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quc'l'ice, Final Exam 1.) (10 pts. each) Diffferentiate each of the following. DO NOT SIMPLIFY answers. 2 a.) y 2 6I cos3(53:) b.) y = arcsin(2x+10g$) «ClJJ = x 2.) pts.) You deposit $2000 in a retirement account earning 12% annual interest
compounded monthly. In how many years will the account grow to $10, 000 ? 3.) (10 pts.) The manager of the Economy Motel charges $30 per room and rents 48 rooms
each night. For each $5 increase in room charge four (4) fewer rooms are rented. What
charge per room will maximize the total amount of money the manager will make in one night ? ' '2
SI? $+l' 4.) (10 pts.) Use the limit deﬁnition of derivative to differentiate f = 5.) (l0 pts.) Use limits to determine the values of the constants A and B so that the
following function is continuous for all values of :c . Bx2+Ax, ifmg —1
f(3:)= 2B—Ax, if—l<m§2
:z:+3, ifx>2. . v l p
I 6.) (10 pts.) Find all points (x, y) on the graph of y = g with tangent lin’es passing
through the point (4, 0). » 7.) (10 pts.) The radius of a sphere is measured with an absolute percentage error of at most 4% . Use differentials to estimate the maximum absolute percentage error in
4
computing the volume of the sphere. ( V = gvrra . ) 3 8.) Consider the equation 27 — :1: = sins: . a.)' (10 pts.) Use the Intermediate Value Theorem to verify that the ,equation is
solvable. ‘ b.) (5 pts.) Use Newton’s method to estimate the value of the solution of the equation
to three decimal places. 9.) Car B is 34 miles directly east of car A and begins moving west at 90 mph. At the
same moment car A begins moving north at 60 mph. a.) (10 pt‘s.) At What rate is the distance between the cars changing after t : % hr. “.7 b.) (10 pts.) What is the minimum distance between the cars and at What time t
does the minimum distance occur ? 10.) (10 pts.) Find the slope and concavity of the graph of $3; + y2 : 3X+l at the point
(0) '1) ' 11.) (10 pts.) Consider all rectangles in the ﬁrst quadrant inscribed in such a way that
their bases lie on the X—axis with the top corner on the graph of y :— V’4 — x . Find the
length and width of the rectangle of maximum area. 12.) (.15 pts.) Consider the function = x e 2 . Determine where f is increasing,
decreasing, concave up, and concave down. Identify all relative and absolute extrema, inﬂection points, X and yintercepts, and vertical and horizontal asymptotes. Sketch the
~—:c —:c
, a: (—‘l I, .r (—)
graph. You may assume that f = (1 — e 2 and f = — 1) e 2 13.) (10 pts.) A lighthouse sits one (1) mile offshore with a light beam turning counter—
clockwise at the rate of ten (10) revolutions per minute. How fast is the light beam racing
down the shoreline when the beam strikes a point on the shore twelve miles south of the nearest point on the shore ? a Write your ﬁnal
answer in MILES PER HOUR. ' 1 m. “J‘Itl‘me
L/> ‘ ’ "é('"0
14.) (10 pts. each) Evaluate the following limits. .3: ‘/
_ _ msinm  O X /
a.) hm —————,, 70 /
r—>0 (arctanas) m
J“
o.
x V”
g  N
b. 1‘ l  ﬂ
9 .38. x m c.) 11520 (mm  *’
h’ ——~ E
8 Each of the following three EXTRA CREDIT PROBLEMS is worth 10 points. These
problems are OPTIONAL. In C
ln B 1.) Show that logBC = ‘ p 2.) Find all values of K for which the the function f (r) «2:3 + Km + x2 is NOT onetoone. 3.) Find a tilted asymptote for the function y = \/:172 + :c . ...
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 Fall '07
 Osserman

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