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Unformatted text preview: MAC 2311 — EXAM 3
FALL 2806 A. Sign your scantron sheet in ink in the White area on the back. ‘ B. Write and encode in the spaces indicated: 1) Your name (last name, first initial, middle initial)
2) Your UF ID number 3) Your discussion section number 0. Under “Special Codes”, encode the test number 3,1 . D. At the top right of your scantron sheet, for “Test Form Code”, encode A. e B C D E E. This test consists of 8 four—point multiple choice questions, 4 twopoint multiple
choice questions, a four—point bonus multiple choice question, and four pages (two
pages front and back) of free response questions worth forty points. The time allowed
is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before you turn in your test, check carefully for transcribing errors. Your
responses cannot be changed after you turn in your exam. 2) You must turn in your scantron and freeresponse portion of the exam to the
proctor. Note: Problems 1—8 are worth four points each. 1. Find the slope of the tangent line to the graph of the function f(33) = minﬂﬂ ~— 3:2)
at the point Where a: = 3. . A. 1 3'3 C. 0 D. wﬁ E. ~18 $2 . . . 1 — e
2. Evaluate the 11111111. $15.55 W A. 1 B. 2 C. 0 D. —1 E. ~2 3. At what value of t on [0, 25] does the function ﬁt) :2 t3 V25 4» t attain its maximum
value? A. 15 B. 16 C. 18 D. 20 E. 22' 4. Suppose the computing time T (in minutes) that a computer requires for a certain
calculation is given by the function T00) = 40+2p+40 111(1 +p) on [0,100}, where
p is the percentage of computer memory that is unavailable. Use differentials to approximate the change in computing time as the percentage of unavailable memory
changes from 9 to 12 percent. A. 6 minutes B. 9 minutes (3. 12 minutes D. 18 minutes E. 24 minutes 5. A television station determines that over a 36~hour period, the median age A of its
viewers is given by the function AU?) : 21 + (ix/t — t, where t is the number of hours
that have elapsed. Find the minimum and maximum median age during the 36—hour period.
A. 18 and 26 B. 18 and 32 C. 21 and 30
D. 21 and 32 E. 26 and 30 6. Choose the value of a: beiow that lies in an interval on which the function Mac) = 4 — 29:  it is both decreasing and concave up. A. —3 B. —1 C. 1 D. 3 E. none of these 7. The graph of f ’(sc) is shown below for the continuous function ﬂat) . Choose the
' TRUE statement. ﬁx) A. ﬁn) has no relative minima
13. ﬁre) has one inﬂection point C. f(33) has exactly two critical numbers D. f (cc) has no vertical tangent lines 8 A train starts at the origin and travels along the curve 9— 2 3:9 so that its velocity' 1s
10 ion/hr in the rat—direction as it passes due north of an observer at the point (2 0)
At what rate is the distance between the train and observer changing at that time? A, 15 ion2 per hour B. 40 km2 per hour 0. 25 km2 per hour D. 20 km2 per hour E. 60 km2 per hour Net{14
t Note: Problems 9—12 ore worth two points each. 9. Suppose the continuous function f (:3) has horizontal tangent ﬁnes at a; w ~2, 3: m 0,
and a: r» 1, and suppose that f”(m) z 2(m —— 1)(2m2 + 23; — 1). At which of the
mvaioes beiow must f (2:) have a local maximum? A. 1 only B. 0 only C. «~22 only
D. 0 and —2 E. none of these 10. Which of the following derivatives are correct (on their domain)? I.%ln(3$)—E II.~ti—mwlnlris—us—$__m2
A. I oniy B. both I and II
C. II oniy D. neither I nor II
93% 1 3
11. The function f(:1:) m g9; has derivative f’(:c) = 2 :1}. Which of the following
_ m n
are true? ' 3mg
I. f(:1:) has two critical numbers II. ﬁx) has no cusps (kinks)
III. f (as) has only one local extremum
A. I only B. III only C. I and III only D. II and III only E. I, II, and III 12. Which of the foﬂowing is the liﬂearization 13(3) of the function 9(32) ﬁ 3111(3)) at acme?
A. L(m) 2 2m —— e B. L(a:) u :$~1 0. 13(16): 2a:
1
33.142322: E.L(m)ue—m+e—l Bonus! !(4 points) 1
33. Evaluate the limit: lim (1 + 33:2 )m . 3—400 MAC 2311—‘EXAM 3 Free Response NA'MEMQ; SECTION UF ID I ‘ ' TEST FORM CODE A ' ‘ YOU MUST SHOW ALL OF YOUR WORK TO RECEIVE CREDITH 1. Exaxxﬁne the function 9(22) # :1: ’52". 3) Calculate g’(w) using logarithmic differentiation. b) Write the equation of the tangent line to the graph of 9(3) at the point with
xcoordinate I. c) What criticallnumber does 9(a)) have on (0, co )? Is it at local maximum or
minimum? if so, which is it and What is the value at that as? 2. A contractor has 100 meters of fencewbuilding material, which needs to be divided
into two portions. One portion is used to create a square fence at a cost of $ 2 per
square meter of area that is enclosed. rI‘he rest is put into storage at a cost of $8 per meter. (See picture)
f/J'Cj giw 9W
loo waive 0'; 'FCﬂQ /N\" “MM W eat“ rewainm not“ a) Write Ian equation for the total cost in terms of 3:. 1)) Find the vaiues of a: and 3; that produce the ieast cost Use the second derivative
test to verify that you have found a. minimum value. c) What is the minimum cost? (1) Under certain assumptions, if E3 A is the cost per square meter of area for the square, and $3 per meter is the storage cost, their the “optimum” cc occurs 23 
when m am am. Suppose that the construction cost A is increasing at $2 per year, and the storage cost B is increasing at $ 1 per year. At what rate is is the
“”optimum as changing when Am 2 and Rm 8? Include units. Is it increasing
or decreasing? Bonus Quickly, given your answer to part (d), what would you estimate to be the
“optimum” a: three years iater? What is its actual value? ' NAME t,   ‘ SECTION 1
4333* 3. Examine the function 9(93) = :1: + 4, and ﬁll in the blanks below.
If there is no answer, write “none”.
' —— 8 .._ 2.2 3.2
Note that g’(m) :: M...“ and 911(3)) m W ' 3%?! (a: + 4)2 933361: + 4)3 9(53) has domain
9(93) has intercept(s) g(:r) has vertical asymptote(s) a: m
9(m) has horizontal asymptote(s) y =2 .
g(sc) has critical numbers :3 H 9(33) is increasing on the interval(s) WW g(3:) is decreasing 0n the interval(s) '  '
g(a:) has a vertical tangent line or cusp at m = g(w) has a horizontal tangent lineat a:
g(a:) has a local minimum at a: =
9(32) has a local mandmum at a: 2 ii 9(33) is concave up on the interva1(s) . .
9(32) is concave demo: on the interval(s) g(m) has inﬂection point{s} at a: m Sketch the graph of 9(3) below, using the above information. Label your local
extreme and inﬂection points. Note that 9(2) m 1, 9(22) 8 0.5, g(—3.2) m m7. YOU HAVE REACHED
THE END OF THE EXAMS! ' THERE ARE NO PROBLEMS w'.
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