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Unformatted text preview: MAC 231 l Test 3
FALL 2004 A. Sign your scantron sheet on the back at the bottom in ink. B. In pencil, write and encode in the spaces indicated:
1) Name (last name, ﬁrst initial, middle initial) 2) UP ID Number
3) Discussion Section Number C. Under “special codes", code in the test ID number 3,2.
l 2 o 4 5 6 7 8 9 O l  3 4 5 6 7 8 9 0 D. At the t0p right of your answer sheet, for “Test Form
C ode”, encode B. A o C D E E. This test consists of l l ﬁvepoint multiple choice
questions, ﬁve onepoint bonus questions and two pages
(both sides) of partial credit worth 25 points. The time allowed is 90 minutes. F. When you are ﬁnished:
1) Before turning in your test, check for transcribing errors. Any mistakes you leave in are there to stay. 2) You must turn in the scantron sheet and tear off sheets
to your discussion leader. 3) Answers will be posted later this evening on the class
website. r4 2 .
1‘ Find the value(s) ofx at which ﬂx) : e "'2x has relative extremal
relative minimum relative maximum
atx=___ atx=__
a. 0 il
b 0 none
0 none 0
d i1 0 2. The deﬂection of a beam of length 4 feet is given by the
function D(x) : x4 — 4x3 + 4x2 for 0 S x S 4 where
x is the distance from one end of the beam. Find the value
of x which yields the maximum deﬂection. Hint: When is
I) a maximum on [0.4]? a.x=:0 b.x:2 c.x=l d.x=4 e.x:3 3. The graph ofﬂx) : 4x5 + 5x4 + 6 is both decreasing and
concave up on __ ___. a. (—3/4, 0) only b. (~00, —3/4) U (0. 00) c. (—1,—3/4) only d. (—l,0) only e. (—60,—1) only 1B 4. Given the graph of f ’(x), which one of the following
statements would be true for the graph afﬁx)? a, ﬁx) is concave down on (0,4) U (4.7). b. ﬂx) has local maxima atx = 2 andx = 6.
c. f(x) is increasing on (0.2) U (4, 6) U (8,00).
d. _/(x) has local minima at x : 4 and x : 7.
0. ﬁx) has inﬂection points atx : 2> x : 6. andx :2 7. 5, If _/(x) 2 (x2 — 4103/5, ﬁnd/(x) and determine which of the
following statements is/are true. P: The graph of/(x) has one horizontal tangent line.
Qzﬂx) has exactly three critical numbers,
R: .1: — 0 is a vertical asymptote of the graph of ﬁx).
S: The graph ofﬂx) has two vertical tangent lines.
a. P and R only b. P and Q only c. P and S only
(1. P, Q, and S only e. Q, R, and S only 6. Find the linearization of/(x) _— 1n(x2 ) nearx : e and use it to
approximate the value of in 9. a. ln9e4+2e b.1n9zg——l c.1n9~+ d. 1119226 0. ln9z8w2e 2B dy 7. Use logarithmic differentiation to ﬁnd at x ~—— 2 if
y 2 xWx).
a.—2 b. 2~2h12 c. l—lr12
d. 4 —41n2 e. 2 8. Find the value of C guaranteed by the Mean. Value Theorem
for/(x) = (x— n3 on [0,3]. ' a. 0:2 b. 623/2 0. 6:0 (1. 021/2 e. Cal 9. The radius of a sphere is measured as 5 inches and this
measurement is used to calculate the volume of the sphere.
Use differentials to approximate the percentage error that
could be made in the calculation if the radius measurement has an error ofi0.01inch. a. :15 % b. i075 % c. $0.6 % d. 14.2% e. i0.5% 10. Use L’Hospital’s rule, if possible, to ﬁnd the following limit. 3&3 w (3 + x) lim x~>0 352
a. 1/6 b. 0 c. 3/2
d. 2/3 c. 1/2 3B 2 . . .
11. The graph of ﬁx) : L + cosx has an inﬂection pomt on 4
[0,270 atx =
a giandﬂ b land—S—E— CZEand—H—II”
' 3 3 ' 3 3 I 6 6
d. % and ~56l e. f has no inﬂection points
Bonus Questions W!!! (1 point each). Be sure to bubble (a) for True and (b) for False. 12. lfﬂx) has a critical number at x z 2. then/(2) must be a
local or relative extreme value of the function. a. True b. False d 2 2x+2
l3. — log (x +2x) =— #————w——.
dx( 2 ) (x2+2x)in2 a. True b. False 14. The graph of/(x) : has an inﬂection point at x z 0. a. True b. False 15 . If g ’(x) h ’(x). then g(x) and 11(x) must be the same
function. a. True b. False 16. Ifﬂx) is continuous on [—3, I], then/(x) must have a
minimum and a maximum value on [T3, 1]. a. True I). False 4B MAC 23] i '1‘EST3B PART II
FALL 2004 N A M E_ ____ _ _ SECTK)N__ MGNATURE_mHum_ UHD___ sum W an WQRK 191992355 FULL “FEE I. An inverted conc shaped container is being ﬁlled with liquid at a
rate ot‘37r cubic inches per second, If‘the height oi‘the cone is twice the radius. how fast is the height of the liquid rising when it is 6 inches high? Him: I7 — «é—mZ/z. ﬂ_ i n/sec
u’r ' "—"'—'— 513 2. Consider the ﬁmctionﬂx) : x3 3(x + 5) and its derivatives: z 'fiu(x) A
Determine the following: (ifnone. write "nonc‘d x intercept(s): x —
y intereept(s)t y 1—
vertical asymptote: x 2 horizontal asymptote: y — critical number('s): .r — __ _ horizontal tangent line(s) at x — _ vertical tangent linc(s) or cusp(s) at X = _ I increasing on decreasing on
relative (local) maximum at x 7 relative (local) minimum at x — _ concave up on concave down on _ in ll ection point(s) at x r ()B NAMEW __ SECTION 3. Sketch the graph ofﬂx) from problem #2. Use all the information
you obtained in problem #2‘ Label all interceptsl local extrema
and inflection points. Hint: I/(AZ) = 4.76. 7B 4. A shipping crate with a square base is to be constructed so that it
has a volume of 250 cubic meters. The material for the top and
bottom costs $2 per square meter and the material for the sides
costs $1 per square meter What dimensions will minimize the
cost Ollcontruction‘? Function to be minimized; _ Constraint: V 2. "1 Use the Second Derivative Test to verify yoiir resultst 8B ...
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