4__(j44JJ=/RdMath 125T Name_________________________
Final
May 8. 1989
I have not violated the Honor Code in any way with regard to this work. Signature
1. (5 points) If sin 2x = f(1,2), what is x?
(Give all such values which are in [0, 2].)
2. (9 points)
a. a(lim,x0) f(7x2 + 9x 1,x3 x + 3) =
b. a(lim,x2) f(2x2 2x 4,x 2) =
c. a(lim,x f(,2)) f(sin x sin f(,2),x f(,2)) =
3. (5 points) g = f(r(x + 1),x 3). g is
a. continuous at 1
b. continuous from the left at 1
c. continuous from the right at 1
d. None of the above
4. (5 points) g(x) = f(r(x21),x 3) . g is
a. continuous at 1
b. continuous from the left at 1
c. continuous from the right at 1
d. None of the above.
5. (6 points) Find an equation of the line which is tangent to the graph of f(x) = 2x3 8 at (2,8).
6. (4 points) g(y) = (9y3 + 1)7 . Find g(y).
7. (6 points) y = 3 sin x cos2 x. Find f(dy,dx)
x = f(5,6)
8. (6 points) f(x) = tan x. Find f (x).
9. (6 points) If 12xy x2 = 2y + y3 x2 + 7, find the derivative of y with respect to x.
10. (8 points) A light is hung 15 feet above a straight horizontal path. If a man 6 feet tall is walking away from the light at the rate of 5 ft/sec, how fast is his shadow lengthening?
11. Let f(x) = f(x,x2 + 1).
a. (3 points) What are the critical points of f?
b. (3 points) Where is f strictly increasing, and where is f strictly decreasing?
c. (4 points) Find the relative extreme values of f.
d. (4 points) Where is the graph of f concave upward? Where is it concave downward?
e. (2 points) What are the inflection points on the graph of f?
f. (6 points) Find the vertical and the horizontal asymptotes of the graph of f.
g. (5 points) Sketch the graph of f.
12. (5 points) Find f(d,dx) ln f(4x 2,7x2 + 1)
13. (8 points) Of all the triangles that pass through the point (1,2) and have two sides lying on the coordinate axes, one has the smallest area. Determine the lengths of its sides.
Hint: Consider the slope of the line which passes through the point (1.2).
14. (7 points) f(d,dx) i( 2, f(2,3) x,a( , )) blc(f(1,t) 3t2) dt =
15. (6 points) i( 1, 1,a( , )) 4x 3 dx =
16. (6 points) i(,,a( , )) t r(t23 1) dt =
17. (6 points) i( f(,6), f(,2),a( , )) cot q dq =
18. (8 points) For each of the following, state whether or not f has an inverse. If it does, find the inverse. If it does not, state why.
a. ff(x) = x + x
b. f(x) = f(x3,8)
19. 6 points) Find (f1) (17) when f(x) = 2x3 + 1.
20 (6 points) g(y) = f(ln y,e2y) . Find g1 (y).
21. (6 points) i( f(1,4),f(1,2), f(e1/x,x2)) dx =
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