Math 115 Review - SOLUTIONS-Fall 2014
1. The graph f(x) = x2+x and g(x) = 2x+2 is below, left. The
solution to f(x) = g(x) is x = -1,
and x = 2. The solution to the inequality f(x) > g(x) is (-¥,-1)U(2,¥).
2. The graph of g(x) = (x+1)4, for -4 < x £-2
is above, right. The solution to g(x) = 25 is x = -4 Ö 25 - 1
= - Ö 5 - 1,
g-1(16)
= -3, and g-1(x) = - 4 Ö x - 1
for 1 £ x < 81.
3. The graph of f(x) = (x -5)/(x 2 -2x
-3). is below. The x- intercept is 5 and the y-
intercept is 5/3. The vertical asymptotes are x= -1 and x= 3. The horizontal
asymptote is y= 0. The domain of log[(x -5)/(x 2 -2x
-3)] is (-1,3) U(5,¥).
4. The graph of f(x) = ex+1-4 is below, left. The asymptote is y = -4. The y-intercept exact is e-4 which is -1.28
accurate to 2 decimal places: the x-intercept exact is ln(4)-1 which is 0.39 accurate to 2 decimal
places. f -1(x)
= ln(x+4)-1.
5. The graph f(x) = ln(x+3)-1 is above, right. The asymptote is x = -3. The y-intercept exact is ln(3)-1
which is 0.10 accurate to 2 decimal places: the x-intercept exact is e-3 which is -
0.28 accurate to 2 decimal places. f -1(x)
= ex+1-3.
6. The domain of g(x) = 2ln(x)+3ln(x+4) is (0,¥). The graph is below, left. The
asymptote is x = 0 (the y-axis). The x-intercept is approximately 0.12. There
is no y-intercept. The solution to g(x) = 1 is x = 0.2 accurate to one decimal
place.
7.(a) The exact solution to 23-x = 52x+1 is
|
x = |
3ln2-ln5 2ln5+ln2 |
; 0.12 |
accurate to 2 decimal places.
(b) x = ± Ö3.
(c) x = 0.23, 0.50.
8. The point A is marked on the unit circle below, left. It is in
quadrant III. The coordinates of A are (-0.65,-0.76) accurate to 2 decimal places.
9. The graph of y = -2cos(2x-p/6)
appears above, right. The period is p,
the amplitude is 2, and the phase shift is p/12
to the right.
10. At 14 years: $5,000 invested at 7%
compounded continuously = 5000e(.07)(14) = $13,322.28. On the other
hand, $4,000 invested at 9% compounded monthly = 4000(1+.09/12)(12)(14)
= $14,035.54. Hence the friend's investment is worth more at 14 years.
11. At the same rate (computed to be ; 9.64%) the population will grow to
300,000 in 2017.
12. About 92.02 gms
will remain at 150 years.
13. If q
= arcsin(x), then
|
tan(q) = |
x
|
.
14. The fires are approximately 10.3 miles apart.
15. The blimp travelled approximately 551
feet during the 3 minutes.