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Unformatted text preview: M4/&/~ FMMM~ F/A/Aéz. EXAM (1) The domain of the function f(x) = 1n(a: + 5) is:
(80 {0,00}
(13) [—5,00)
(C) [5:00)
((0 [4,00) (6) [4: 00) (2) Let f (22;) = m. Which of the following is f *1(m)?
(a) (2 — $)3
(b) (m — 2P (C) 2 + $3 3 $2 + 9 for a: < 1
3 If = _
( ) f(x) {12:12 — (2:62 for x > 1 uous at all values of (1:. determine all values of a so that f (in) is contin— (®a=0
(ma=1
(®a=2
(d)a=3 (6) There are no such values of a (4) If f (m) = 32 team, the slope of the tangent line at (g, f (g) is: 4%2 + (ix/5W
27 4W2 + 6\/§7r (a)
(b) (6) If f(:n) = e493 evaluate lim w h—+0 h (a) 612 $2+1 (7) The function f (:12) = $3 + 8 has: (a) no vertical or horizontal asymptotes (b) 1 vertical and 1 horizontal asymptote
(c) 2 vertical and 1 horizontal asymptote
(d) 1 vertical and 2 horizontal asymptotes (e) 1 vertical and no horizontal asymptotes (8) A particle moves on a line with velocity v(t) = t—ln(t2 + 1). What is its maximum
velocity on the interval 0 S t S 2? (a) 1 —ln2
(b) o (c) 2—1115
(d)1n2——1 (9) Assume that; f and g are differentiable functions deﬁned on (—oo,<>o), f(0) =
6,f’(0) = 10,f(2) = 5,f’(2) = 4,g(0) = 2, and g’(0) = 3. Let h(a:) = f(g(a:)). What is h’(0)? 
(a) 4
(b) 8
(c) 10
(d) 12 (e) 30 (10) Assume that y is deﬁned implicitly as a differentiable function of x by the equation 2:133 +3323; — xy3 = 2. Find % at (1,1).
—3 (a) "2“
7 (b) 5 (0) 0 (d) 3 (11) Evaluate lim Mi
m—eO £17 (a) —2 (12) Water is withdrawn at the constant rate of 2 ft3 / min from a cone—shaped reservoir ' which has its vertex down. The diameter of the top of the tank measures 4 feet
and the. height of the tank is 8 feet . How fast is the water level falling when the
depth of the water in the reservoir is 2 feet? (Recall that the volume of a cone of
height h and radius r is V = grzh). (a) :ft/min (b) Sft/min (c) :ft/min (d) ga/nﬁn ' (e) gift/min (13) At the beginning of an experiment a colony has N bacteria. Two hours later it
has 4N bacteria. How many hours, measured from the beginning, does it take for
the colony to have lON bacteria? ln5N
(a) ln2 N1n5
(b) 21112 1115
(0) E2 In N
(d) 411—2" In 10
(6) F2 (14) The approximate value of (16.32ﬁ given by linear approximation is equal to
(a) 2.01
(b) 2.10
(c) 2.02
(d) 2.20 (e) 2.06 (15) Find the critical numbers of f (m) = e” sing: for 0 _<_ :1: S 27r.
(a) 77/4 and 57r/4
(b) 37r/4 ﬂaw/4
(c) 7r/4 and 37r/4
(d) 7r/4 and 77r/4 (e) 7r/4 and 7r/2 (16) Compute [I 4(f — 3‘5) da:
(a) 2\/'2’ — 10/3
(b) \/'2‘ — 1/3
(c) \/9: + 4/3
((1) ‘2\/§ + 14/3 (6) 8/3 10 2:2:
(17) Evaluate é:— ( 0 arctant (it) at a: = %. (a) 7r/3 (b) 1 (18) A certain function f(:c) satisﬁes f”(x) = 2 —— 3m. We also know that f’(0) = —1
and f(0) = 1. Compute f(2). (60 ~1 0?) —3 11 (19) Compute lim(1 — 2:)3 {It—+0 (20) The derivative of a function f is given by f’(m) = (a:  1)2(93 — 2)3(:c — 3). Which of the following are correct?
I) f(2) is a local maximum and f(3) is a local minimum of f(a:).
II) f (:12) is increasing on the interval (1, 3).
III) f (11:) is decreasing on (—00, 1) and increasing on (1,00). (a) only I is correct (b) only I and III are correct (0) only II is correct ((1) only II and III are correct (6) only III is correct 12 (21) A rectangle is centered at the origin, its sides are parallel to the axes and all of its
vertices lie on the curve ll.’z:2+y2 = 8. What is the maximum area of such rectangle? (a) 4
(b) 8
(c) 4\/§
(d) 2V5
(e) 2 3x2 1
22 C 1; _.__._da:
( ) ompue/0 $3+1
(a)3\/§~3 (b) 2N5 — 1)
(C) 2 (d) 2N5 — 1)
(e) (ix/3: — 4 13 (23) On What intervals is the graph of f(:c) = :04 +4x3 — 189:2 — 6o: concave downward?
(a) on (—3, 1) and (2, 3)
(b) on (—00, ——3) and (1,00)
(0) only on (—00, 11)
(d) only on (3, 00) (e) on (—3, 1) .(24;) The ﬁgure below illustrates the graph of the derivative of a differentiable func
tion f which is deﬁned in (—4, 4). We can conclude that f (:13) achieves local maxima
and minima at the following points: (a) local maxima at —3 and 2 and local minima at —1 and 3 (b) local maxima at —1 and 3 and local minima at —3 and 2 (c) local maxima at —l and 3 and local minimum at 2 (d) local maxima at —3 and 2 and local minimum at —1 (e) local maximum at a point between —3 and —l and a local minimum at 0. 14 (25) The graph of the function f(:z:) = ~§x3 — §x2 + 2:1: + 2 looks mostly like ...
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 Fall '10
 Purzer

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