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Unformatted text preview: MATH 140 Dr. Wolfe FINAL EXAM July 25, 2008 1. (25 points) Determine whether each limit exists as a real number, as 00 or —00, or fails
to exist. If the limit exists, evaluate it. Give reasons for your answers. (a) lim @ (b) 11111 W
$—>oo x [IL—+0 h 296 — 6, a: < 1
(c) lim1 f(a:), where f(:1:) = 0, cc 2 1
“3—) —(ac + 1)2, 33 > 1 2. (15 points) Water leaking onto a ﬂoor creates a circular pool whose area increases at a rate of 3 square inches per minute. How fast is the radius of the pool increasing when the
area of the pool is 647r square inches? 3. .(30 points) Let
1 1
me) = — — —— a; $2. (a) Find the intervals on which f is increasing and those on which f is decreasing. (b) Find all local maximum and minimum points. ' (c) Find the intervals on which the graph of f is concave up and those on which the graph
of f is concave down. ((1) Find all points of inﬂection of the graph of f. (e) Find any horizontal and vertical asymptotes. (f) Find all x and y intercepts. (g) Sketch the graph of y = f Indicate all features found above. 4. (20 points) A manufacturer wishes to produce rectangular containers with square bot toms and tops, each container having a capacity of 250 cubic inches. If the material used for the top and bottom costs twice as much per square inch as the material for the sides,
What dimensions will minimize the cost? 5. (25 points) Evaluate the following integrals: (a) leg—Erich?) (b) /e‘y(1+e2y) dy (c) / 5+1 dw 6. (20 points) (a) Sketch the ellipse 4x2 + 93;2 = 36 and locate the foci and vertices. (b) Find the equation of the line L which is tangent to the ellipse at (—3\/§/ 2, 1) (c) Express the area A of the region inside the ellipse of part (a) as an integral. Do not
attempt to evaluate the integral. 7. (15 points) Let f(ac) = a: — 5111113. (a) Show that there is exactly one number a with 1 < oz < 2 such that f (a) = 0. Note:
ln2 % .69. (b) We Wish to ﬁnd a by using the Newton—Raphson method. If our initial guess is 930 = 1,
What is $1 ? ...
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 Spring '08
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 Algebra, Derivative, Inch, Convex function, Square inch

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