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Unformatted text preview: Math 171 — Calculus with Algebra and Trigonometry. I
Fall 2006 — Final Exam Name:
TA: VERSION A (a) The exam has 6 problems. (b) Show your work for every problem. Answers without justiﬁcation will not receive credit. (0) Calculators are not allowed.
(d) There is a blank page at the end of the exam for if you need it. 1. Compute the following limits.
(3) (10 points) (b) (10 (points) (0) (10 points) cos(:1:) — tan(x) . t2—5t+6
hm ————.
t—’3+> 3—t £1113 cot(x) — csc(a:). Iim sin (———————~——8m($) * 5 ) . 2. (a) (10 points) Let f(m) = sin(9:3 + 3). compute f’(ac). (b) (10 points) Let g(a:) = W. Compute g’(:c) .1! (c) (10 points) Let h(a:) = 003(333). Compute h<10)(93). [Note that you are askedto compute the
tenth derivative of h] ' 3. Consider the following equation: sin(2y) + cos(3ac) = O.
(a) (10 points) Show that (7741, g) lies on the graph of this equation. (b) (20 points) Compute the tangent line to the graph at the point (g, g). (c) (10 points) Explain why the graph of this equation cannot be the graph of a function y = f (3:) 4. Recall that lim Sin(h)
h—~>0 h (a) (20 pointS) Use the above limit to deduce that Ilmh #00802 ~ 1 = =1. 0. (b) (20 points) Use the formal deﬁnition of the derivative to Show that if f(:v) = sin(2a:) then
f’(m) 2 2008(2113) .5. Suppose that the temperature (in Fahrenheit) in Madison t hours after spring equinox is given
by the function 7r it ~—  —) + 40; T(t) = —1Ocos(—1~7r§t) + 40 sin(12 365 (a) (15 points) Is the function T periodic? If so, What is its period? If not, why not? (b) (15 points) At What rate does the temperature change when t = 0? (c) (10 points) Is it morning, afternoon, evening or» night when t ,: 0? Brieﬂy explain your
answer.  6. (20 points) Let f (:0) = cos(2:1:). Show that f (as)  f’ (at) = —— sin(4x).
[You may use that cos(u + v) = cos(u) 003(2)) — sin(u) sin(v) and sin(u + v) = cos(u) 3111(1)) +
Sin(u) cos(v).] ...
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 Fall '07
 GOMEZ,JONES
 Calculus, Algebra, Trigonometry

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