interval 2 , 5 , there is a value r such that ( f(c) > f ( x ) for all x in]2,5|- 9. Let ■ ' be the function defined by / ( z ) z 3 ftz’ - 9 z ■ 4 for O ' z 3. Which of the following statements is true?

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    Unit 5 Progress Check: MCQ Part A (A) f is decreasing on the interval ( (), 1 I because y'j , r - ) < (J on the interval ( (J, | . . (B) y is increasing o n the interval i 0 T I because _r ) .. lion the interval ((), | . (C) / is decreasing on the interval i 0. 2 1 because ,r ) ■ (j o n the interval 0 . 2 ) . (D) f is decreasing on the interval i ] . 3 1 because ’ ' i r i 0 on the interval i 1 . 3 : . 10. Let f be the function defined by / ( J?) J" In J for ,r ■ (). On what open interval is / decreasing? (A) 0 < £ < | only (B) 0 < j < 1 (C) » > 1 (D) There is no such internal. 11. Let ■ ' be a function with first derivative given by f ( r ) - j — 5 ) ‘ ( j: -I- 1 J . At what values of j? does / have a relative maximum? (A) 1 only (B) 0 only (C) 1 and 5 only (D) ] , 0, and 5 only y G r a p h o f f ' The graph of f. the derivative of the function f. is shown above for (I ... j? < <J. XMiich of the following statements is true for [) < j? < 9 ? Unit 5 Progress Check: MCQ Part A (A) f has one relative minimum and two relative maxima. (B) f has two relative minima and one relative maximum. (C) f has two relative minima and two relative maxima. (D) f has three relative minima and two relative maxima.

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    Unit 5 Progress Check: MCQ Part B The second derivative of the function / is given by f *' (> ) x 3 cos '' , i . At what values of x in the inter. _ al ■ 1. 3 > does the graph of f have a point of inflection? (A) 2.229 only (B) 0 and 2.229 (C) —2.357 and 0.987 (D) 3.259,0, and 1.603 The second derivative of the function ,< is given by f ( x | ■ s i n ( ) 2 cos ;r- The function / has many critical points, two of which are at j (J and j- 6.949. Which of the following statements is true? (A) ( has a local minimum at j- . (I and at j* 6-919- (B) f has a local minimum at r 0 and a local maximum at r 6.9-19. (C) / has a local maximum at r — 0 and a local minimum at j— 6,949- (D) f has a local maximum at j Q and at j = 6,949. v 3. Let f be the function given by / 1 x J 2_r ■ ’ -I 3x - 1 ■ What is the absolute maximum value of / on the closed interval - 3 , 1| ? (A) 1 (B) 2 (C) <5 (D) 26 4. Let f be the function defined bv f ( x ' ■ = sin x cos ,r. What is the absolute minimum value of / on the interval (A) 2 (B) - v 2 (C) - 1 (D) 0 5. Let be the function defined by y]'z) | r 2 — X + 1 ' c r ■ What is the absolute maximum value of y on the interval -1,1 ? (A) 1 (B) e (C) | (P) £

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    Unit 5 Progress Check: MCQ Part B The graph of f', the derivative of the function f, is shown above. On which of the following open intervals is the graph of f concave down? (A) 7 - 5 , - 3 ) and ( h 6 ) (B) 7 - 3 . t | and 6 , 8 ) (C) ( - 1 . 4 ) (B) (4,8)

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    Unit 5 Progress Check: MCQ Part B Let f be the function defined by f ; j: x ' 1 ()j . The graph of f f . the derivative of f 3 is shown above. On which of the following intervals is the graph of f concave up? (A) j < v 3 and 0 ■ j* < y'3 (B) v'''3 x ■ (J and x v 3 ( c ) JC < - v 6 ™d a; > G (D) VO < x V'6 3, The Second Derivative Test cannot be used to conclude that 2 is the location of a relative minimum or relative maximum for which of the following functions? (A) f ( j ] ~ 2 ) : where / ' ( > } - sin fj' 2 ) (B) /(jr) _ J-f : , where c 7 4 (C) /( X ) - P 4 x 2:Where/'(j) 4 CD) / ( x ) - x a - & r 2 + 12x - L where / { z ) - L2x 4- 12

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