MATH 139 Midterm: MATH139 Fall 1999 Exam
31 Jan 2019
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Final Examination Friday, 10 December 1999 Mathematics 189-139A
1. Find the derivative function of
(a) f(x) = cos(x3) (b)f(x) = cos3x(c) f(x) = x2+ 3x+ 4
sin x
2. A function y=f(x) is known to satisfy the equation 3xy2−4x2y+y3= 5. Find y′
in terms of xand y.
3. Find all the critical points of the function f(x) = sin x−xcos xin the interval (−π, 3π)
and classify them.
4. Find the first and second derivative and all the critical points of the function
f(x) = e−x(x3−x2+x+ 1).
Classify the critical points and sketch the graph. Include the behaviour at ±∞.
5. What point on the line 3x−2y= 4 is nearest to the point (1,−1). (In minimizing a
distance, you can minimize the square of the distance.)
6. An open topped square box with a volume of 4 cubic meters is to be constructed using
a minimum of material. What should its dimensions be?
7. Use the differential to find good approximations to
(a) √101 (b) 28
1
3(c) log10 1002
8. Find the following limits if they exist or explain why they do not. (Do not use
L’Hospital’s rule).
(a) limx→2
x−2
x2−4(b) limx→0
sin x
|x|
(c) limx→∞
4x3−3x+ 1
5x3−4x2+ 7
1
80
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