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**Unformatted text preview: **Math 41, Autumn 2009 Final Exam — December 7, 2009 Page 1 of 18 1. (9 points) Find each of the following limits, with justiﬁcation. If there is an inﬁnite limit, then explain
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strong wind blowing the kite away from the child, in such a way that the kite is moving horizontally
at a speed of 7 ft / sec. (a) At what rate is the child releasing string when 300 feet of string have already been released? Le+ 'f. Leth army/11L Shivg Nimséiij 'n“ is, ‘Hua OpiS’I'an’e MIA/86h ’HLQ K chi/ﬁe hypafenuse "m 11% Jfajmmf [.511 x be ’f/Li horizow’fa, LJ/S'f‘anté Wm #45:, md #49, Poin‘f‘ ﬁlmed/y Lelaw 44¢ on Jmuwf. gem ﬂuz Prob/Pm) 31. ,— 7 fazed) I 14‘
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to the equation f = 0. (Note: you don’t have to solve the earlier parts of this problem to solve
the subsequent ones; just cite any part’s stated result if you need it.) (a) Show that 304 — 3:10 + 1 = 0 for at least one at-value in the interval [1, 2]. Explain your reasoning
completely; however, you don’t have to ﬁnd the exact value. Node 5—‘ l#3+/;:{<O‘J 01W}?
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radius 7". (There is no material used for the bottom.) Thus, a formula for its surface area is A5110 2 27rrh + 27rr2. Suppose there is enough material available to build a silo with surface area equal to 500 square feet.
What is the maximum possible volume of such a silo? Justify your answer. (Note: if you use the
volume formulas on the reference sheet, remember that a hemisphere is half a sphere.) We (We Wat/#5500; We "f’i’uim P>O mi ‘20.
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is necessary. The statement “ling—10 + 6:12) : 2” can be rigorously justiﬁed by TRUE
13—) saying that for any positive 6, we have {(—10+ 6m) —2| < 6 whenever lac — 2| < 6/2. jig—X4 “wayska is aljebmkaily eyyimkmf to 1*?! <‘ ‘79 L04” you con/ﬂ Say “ Ix‘gkg/Q‘ WIVWW’V “‘314 ff: 0 [mews-Q
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The statement “lim (—10 + 6x) = 2” can be rigorously justiﬁed by @ FALSE :z:—>2
saying that for any positive 6, we have |(—10 + 6x) — 2| < 6 Whenever la: — 2| < 6/60.
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(Here a and b are constants.) cam (12+b2 /e‘m sin bx d3: = (a sin bx —- b cos M) + C’ [24 6D" ‘
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derivative 1” m||.1|2 345|6
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lines y = 0, m = 1, and a: = 4. Find the area of R by evaluating the limit of a Riemann sum that
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50.051111: c 742‘s Wflt’iés 1%, MP --. Math 41, Autumn 2009 Final Exam — December 7, 2009 Page 13 of 18 11. (10 points) A hybrid vehicle is traveling north from Los Angeles on Interstate 5; suppose we measure
its position by the number of miles north of LA it is located. Let f (2) be the rate, in gallons per
mile, at which the vehicle is consuming fuel with respect to position when the vehicle is located 2
miles north of Los Angeles. (a) What does the quantity f3500 f (z) dz represent? Express your answer in terms relevant to this
situation, and make it understandable to someone who does not know any calculus; be sure to
use any units that are appropriate. (0 i5 1%, 64¢“) Miami Bl) 4276/ Con/’5qu ﬁn Sal/0‘15) MVP/6h
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suppose 3(0) = 0. Deﬁne the function 3(t) H(t) = 0 f(z) d2. What does H (25) represent? Again use simple terminology relevant to this situation, and any
appropriate units. 1L6 ﬁlms {We amemmL 0‘? 48d cal/wwa j“ WA
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true and sometimes false, depending on the situation (“MAYBE”). For each part, decide which. You
must give a brief justiﬁcation in order to receive credit. 9 is a differentiable function of x. @ F MAYBE Simth has a Pc?3l+lv‘1 Jewva'l‘liﬁ? n‘l Why ’plf (I’Q’dml‘mué Mrywlmw)
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