Problems:
1) For each function, determine the domain:
a) f(x) =√( x - 7)
b) f(x) = 1/x - 2
2) For each function, determine the range:
a) f(x) = √(x+6)
b) f(x) = 7+x2
3) Consider the function f(x) = x2 -2x-3
a) Present a graph of the function.
b) List all x-intercepts
c) List all y-intercepts
d) Identify the vertex
4) The revenue for selling x units of a product is R = 45.95x. The cost of producing x units is C = 24.75x + 848.
a) For what values of x will this product return a profit?
b) What is the marginal cost and what is the marginal profit for this product?
c) What is the cost when 115 units are produced?
d) How many units are produced if the cost is $14904?
e) How many units must the company sell and produce to break even?
5) The current pay scale for public school teachers in a school system is shown in the table below:
|
Years of Service
|
Salary
|
|
0
|
$34780
|
|
5
|
$36444
|
|
10
|
$38108
|
|
15
|
$39772
|
|
20
|
$41436
|
a) Plot the data set
b) Find a linear model for the data
c) Interpret the meaning of the slope and y intercept
d) Find the value of c for which f(c)= $43100
e) Find and describe the meaning of f(17)
6) A pebble is thrown upward with an initial velocity of v0 = 80 ft per second and leaves the bridge with an initial height of h0 = 3 ft.
a) Write a formula s(t) that models the height of the pebble after t seconds.
b) How high was the pebble after 2 seconds?
c) Find the maximum height of the pebble
7) Consider the function f (x) = x2 -2x
a) Present a graph of the function.
b) Describe the increasing and decreasing intervals.
c) What is the domain and range of this function?
d) What is the vertex?
8) The function f (x) = √0.75x+10 , 0 ≤ x ≤ 10, approximates the population of a small town in thousands where x is the year with x = 0 representing 1990.
a) Present a graph of the function.
b) Find f(6)
c) Find the year when the population will reach 5000.
9) A company is paving a rectangular parking lot that must have an area of at least 500 sq m. The parking lot has a perimeter of 200 meters. Within what bounds must the length of the rectangle lie?
10) The population of a town increases according to the model P (t) = 100e0.2196t, where t is the time in years.
a) Present a graph of P
b) Find P(5)
c) Find the time when the population is 900.
11) Consider the function f(x) = log4 x.
a) What is the domain?
b) What are/is the x-intercept(s)?
c) What is the vertical asymptote if it exists?
d) Present a graph the function.
12) Evaluate:
a) 3ln 0.32
b) log 12.5
c) ln (1/2)
13) Suppose that $2500 is invested at 8% for 20 years. Find the total amount present at the end of this time period if:
a) The interest is compounded monthly
b) The interest is compounded continuously.
14) Solve the exponential equation
23-x = 565
15) Determine the principal that must be invested at a rate of 7-1/2 % compounded quarterly so that the balance in 20 years will be $35000.
16) The population P of a city is P = 2500ekt where t = 0 represents the year 2000. In 1965, the population was 2550.
a) Find k.
b) Present a graph of P
c) Predict the population in the year 2010.