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Unformatted text preview: Math 100, 120, 153 Common Exam/Dec. 1995 Math 100, 120, 153 Common Exam Dec. 1995 Instructions: Students may bring and use one 8.5” x 11” information sheet.
Calculators are allowed.
Show all calculations for your solutions. Do All Eight Questions In Part A. l. [8 marks] Use the deﬁnition of derivative to determine the derivative of “3) = 22:1 Note: no credit will. be given for using differentiation formulas. 2. [12 marks] Consider the functions 282:
“3)  W’ (a) Show that the tangent lines to the graphs of f (1:) and g(:r) at a: = 0 are
parallel. ' (b) For what value of I: will the tangent lines to the graphs of f (1:) and h(2:)
at a: = 0 be perpendicular? 3. [10 marks] The equation y5 + my2 + 1:3 = 4:2: + 3 defines y implicitly as
a function of :1: near the point (2, 1). (8) Determine the values of y’ and y” at this point.
(b) Use the tangent line approximation to estimate y when a: = 1.97. (c) Make a sketch showing how the curve relates to the tangent line at the
point (2,1). g(x) = 1+tan‘1(ln(1+3m)), h(z) = sin(k:r)+sin‘l(k:r). Page 1 4. [10 marks] A closed box has a square base and a volume of 320cm2. The
material used for the base and for the top is five times as expensive per unit
area as the material used for the four sides. Determine the dimensions of the
box for which the total cost of materials used is minimum. 5. [10 marks] When sugar is dissolved in water, it dissolves at a rate pro
portional to the amount of undissolved sugar present. After 1 minute, 75%
of an initial portion of sugar is still undissolved. (a) How long does it take for 75% of an initial portion of sugar to dissolve? (b) After 2.5 minutes, there are 10gm of undissolved sugar left. How much
sugar was there initially? 6. [10 marks] An aircraft climbing at a constant angle of 30° above the
horizontal passes directly over a ground radar station at an altitude of 1 km.
At a later instant the radar shows that the aircraft is at a distance of 2 km
from the station, and that this distance is increasing at 7 km/min. What
is the speed of the aircraft at that instant? (Note: the law of cosines: c2 =
a2 + b2 — 2ab cosC may be useful.) 7. [10 marks] The function f(:1:) = (:1:3 — 5)e" has one critical point in
the interval a: > 0. Find this critical point with error within $0.005 by
applying Newton’s method to an appropriate function. Choose your initial
approximation carefully, and show all intermediate approximations. 8. [10 marks] The function f (1:) is given by f (1:) = $5 — 10km:4 + 2515233,
where k is a positive constant. (a) Determine the intervals on which f (:r) is either increasing or decreasing.
Determine all local maxima and minima. (b) Determine the intervals on which the graph is either concave upward or
downward. What are the inﬂection points of f (2:)? (c) Sketch the graph of f (2:). Math 100, 120, 153 Common Exam/Dec. 1995 Page 2 Part B Consists Of Four Questions. You Are To Do Two Of Them. (a) Show that f’ (0) exists and ﬁnd its value. You Must Specify Which Two You Want Marked.
(b) Show that f’(l/3) does not exist. 9. [10 marks] A car is traveling at night along a highway curved in the shape c For what values of a: does I a: fail t  7
o e ist.
of a parabola with northsouth axis and with its vertex at the origin. The ( ) f ( ) x
car starts at a point 100 m west and 100 m north of the origin and travels in
an easterly direction. There is a statue located 100 m east and 50 m north of the origln (i) f " (z) is continuous for all 1:. (ii) f(a:) is constant on the interval :1: < 0. (iii) f(1) = l and f(2) = 2. 12. [10 marks] Let [(2) be a function satisfying: (a) At what point on the highway will the car be when its headlights illu
minate the statue? . (b) At what point on the highway will the car come closest to the statue? . _
> (a) Show that there 18 a pomt a with f’(a) = l. . k ' h ' :
1° [10 mar 51 0mm!" t 9 ﬁgure at the "gm (b) Show that for all k with 0 < k g 1/2, there exists 0 with f”(c) = k. The distance between A and B is 2 m. A The distance between B and C is l m. The distance between C and D is 10 m,
as is the distance between D and E. Find the largest possible value of 0 =
lAPB, where P is some point on the
line segment CE. Explain why the angle you have found
is the absolute maximum. I A D E 11. [10 marks] Consider the function f (1:) defined by when a: = 0. rm={ 311505531 W ...
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