Practice Problems for Exam 3 (Math 71)
1. Find the following integrals:
" 3 3 5 2
%
−2 x
+ 8'dx
(a) ∫ $ x − x + + e
#
&
x
(b)
! x 2 +1 $
(e) ∫ # 3
&dx
" x + 3x %
(e x − 1)(e x − x)7 dx
(d)
∫
(g)
∫ xe
− x2
dx
∫
(h)
1− x
dx
x
2
[ln(x)]2
(j) ∫
dx
x
(k)
∫
0
2
(m)
∫
(x −1)2
dx
x
25
2
∫ (1− 8x) dx
(n)
0
∫ (x
x2
3
x +1
1
2
2. Find the area of the region y =
1
∫ x dx
(c)
−2
! x 3 + 3x $
(f) ∫ # 2
&dx
" x +1 %
(i)
∫ (x
2
− 2)(x 3 − 6x)dx
−1
dx
+x
0
−1
(l)
∫ (x
−1
+ e)dx
−2
1
4
)dx
4
(o)
∫
0
e2 x
dx
e2 x + 7
x−5
.
x
3. Find the exact value of the area bounded by the x-axis and the graph of
f (x) =
x + x, 1 ≤ x ≤ 4
by evaluating an appropriate definite integral using the Fundamental Theorem of Integral Calculus.
4. Find the area of the region bounded by the graphs of the equations.
y = ex, y = 0, x = 0, and x = 3
5. Set up, but do not evaluate, integrals for the area:
(a) Between the curves y = x2 – 2x and y = 2x
(b) Bounded by the curves y = x3 and y = x.
6. Find the average value of f(x) = 1
x3
on [1,3].
on [-1,2].
dC
= 12x 2 + 3e−0.1x where x is the number
dx
of units, and fixed costs are $100. Find the cost function, C(x).
7. A company’s marginal cost for a product is modeled by
8. Find the demand function p = f (x) that satisfies the initial conditions.
dp
400
=−
, x = 10,000 when p = $100
dx
(0.02x −1)3
9. Because of an insufficient oxygen supply, the trout population in a lake is dying. The population's
rate of change can be modeled by
−t
dP
= −110e 20 , where t is the time in days. When t = 0, the population is 2200.
dt
(a) Find a model for the population. (Round your constant term to two decimal places.)
P(t) =
(b) What is the population after 11 days? (Round your answer to the nearest integer.)
10. The rate of change in revenue for Under Armour from 2004 through 2009 can be modeled by
dR
284.653
, where R is the revenue (in millions of dollars) and t is the time (in years),
= 13.897t +
dx
t
with t = 4 corresponding to 2004. In 2008, the revenue for Under Armour was $725.2 million.
(a) Find a model for the revenue of Under Armour. (Round your constant term to two decimal
places.)
R(t) =
(b) Find Under Armour's revenue in 2004. (Round your answer to two decimal places.)
11. A company purchases a new machine for which the rate of depreciation can be modeled by
dV
= 10, 000(t − 8) , 0 ≤ t ≤ 5 , where V is the value of the machine after t years. Set up and
dt
evaluate the definite integral that yields the total loss of value of the machine over the first 2 years.
Please review ‘Study Sheet (16.3 – 16.4)