MA-139 Applied Calculus - Exam 2 SAMPLE ANSWERS
March 25, 2009
1. y=๎1 + 13
6(x๎1).
2. Find the โฆrst derivative of the following functions:
(a) f0(x) = 4
3x1=3(b) f(x) = ๎1
x2
(c) f(x) = ๎x2+ 4x+ 4๎(2x๎1)๎1=2๎(2x+ 4)p2x๎1
2x๎1(d) f0(t) = ๎1
(t๎1)2
(e) f0(x) = 1
8x๎1=2๎2x๎3=2(f) p0(s) = ๎1
2s๎3=2
(g) dy
dx = 31
2=7
2(h) dy
dx = (x๎1) ๎2x+x๎1=2๎+๎x2+ 2px+ 7๎
(i) y=๎x๎x2+ 1๎๎1=2(j) f0(x) = 2 ๎x๎1๎x๎๎๎x๎2๎1๎
3. Complete the table below, given that h(x) = f(g(x)) and p(x) = f(x)g(x):
x f (x)f0(x)g(x)g0(x)h(x)h0(x)p(x)p0(x)
1 2 3 2 ๎2
3๎4๎2 4 14
3
2๎4 2 1 5 2 15 ๎4๎18
4. (a) Find dy
dx =2x+ 2xy
3๎x2.
(b) Find dy
dx =
1
2x๎1=2y๎1=2
1
2x1=2y๎3=2๎1
=1
xy๎1๎2x1=2y1=2=y
x๎2x1=2y3=2.
5. ds
dt =48
12 ๎hwhere his your height.
6. Critical points: x=๎p2(neither), x= 0 (relative maximum), and x=p2(neither).
7. f00 (x) = 12x2๎36x; concave up: x < 0and x > 3; concave down: 0< x < 3.
8. x= 10;500.
9. (This one will only ask you to label critical points, and increasing and decreasing areas.)
10. The order is g(x),f(x),f0(x), and f00 (x).
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