c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Exam 3 Review
1. Approximate the area under the graph of f (x) = x2 + 1 on the interval [0, 8] by using the partition
P = {0, 2, 3, 6, 8} and where x∗i is taken to be the left endpoint.
2. Approximate the area under the graph of f (x) = ln x on the interval [1, 11] using 5 equal subintervals
and taking x∗i to be the midpoint of the subinterval.
3. Set up the limit to evaluate the exact area under the curve f (x) = 25 − x2 between x = 2 and x = 4
using n equal subintervals and taking x∗i to be the right endpoint.
4. Evaluate log6 15 + log6 2 − 2 log6 3 − log6 20
5. Solve the following equation: 2 log9 (x − 2) − log9 (8 − x) =
1
2
6. Find the derivative of f (x) = 7arccos x + log5 (x3 + 2x) + x2 ln(3x + 1)
7. Find the derivative of f (x) = (tan x)3x .
8. Evaluate the following.
(a) arcsin(sin 5π
4 )
(b) tan(cos−1 29 )
(c) sec(tan−1 x)
9. Calculate lim arccos
x→∞
1 − 3x2
6x2 − x
!
10. Find the derivative of f (x) = arctan(ln x) + x sin−1 (e2x ).
11. The population of a bacteria quadruples every 3 days. How long will it take for the population to be
10 times its initial size?
12. A thermometer is taken from a room with a temperature of 70◦ F and placed in a room with temperature 50◦ F. After 8 minutes, the reading on the thermometer has dropped 3◦ . What is the reading
on the thermometer after t minutes?
13. Calculate the following limits.
arctan x − x
x3
(b) lim+ 2x csc x
(a) lim
x→0
x→0
(c) lim
x→∞
5
1+
x
x
1
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
14. Given the graph of the DERIVATIVE of a function f below, identify the locations of all local extrema
and inflection points.
15. Find the critical numbers of the function f (x) = (x − 2)1/7 x2 .
16. Find the abosolute maximum and minimum values of the function f (x) = sin2 x + cos x on the interval
[0, 3π
4 ].
17. For the function f (x) = 15x4 − x5 , find the intervals where f is increasing/decreasing, the x-values of
any local extrema, intervals of concavity, and the x-values of any inflection points.
18. For the function f (x) = xex
of any local extrema.
2 +3x
, find the intervals where f is increasing/decreasing and the x-values
19. A box with an open top is to be constucted from 30 m2 of cardboard. The length of the box is to be
twice its width. Find the dimensions of the box that will maximize the volume.
20. Find the point on the parabola y = 21 x2 which is closest to the point (5, 1).
21. Find the most general antiderivative of f (x) =
22. Find f (x) if f ′ (x) =
6
x
√
1
3
3
+ sin x + x7 .
x+ 5 + √
x
1 − x2
+ ex − 4 and f (1) = 6.
23. If the acceleration of a particle is given by a(t) =< cos t, t >, find the position function r(t) if it is
known that v(0) =< 2, 3 > and r(0) =< 1, 1 >.
24. Compute
6
X
(−1)i (i − 5)2
i
i=3
25. Calculate lim
n→∞
"
n
X
5
i=1
n
5i
2+
n
2
#
+1
2