Prof: M. Nasab
Review Problems for the Final Exam (MAT125 - 04/Cal Poly Pomona Spring 2014)
The final exam is Tuesday, June 10 (7:00 am to 9:00 am). The exam is comprehensive and will cover
chapters 1 through 5. The exam problems will be similar to your quiz and midterms problems. Thus, it is
strongly recommended that you study your lecture notes, mid-terms, and the quiz problems plus the
problems given in this handout.
The following sample problems are for the chapters which we have covered since Midterm 2 and they are
meant to be as a guide, but there may be a topic or technique we discussed that is not included here. Please
don't assume that means it will not appear on the exam. Remember, the final is COMPREHENSIVE!
1.
a.
Find the derivative of the following functions and simplify your answers as much as possible.
f ( x ) = 4e
e. f ( x) = ln(
2.
b.
x2
)
x +1
c. f ( x) = 1−10
2 ex
f ( x) = x 2e x
f. f ( x ) = ln(e
2 x +1
d. f ( x ) = ln(3 x )
2
g. f ( x ) = ln( x + 1)
3
)
3
h. f ( x) = ln( x
2
x + 1)
Using the Calculus-method, graph the following functions: [Relative max/min. point of inflection, asymptotes and etc..]
f ( x ) = x 3e x
a.
3.
x
b.
f ( x) = ln( x + 3)
Integrate/Evaluate the following integrals.
a.
e.
i.
∫ −9 dx
∫ ( x − 6)
∫e
−5x
4
3 dx
dx
b.
∫ 3x
f.
∫ (1 + 5 x)
j.
1
∫ x − 6 dx
0
m.
3
∫ (2 + x) dx
−2
2
dx
2
dx
c.
∫ (5 − 6 x
g.
∫ x (1 − 4 x
k.
x2
∫ 1 − x3 dx
1
9
n.
5
∫3 x dx
o.
x
∫ 3xe
−1
2
2
−1
) dx
2 3
) dx
4
+ x ) dx
x
d.
∫(
h.
∫ 4e
4x
dx
4
l.
∫ (2 + x) dx
0
2
dx
p.
∫ (x
4
+ 2 x 2 − 5) dx
−2
dC 1 5
=
x + 90 and fixed cost of $2000 (for x = 0).
dx 20
4.
Find the cost function for the marginal cost
5.
Find the supply function x = f ( p ) that satisfies
dx
= p p 2 −25 and the initial condition x = 600 when p = $13 .
dp
1
6. Application Problems:
a. With an annual rate of inflation of 4% over the next 10 years, the approximate cost of goods or services during any
C (t ) = P(1.04)t , 0 ≤ t ≤ 10 where is the time (in years) and is the present cost. The
year in the decade is given by
price of an oil change for a car is presently $24.95.Estimate the price 10 years from now.


b. The demand function for a product is modeled by p = 3000  1 −
3
3+ e
−0.0002 x

 . Find the price of the product if the

quantity demanded is x = 200. Round your answer to two decimal places where applicable.
c.
Future value. The future value that accrues when $500 is invested at 5%, compounded continuously, is
s (t ) = 500e0.05t , where t is the number of years. At what rate is the money in this account growing when t = 7 ?
d. The cost of producing x units of a product is modeled by C = 900 + 200 x − 200 ln x,
x ≥ 1. Find the minimum
average cost analytically. Round your answer to two decimal places.
e.
The rate of depreciation of a building is given by D '(t ) = 3, 700(25 − t ) dollars per year, 0 ≤ t
integral to find the total depreciation over the first
f.
≤ 25. Use the definite
25 years.
The demand function for a product is p = 100 − 2 x , where p is the number of dollars and x is the number of units. If
the equilibrium units is 40, what is the consumer’s surplus?
x = 40
Consumer surplus =
∫
( demand function − price) dx , and price is found by replacing x with $40 in the demand
x =0
function.
g.
Find a. the consumer and b. producer surpluses by using the demand and supply functions, where p is the price (in
dollars) and x is the number of units (in millions), where Demand Function is given by
supply function is given by
p = 42x .
p = 975 − 23x
and The
Note: Equilibrium price: demand function = supply function
Note: The equilibrium units is 15, and price is found by replacing x with 15 in either the demand function or the supply function.
x =15
a.
Consumer surplus =
∫
(demand function - price) dx
x =0
x =15
b.
Producer surplus =
∫
(price- supply function) dx
x=0
Consumers' surplus is the monetary gain obtained by consumers because they are able to purchase a product for a price that is less
than the highest price that they would be willing to pay.
Producer surplus or producers' surplus is the amount that producers benefit by selling at a market price that is higher than the least
that they would be willing to sell for.
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3
4