4LZZZZZhXL..DDF=/RdCalculus 125T Name___________________
Test II
March 20, 1989
1. (12 points) Find f(x) for each of the following:
(a) f(x) = 6x3 cosx
(b) f(x) = f(x2 + 1,x3)
(c) f(x) = sin3 x
(d) f(x) = r(2x 3x2)
2. (3 points) What is the third derivative of g(x) = f(1,x) ?
3. (6 points) Suppose xy3 +2y2 x2 = xy +8. What is the slope of the line which is tangent to the graph of this equation at the point (0,2)?
4. (5 points) Approximate r(3,65) using differentials.
5. (6 points) If f(x) = cos x and f(f(,2)) = 3, what is f(x) ?
6. (8 points) Let f(x) = 2x3 3x2 12x + 1. What are the extreme values of f on [4,4]?
7. (9 points) Find the critical points of
(a) f(x) = 2x3 3x2 36x + 1
(b) f(x) = f(1,x2)
(c) f(x) = xsup8(f(3,4))
8. (8 points) Use the first derivative test to find the relative extreme values of
f(x) = x3 + 6x2 15x +6.
9. (8 points) Use the second derivative test to find the relative extreme values of
f(x) = 3x4 10x3 + 6x2+ 3.
10. (9 points) Find the antiderivatives of:
(a) f(x) = x3
(b) f(x) = cos x
(c) f(x) = sin f(1,2) x
11. (7 points) Let f(x) = x3 6x2 36x+ 4. Where is f strictly increasing, and where is f strictly decreasing?
12. (8 points) Notre Dame Ave. and Howard St. are straight and perpendicular to each other. Suppose Abigail is running north on N.D. Ave. at a pace of 8 mph, and Harold is running east on Howard St. When Abigail is 4 miles from the intersection and Harold is 3 miles form it, the distance between them is decreasing at a rate of 10 mph. How fast is Harold running?
13. (11 points) Let f(x) = x3 3x2 105x + 6.
(a) Find the relative extreme values of f.
(b) Sketch the graph of f.
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