4PZZd" Lllc=/RdCalc. 125T Name_________________________
Exam III
April 26, 1989
1. (8 points) Find the inflection points of the graph of
f(x) = x6 6x5 + f(15,2) x4 + 3.
2. (6 points)
a. Find a(lim,x) f(4x2 2,6x3 + 3x + 4)
b. Find a(lim,x) f(3x6 + 4x4,5x6 10x3) .
3. (10 points) We have a piece of wire 10 feet long and wish to cut it so that one piece can be formed into a circle, and the other into a square. How should the wire be cut so that the combined area of the two figures is as large as possible?
4. (8 points) Let f(x) = x5 + 5x4 60x3 + 17x 3.
Where is the graph of f concave upward, and where is it concave downward?
5. (3 points) Let f(x) = ln(3 x2)
a. What is the domain of f?
b. What is f(x)?
6. (7 points) Let f(x) = 3x2 + 1, P = {2, 1, f(1,2), 1,2}.
What is the lower sum of f with respect to the partition P?
7. (3 points) If f is continuous on [a,b] and F is an antiderivative of f, then
i( a, b,a( , )) f(x) dx = _________________________________.
8. (20 points) Evaluate the following integrals.
a. i( f(,3), f(,2),a( , )) (cos x + f(1,x3) ) dx
b. i( 1, 0,a( , )) (3x4 4x2 + 6x 1) dx
c. i( 0, 1,a( , )) (5x4 3x)(2x5 3x2 + 1)17 dx
d. i( f(,2), f(3,2),a( , )) sin5 t cos t dt
9. (15 points) Evaluate the following integrals.
a. i(,,a( , )) 5t3 cos (t4) dt
b. i( 0, 2,a( , )) f(2x3,x4 4) dx
c. i(,,a( , )) y5 (4y3 + 3)5 dy
10. (10 points) Let G(x) = i( 2, x, f(1,t2 1)) dt + i( f(1,2), f(1,x), f(1,1 t2)) dt.
Show that G is a constant function.
11. (10 points) Sketch the graph of g(x) = sin2x for x in [0, 2]. List all relevant information.
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