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Unformatted text preview: Math 140 FINAL EXAMINATION Fall 2007 Answer each of the 9 numbered problems on a separate answer sheet. Each answer sheet
must have your name, your TA’s name, and the problem number (= page number). Show
all your work for each problem clearly on the answer sheet for that problem. You must show
enough written work to justify your answers. No calculators or electronic equipment allowed.
Don’t forget to sign the Honor Code on the last of the answer sheets. Good luck! 1. (5 points each) In each of the following, determine whether the limit exists as a real
number, as 00 or —00, or fails to exist. If the limit exists, evaluate it. _ sin(a:+2) _ x/9—t2 _ |x——5|
(a) $1392 902 — 4 (b) ting— t~ 3 (C) $13100 2115 — 10 2. (a) (10) Use the tangent line approximation to estimate (26)1/3. (b) (10) Let f(a:) = x3 + as — 3. Find an interval of the form [0, c + 1] in which f has
a zero. Support your answer by citing an appropriate theorem. 3. (10 points each) Compute the following derivatives. (Don’t simplify.) d tan(a:2) d2 d 3-—2m
__ 2 3y _ 3
d3: x3+$ (b) dy2 (y e ) (c) dx/W? ln(t +5) dt (a) 4. (a) (10) Consider the equation x4 - my +y2 = 7. Show that (—1, 2) is on the graph of
the equation, and then ﬁnd an equation for the line tangent to the graph of the
equation at the point (—1, 2). (b) (10) The altitude (i.e., height) of a triangle is increasing at a rate of 2 inches per
minute while the area of the triangle is increasing at the rate of 4 square inches per minute. At what rate is the base of the triangle changing when the altitude
is 10 inches and the area is 100 square inches? :52 (3x + 6) 5. (a) (5) Let M) = m asymptotes. . Identify all horizontal asymptotes and all vertical (b) (5) Give an example of a function f that is continuous for all real :5 but is not
differentiable at m = 71'. PLEASE TURN OVER FOR THE REMAINING PROBLEMS 6. A rectangle is to be drawn so its base is on the :r axis, its left side is on the y axis, and
the upper right vertex is on the parabola given by y = 12 — 5132. (a) (10) Draw a sketch of this situation, and ﬁnd a formula for the area A as a function
of the length of the base. (Don’t forget to include the domain of A!) (b) (15) Then determine the area A of the rectangle that is as large as possible,
identify the upper right vertex of that rectangle, and justify your answer as the
maximum possible area. —(33 ~ 2)(3x —~ 2). 7. Let $205 _ 1)2 H 4 3 ~4
ln1>. Then f’(x) = $73 and f”(a:) =
(a) (3) Find the domain of f. (b) (6) Find all horizontal asymptotes and all vertical asymptotes (if any), and label
them. (C) (3) Find all intervals on which f is increasing (if any). (d) (3) Find all relative maximum values and all relative minimum values (if any),
and identify each. ‘ (e) (6) Determine all intervals on which the graph of f is concave upward (if any).
(f) (4) Determine all inflection points (if any).
(g) (10) Sketch a graph of f, indicating all pertinent information on the graph. 8. (10 points per part) Perform the integrations: (a) /$466m5dx (b) AW—ﬂ—dzﬁ (c)/_2|a:+1ldx 3+cost 2 9. (a) (10) Let f(a:) = x3 + a: — 1 and g(33) = x3 — 3:2 + 2116 + 1. Find the area A of the
bounded region determined by the graphs of f and g. (C d 37
(b) (5) Simplify the following expression, showing your work: / f’ (t) dt — E / f (t) dt
1 2 END OF EXAMINATION — Good luck! ...
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This note was uploaded on 04/04/2010 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08