1. No category

Math 226
Final Exam Sample Problems
The final will be cumulative, and will cover the following sections:
1.1 − 1.3, 1.5, 1.6, 2.1 − 2.6, 3.1 − 3.6, 4.1, 4.2, 4.4 − 4.6, 4.8, 5.1 − 5.6
Please refer to the previous sample problems, quizzes, exams, and wkshts for material
from previous sections. This review, although containing some problems that relate to
previous material, is largely focused on recent material, with an emphasis on chapter 5,
as we have not had any sample problems from chapter 5 up until this point.
2
Z
1. Let I =
f (t) dt. Assume f is increasing and concave down and f has values
−1
shown in the table
x
f(x)
-1
0
-0.25
2.65
0.5
4.88
1.25
6.83
2
8.67
(a) Compute L4 . Is L4 an overestimate or an underestimate for I?
(b) Compute R4 .
(c) Compute M2 . Is M2 an overestimate or underestimate for I?
4
Z
2. Let I =
(x + 1) dx. Find I by taking the limit of right Riemann sums, Rn .
1
5
Z
3. Let I =
(x2 − 2x + 5) dx. Find I by taking the limit of left Riemann sums, Ln .
0
4. Let f (x) = 2x + 3 and g(x) = x. Find the area between f and g on the interval [0, 3].
5
Z
√
x2 − 9 dx. Estimate the value of the integral
5. Consider the following integral:
3
using a left endpoint Riemann sum with n = 6 subintervals. Ie, calculate L6 . Is this
an overestimate, or underestimate?
6. The graph of y = f (t) below is made up of straight segments and a quarter circle.
2
Z
f (t) dt
(a) Determine
0
−2
Z
f (t) dt
(b) Determine
0
1.5
Z
7. In the graph of f (x) below, note that
0
2
Z
a. Find
1.5
Z
b. Find
f (x) dx
0
1.5
f (x) dx
2
Z
f (x) dx = 10, and
f (x) dx = 8.
0
8. Compute each quantity: If you use u-substitution, show your work. Also, do not leave
your antiderivatives in terms of u
Z 3
a.
x3 − 2x + 1 dx
−1
Z
sec2 (θ) dθ
b.
π/4
Z
cos θdθ
c.
0
1 + x3
Z
d.
x
dx
1
Z
2ex dx
e.
0
x2
Z
d
f.
dx
1/2
ln
√
1 + t dt
!
2
d
t2
g.
+ π dt
dx x 2
Z
h.
e2r dr
Z
π/2
Z
i.
cos 2θ dθ
0
ln 1
Z
j.
−1
dx
ex
Z
cos3 (t) sin(t) dt
k.
Z
1
l.
x+1
1
Z
m.
dx
20
x 1 + x2
dx
0
1
Z
x2 (1 + x)20 dx
n.
0
Z
o.
dx
+9
x2
9. The speed of a runner increased steadily during the first three seconds of a race. Her
speed in half-second intervals is given in the table below.
time (s)
velocity (ft/s)
0
0
0.5
6.2
1
10.8
1.5
14.9
2
18.1
2.5
19.4
3
20.2
Use a right-endpoint Riemann sum with 6 subdivisions to approximate the distance
the runner traveled during the first 3 seconds.
10. Find the solution of the initial value problem.
(a)
dy
= 6x2 + 4x, y(2) = 10
dx
(b)
dP
= 10et , P(0) = 25
dt
(c)
ds
= −32t + 100, s = 50 when t = 0
dt
11. A construction worker pulls a five meter plank up the side of a building with a rope
tied to the top of the plank. Assume the opposite end of the plank drags along the
ground. If the worker pulls the rope at a rate of 0.15 m/s, how fast is the end of the
plank sliding along the ground when it is 2.5 meters from the wall of the building?
12. Determine, from the limit definition, the derivative of the function f (x) = x2 − 2x
13. Determine each limit:
x2 − 2x
x→2 ln(x − 1)
a) lim
 2

x +3



7
b) lim− f (x) where f (x) = 


x→2
 √x + 1
√
x + 16 − 4
c) lim
x→0
x
x<2
x=2
x>2
14. Use a local linear approximation to estimate 4.031/2
15. Find g0 (x) where g(x) = x2 ln(x)
16. A piece of wire 200 cm long is going to be cut into several pieces and used to construct
the skeleton of a rectangular box with a square base. What are the dimensions of the
box with the largest volume?
17. If a functions exists with all of the properties listed below, sketch its graph. If such a
function does not exist, explain why.
a) f is continuous on [−2, 4] and differentiable on (2, 4)
b) f 00 (x) > 0 on (−2, 0)
c) f 00 (x) < 0 on (0, 4)
d) f 0 (x) > 0 on (−2, 2)
e) f 0 (x) < 0 on (2, 4)
18. Find
dy
if y3 + xy − y2 = 1
dx
19. Find f 0 (x).
√
a) f (x) = x8 − 3 x + 5x−3
b) f (x) = tan x + 2 cos3 x
c) f (x) =
d) f (x) =
3x + 1
3
x2
1
2x + sin3 x