MA-139 Applied Calculus - Exam 3 SAMPLE AN SW ERS
April 24, 2009
Be sure to show work for every problem.
1. y= (40 + x) (16 ๎2x))x=๎16 (24 trees per acre).
2. F(x) = 3x+ 1200x๎1)x= 20,y= 30.
3. Find the โฆrst derivative of the following functions:
(a) f0(x) = 0:4e0:4x(b) g0(t) = 2t
t2๎7(c) f0(x) = ex
x+exln x
(d) f0(x) = 1 ๎1
x(ln x)2(e) dy
dx = 2ex(ex๎5) (f) f0(x) = 0
(g) f0(x) = (ex+ 3) ๎1
x+ ln x๎(h) f0(x) = 1
2ex=2=1
2(ex)๎1=2ex(i) f0(๎) = 2 + 6๎๎2๎e(๎2)
(j) f0(x) = x1
xex(xex+ex) + ln (xex)
4. (a) x=r3
2.
(b) Absolute minimum at x=๎5; Absolute maximum at x= 3.
5. y=e2
2+e2
4(x๎2).
6. (a) Q(t) = 100eln(1=2)=53000t.
(b) Q0(t) = ln (1=2)
530 e5 ln(1=2)=53 ๎ ๎1:225 ๎10๎3.
7. Show that F0(x) = xe3x.
8. Evaluate the following integrals (remember the โ+Cโ):
(a) ex+1
2x2+C(b) 1
3t3๎7t+C(c) x๎ln x+C(d) 0:1x2๎1
x+C
(e) et๎3
4t4=3+C(f) x๎e2๎2e๎+C(g) 2
3x3=2+C(h) 2 ln v+1
2v2+C
(i) x+ 2 ln x+2
3x3+1
4x4+C(j) 1
6x3+1
12x4+C
9. f(x) = 2px๎3 ln x๎1 + 3 ln4.
10. On the graph below, label all inโกection points with a F, all areas of positive concavity with a [and
all areas of negative concavity with a \.
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