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Unformatted text preview: Math 100 Exam/Dec. 1993 Math 100 Exam Dec. 1993 PART A: DO ALL 8 QUESTIONS IN PART A. J 1. [12 marks] Find the derivatives of the following functions. You do not
have to simplify your answers. (a) y = J21 +1 tan21:
(b) y = tan“‘(a:"’) Note: tan‘1 5 arctan
(c) u = (2+ 1)“ (d) “191%;351) 2. [10 marks] Find the derivative of 1(3) = \lz’ + .1: directly from the
J definition. No credit will be given if you use the rules of differentiation. 1 3. [10 marks] The equation 2:”; + 21113 = 8 deﬁnes y as a function of 1:,
\] y=f(:r),nearz=2.y=l. (a) Find the slope of the curve :21; + 2x113 = 8 when a: = 2, y = l. (b) Use the linear approximation (tangent line approximation) to ﬁnd an
approximate value for f (1.92). (c) Given that y” = % for a: = 2 and y = 1, does the tangent line approxi
mation yield a bigger or smaller answer than the actual value in some small
neighborhood of a: = 2? You must give reasons for your answer. 4. [8 marks] A half mile race track consists of two opposite sides of a rect Page 1 angle with semicircular ends as shown in the following diagram. Find the dimensions a: and y which will maximize the area of the rectangle. 5. [8 marks] A 50 foot ladder is placed against a wall of a large building.
The base of the ladder, resting on slippery ground, slips away from the wall
at a rate of 3 feet per second. Find the rate of change of the height of the
ladder top when the base of the ladder is 30 feet away from the wall. 6. [12 marks] Evaluate the following limits. t_ i
J?
(‘l 332.. ——1‘
s/i + —
ﬂ
sin" l — 1  (b) hm ———23’—§ Note: sin"l .=.. arcsin 2—01 I — l . . 1/,
(c) [1330 + sin 2:) . a'
id) 21320 1 “1,, the value of a. where a > 0 is a constant. Hint: The answer depends on 1 7. [8 marks] Let f(:r) = ze' — 2. (a) Prove that f (1:) = 0 has exactly one solution r between 1: = 0 and z = l. Math 100 Exam/Dec. 1993 (b) Give the “Newton’s method" iteration formula for ﬁnding 1'. You do
not actually have to ﬁnd the root. (c) Show that for any initial value so, 0 5 1:0 3 1, the value of 11 from
Newton's method satisﬁes 2:1 2 r. 8. [12 marks] Let y = 22"” + 22/3. (a) Find all intervals on which the function is increasing, decreasing, concave
up or concave down, and ﬁnd all inﬂection points (if any). (b) Determine the coordinates of all local or absolute maxima and minima
(if any). (c) Determine the asymptotes of the graph (if any). ((1) Plot the graph. Notes: You must show all your work to get full marks. No marks will be
given for a graph that does not match the calculations in parts a, b and c. PART B: CONSISTS OF 4 QUESTIONS. YOU ARE TO DO 2.
YOU MUST SPECIFY WHICH 2 YOU WANT MARKED. l
2 a: sin— forty“) 9. [10 marks] Let f(z) be deﬁned by f(a:) = {0 x f o
01' 3 = . (a) Use the deﬁnition of the derivative to ﬁnd f’(0). (b) Explain why f'(:r) is not continuous at z = 0. 10. [10 marks] Prove that: (a) sinzzzsfor031sg. Page 2 2 (b) coszgl—iforOst 1.
1r 2 11. [10 marks] Let f (2:) be a function deﬁned on the real line such that f’(::)
and f"(::) exist for all 2. Suppose f(1) = 0, f(2) = 3, f(3) = 2 and f(4) = 4. (a) Prove that f'(a) = 0 for some a. (b Prove that f”(b) = 0 for some b. 12. [10 marks] A lake of constant volume V gallons contains Q(t) pounds of
pollutant at time t, evenly distributed throughout the lake. Water containing
a concentration of 1: pounds per gallon of pollutant enters the lake at a rate
of r gallons per minute, and the well mixed solution leaves at the same rate. (a) If 00 is the amount of pollutant in the lake at time t = 0 ﬁnd Q(t) for
any time I. (b) Determine the limiting amount of pollutant in the lake (i.e. ﬁnd
guano 0(a)). (c) If k = 0 ﬁnd the time T for the amount of pollutant to be reduced to
50% of its original value. ...
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 Math, Derivative, 3 feet, Convex function

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