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Unformatted text preview: Math 100, 120, 153 Common Exam/Dec. 1994 Math 100, 120, 153 Common Exam Dec. 1994 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. 1. [15 marks] (8) Find f’(1) if f(:c) = (22: + 3)e'. (b) If y = ln(a: + «EL—1 , ﬁnd dy/dz. (c) Find the slope of the tangent line to the curve y = W at z = 1. (d) Find the :c-coordinate of the point where the curve y = 2'22 — 2’ has a
horizontal tangent line. 2. [6 marks] Use the deﬁnition of derivative to calculate f’ (2) for the function
f(a:) = x/z’ + 5. No marks will be given for using any differentiation rules in this question. 3. [10 marks] The equation 3/3 + z’y = 10 has a solution y = f (1:) that
satisfies f(3) = 1. (:1) Find f’(3). (b) Check that f” (3) = 557. Page 1 (c) Use the results of parts (a) and (b) to make a sketch showing the
relationship between the graph of y = f(a:) near (3,1) and its tangent line
at that point. Give reasons to justify your sketch. 4. [10 marks] At time t the biomass M = M (t) of a cell culture is given by 4 M=1+3e-¢‘ (a) Find lim M(t) and tlim}M(t). t—i—oo b Calculate m and verify that w— = 1M (4 — M).
t“ dt 4 (c) What is the maximum rate of growth of M? 5. [10 marks] Two variable resistors, R and S, are connected in parallel, so
that their combined resistance C is given by 1-1+;
0—H 5' At an instant when R = 250 ohms and S = 1,000 ohms, R is increasing at a
rate of 100 ohms/ minute. How fast must .3 be changing at that moment if
C’ is increasing at a rate of 10 ohms/minute? 6. [10 marks] Four identical squares are cut out of a rectangle of cardboard
5 ft by 8 ft as shown in the ﬁgure, and the remaining piece is folded into
a closed, rectangular box, with the two extra ﬂaps tucked in. What is the Math 100, 120, 153 Common Exam/Dec. 1994 largest possible volume for such a box? 7. [10 marks] The function f has the following properties: ,lésmﬂz) =1, f(-1)= 0. f<o)=1. m) = 2.
f'($) = €514.53 for all real numbers 1:. (3) Find the intervals on which f is increasing and decreasing, and any local
maximum or minimum values of f. (b) Find the intervals on which the graph of f is concave up and concave
down. At what values of a: does the graph of f have inﬂection points? (c) Sketch the graph of f. 8. [9 marks] Evaluate the following limits: Page 2 ‘ . lna:
(a) sin(1rx)' , 1 1
0’) 321367;)- . 1/12
(c) 1135 (sec 2:) . Part B Consists Of Four Questions. You Are To Do Two Of Them.
You Must Specify Which Two You Want Marked. 9. [10 marks] Suppose that the function f satisﬁes lim “1) — 5 = 2-42 2:2 — 4 3' (8) Find f(a:). (b) Given that f is continuous at a: = 2, show that f is differentiable at
:c = 2 and find f’(2). 10. [10 marks] When ﬁred from rest at ground level, a small rocket rises
vertically so that its acceleration after t seconds is 6t m/sz. This continues
for the 10 seconds that its fuel lasts. Thereafter, the rocket’s acceleration is
9.8 m/s2 downward, due to gravity. In other words, the acceleration a(t) is
—9.8 if 10 < t < T, where T is the time at which the rocket strikes the ground after falling back.
Find explicit formulas for v(t), the velocity of the rocket, and y(t), the height
of the rocket, t seconds after it is fired. Both formulas should be valid for
0 s t g T. 11. [10 marks] Suppose that the function f (2:) satisﬁes , l H l
— _. .__. < < .—
f — 1, and 2 f (2:) for all z > f0) = 4, Math 100, 120, 153 Common Exam/Dec. 1994 Page 3 (3) Find the best linear (tangent line) approximation you can for the value of f (3). (b) Find the smallest interval that you can be sure contains f (3). 12. [10 marks] A function [(x) is continuous on the interval [a, b] and satisﬁes
a5f(:c)$b for aSsz. (a) Prove that there exists at least one number c in [a, b] such that f (c) = c. (b) If f also satisﬁes |f’(z)| < 1 for a g a: 5 b, prove that there is exactly
one number c in [a, b] such that f (c) = c. ...
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