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Unformatted text preview: Math 140 Final Examination Fall, 2006 Instructions: Answer each of the 10 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 l. (8 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.
Sing: 36433 — 3 a) lim b) lim xsina: 0) lim
x——>oo :1: m—>oo x—>0 4:): 2. (8 points each) Compute the following derivatives (you do NOT need to simplify your answers):
a) i ytany b 1 t2 \/ x3 + 1 dx c) (cos4((3t + 4)10))/
dy y2 + 1 dt 3
3. (15 points) Find all asymptotes (both horizontal and vertical) of the function
9:2 — 3:): + 2
“a” — "ﬂ— 4. (10 points each) A ball shot directly upward from the ground with an initial speed of
k feet per second has a height of h(t) = kt — 16152 for all times t 2 0 until the ball hits
the ground again. a) Find the initial speed k if the ball is to attain its maximum height 2 seconds after
it is shot upward. b) At what speed is the ball moving at the instant it attains its maximum height? 5. (20 points) A helicopter ﬂies parallel to the ground at an altitude of 1/2 kilometer and
at a speed of 2 kilometers per minute. If the helicopter passes directly over the math
building, at What rate is the distance between the helicopter and the math building
changing 3 minutes after the helicopter ﬂies over the math building? please turn over —-> ._._.______.._._._._..__._._..__—_. _.__.—.—_———___._._.____________—_________-_.—__—_—__—.
v 6. (5 points each) Let f(a:) = Ice—3”.
a) Find all relative maximum values of f (if any).
b) Find all relative minimum values of f (if any).
c) Find the intervals on which the graph of f is concave upward (if any). (1) Find all inﬂection points on the graph of f (if any). 7. The point (0, 7r) is on the graph of ycos(:c —— y) + 7r 2 3x.
a) (10 points) Find the slope of the line tangent to the graph at the point (0, 7r). b) (5 points) Find an equation (in any form) of the line tangent to the graph at the
point (0,7r). 8. (20 points) For an electron in the 21) state of an excited hydrogen atom, the probability
density function P of the distance 7" between the electron and the center of the atom is given by
P 7' = —— e‘T/a
for some constant a > 0. Find the distance 7" that maximizes the probability density
function. 9. (9 points each) Evaluate the following integrals: 1 4x3 2 1 2
a) / |y+1|dy b) /w2+1da: c) /1(w+——) dw —2 10. (15 points) Suppose that the velocity at time t of an object moving along a straight
line is given by v(t) = sint + seczt cm/sec for all times 0 g t < 71' / 2. Find the total
distance D travelled by the object during the ﬁrst 77 / 3 seconds. END OF EXAM —- GOOD LUCK! ...
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- Spring '08