Practice Problems: Final Exam
Z 1
1. Estimate the integral
0
1
dx using five rectangles and the Midpoint Rule.
1 + x3
2. Evaluate the following integrals.
Z 2
x(x + 2) dx
(a)
(b)
Z 4
√
x dx
1
1
Z 2
(c)
0
Z 1
(e)
1
dx
5x + 1
(d)
(1 − x)5 dx
(f)
Z π/2
Z π/4
−1
Z 2
(g)
sec2 x dx
0
Z 1
(ex + 1)2 dx
x3 cos x + 3x2 sin x dx
(h)
0
0
Z 4
(i)
cos(3x) dx
0
2x
Z 1
p
x2 + 9 dx
(j)
0
0
e2x
dx
1 + e2x
3. The following table shows the velocity of an object over the course of one second.
t (seconds)
0.0
0.2
0.4
0.6
0.8
1.0
v (meters/sec)
3.6
4.0
4.3
4.6
4.8
4.9
(a) Estimate the distance traveled by the object between t = 0 sec and t = 1 sec.
(b) Estimate the acceleration of the object at t = 0.4 sec.
(c) Estimate the velocity of the object at t = 0.43 sec.
4. Evaluate the following limits:
(a) lim
x→∞
x
x+5
(b) lim
x→0
sin x
.
x
ln x
(c) lim √
x→∞
x
(d) lim x − x2
1 x
(e) lim 1 +
x→∞
x
√
x−2
(f) lim
x→4 x − 4
x→∞
5. Two boats are both sailing away from a harbor, as shown in the picture below.
land mass
boat A
B
harbor
C
N
P
W
E
boat B
S
Boat A is 80 miles east of the harbor, and is sailing due east at 15 miles/hour. Boat B is 60 miles south
of the harbor, and is sailing due south at 20 miles/hour. How quickly is the distance between the two
boats increasing?
6. Suppose that f (3) = 5 and f 0 (3) = 2. Based on this information, estimate the value of f (3.04) as
accurately as you can.
7. The following graph shows the derivative of a function f (x):
f 'HxL
1
0
1
2
3
x
4
-1
(a) Given that f (0) = 1/2, evaluate f (1) and f (4).
(b) Find the maximum value of f (x) on the interval [0, 4].
(c) For what values of x is f 00 (x) positive? For what values of x is f 00 (x) negative?
(d) Sketch a graph of the function f (x):
f HxL
2
1
0
0
1
2
3
4
x
8. (a) Find g 0 (u) if g(u) =
√
1 + u2 .
dx
1
θ
(b) Find
if x = sec
dθ
5
2
(c) Find
dy
if y = ln(sin x).
dx
(d) Find
Q
dP
if P = 2 .
dx
R
(e) Find f (x) if f 0 (x) = x2 + e2x and f (0) = 4.
(f) Find f 0 (x) if f (x) =
x
x2 + 1
.
9. A piece of wire 34 cm long is cut into two parts:
x
The first part is bent into the shape of a rectangle whose length is equal to twice its width, while the
second part is bent into the shape of a square.
(a) Find a formula for the total combined area A of the two shapes in terms of x.
(b) What is the minimum possible value of A?
10. Let f (x) = 3x .
(a) Write a limit for f 0 (2) involving the average slope between x = 2 and x = 2 + h.
(b) Use a table of values to estimate f 0 (2) to within 0.01. You must show your work to receive full
credit.
11. In physics, the kinetic energy of a moving object is given by the formula
K=
1 2
mv
2
where m is the mass and v is the velocity. Suppose an object with a mass of 2.00 kg is accelerating
at a rate of 6.00 m/s2 . How quickly is the kinetic energy of the object increasing when the velocity
is 23.0 m/s?