Prof: M. Nasab
Review Problems for Mid-Term 1 (MAT125/Cal Poly Pomona Spring 2014)
5
2x 2 + x
−
1
2
1.
Factor completely
2.
Find all real zeroes of 3 x − 8 x + 4 [Hint: Find x]
3.
Find all real zeroes of
4.
Add and reduce
5.
Rationalize the denominator
6.
Graph the following equations:
2
( x + 1)2 − 36 [Hint: Find x]
2
3x − 2
+ 2
x +1 x − 2x − 3
6
x +1
a. y = −3 x + 2
b.
y = x3 + 1
c.
d. y =
e.
y = x +1
f. y =
x +1
y = − x2 + 1
y = 16 − x 2
1
x −1
2 x2 y + 8 y − x2 = 1
7.
Find the x- and y- intercepts of the graph of the equations:
8.
A manufacturer of DVD players has monthly fixed costs of $8600 and variable costs of $75 per unit for one particular
model. For this model DVD player, a) Find the function
a.
b.
C ( x ) for monthly total costs where x denotes the number of
units produced and sold. b). Interpret the slope of the cost function and its y-intercept in your own words.
9.
A small business recaps and sells tires. The business has a revenue function R ( x ) = 115 x and a cost function
C ( x ) = 3500 + 80 x , where x represents the number of sets of four tires recapped and sold. b) Find the number of
sets of recaps that must be sold to break even.
10. Find the market equilibrium point for the following demand and supply functions below, where p is price per unit
and q is the number of units produced and sold. Demand: p = −2q + 320 ,
Supply: p = 6 q + 2
11. Write the equation of the line passing through the given pair of points (- 6, 5) and (5, 6).
[Leave your answer in y = m x + b form]
12. Personal income (in billions of dollars) in the United States was 9937 in 2004 and 12,175 in 2009. Assume that the
relationship between the personal income y and the time t (in years) is linear. (Let t = 0 represent 2000.) a. Write a
linear model for the data. b. Estimate the personal income in 2006.
13. Simplify the expression
f ( x ) − f (1)
x −1
given the function f ( x ) = −13 x − 14 .
14. Find the following limits:
a.
lim x →−1 3 x + 2
b. lim x → 2 ( − x + x − 2)
2
c. lim x →3
t2 + t − 2
3x + 1
x −1
d. limt →1
e. lim x →1
2
2− x
t −1
x −1
15. Suppose that lim f ( x ) = 8 and lim g ( x ) = –11 . Find the following limit:
x →c
16. Let
x →c
lim [ f ( x) + g ( x) ]
x →c
 x 2 + 4, x ≠ 1
f ( x) = 
.
x =1
 1,
Determine the following limits. [Hint: Use the graph of the function.]
a) lim f ( x )
b)
x →1
lim x→0 f ( x)
17. Find the derivative of the following function using the limiting process. [Definition of the derivative]
a.
f ( x) = –2 x 2 – 9 x
b.
f ( x ) = 10 x − 14
c.
f ( x) = x
d.
f ( x) =
1
x
18. Find the slope of the tangent line to the graph of the function below at the given point.
f ( x ) = 2 x –10, (3, –4)
19. Find the slope of the tangent line to the graph of the function at the given point.
f ( x) = 2 x 2 + 6, (3, 24)
20. Draw the tangent lines at the given points (Dots) on the graph.
21. a. Graph
y = 2x
1
5
b. Graph y = ( ) x
22. Find the accumulated amount in an account with a principal of $5000 and the interest rate of 2.5% compounded
r n.t
quarterly for the next 10 years. [Hint: use A = P (1 + ) and n = 4]
n
23. With an annual rate of inflation of 10% over the next 10 years, the approximate cost C of goods or services during any
year in the decade is given by
C ( x) = P (1.10)t , with 0 ≤ t ≤ 10 where t is the time (in years) and P is the present
cost. The price of an oil change for a car is presently $24.95. Estimate the price 10 years from now.
24. Using the rules of the differentiation, find the derivative of the following functions and simplify your answers.
1
a.
f ( x) = − x 3 + x 2 − 9
e.
f ( x) = 3 x 3
b.
2
2
f.
f ( x ) = ( x + 5) 2
f ( x) = x 3 (3 x 2 + 7)
8
x5
c.
f ( x) =
g.
f ( x) = −.05 x 3 + 30 x 2 − 164.25 x − 1000
d.
f ( x) = 7 x 2
25. Application problems for the derivative of a function:
a.
Find the marginal cost for C ( x ) = 3 x + 24, 000 at x = 10 units.
b.
Find the marginal revenue for
c.
The demand and cost functions for a product can be modeled by p = 211 − 0.002 x and C ( x ) = 30 x + 1, 500, 000
3
R ( x ) = −4 x 3 + 2 x 2 + 100 x
where x is the number of units produced.
1. Write the profit function P(x). [Hint: R(x) = x . p, and P(x) = R(x) – C(x)]
2. Find the marginal profit when 10,000 units are produced. Interpret your answer(s).
d. The revenue R in dollars from renting x apartments can be modeled by R ( x) = 2 x (900 + 32 x − x )
2
1. Find the additional revenue (Average revenue) when the number of rentals is increased from 14 to 15.
2.
Find the marginal revenue (The instantaneous revenue) when x = 14.
3. Compare the results of parts (a) and (b).