math 112 final

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x) = 100x − 0.2x2, where the revenue R(x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

$12,500 at 250 units

A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells x cans of soda pop in one day, his profit (in dollars) is given by P(x) = −0.001x2 + 3x − 1900.

$350 at 1500 cans

r(t) = 6 − 16t; t = 6, t = 12 (a) Determine the net change between the given values of the variable. (b) Determine the average rate of change between the given values of the variable.

-1, -1/6

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). (Enter your answers as a comma-separated list.) T(x) = 4x^6 − 4x^2 − 11

-1, 1, -11, 11, -1/2, 1/2, -11/2, 11/2, -11/4, 11/4

Find all real solutions of the equation by factoring. (Enter your answers as a comma-separated list.) 6x(x − 1) = 4 − x

-1/2, 4/3

Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) |4 − 2x| + 7 = 12

-1/2, 9/2

Find all real solutions of the equation by factoring. (Enter your answers as a comma-separated list.) 5x2 − 14x − 3 = 0

-1/5, 3

All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) P(x) = x^4 − 5x^2 + 4

-2, 2, -1, 1

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). (Enter your answers as a comma-separated list.) R(x) = 2x^5 + 3x^3 + 3x^2 − 6

-6, -3, -2, -3/2, -1, -1/2, 1/2, 1, 3/2, 2, 3, 6

Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) 36x − 34x − 4 + 4 = 0

-9/2, 8

Evaluate the quotient, and write the result in the form a + bi. (Simplify your answer completely.) 3 − i9 + 5i

11/53- 12i/53

f(z) = 4 − 5z2; z = −2, z = 0 (a) Determine the net change between the given values of the variable. (b) Determine the average rate of change between the given values of the variable.

20, 10

If $1000 is invested at an interest rate of 3.5% per year, compounded continuously, find the value of the investment after the given number of years. 2 4 12

2:1072.18 4:1150.27 12:1521.96

When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after t hours is modeled by D(t) = 40e−0.1t. How many milligrams of the drug remain in the patient's bloodstream after 2 hours?

32.7

Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)D(x) = Q(x) + R(x)D(x). P(x) = 3x3 + 9x2 − 7x − 1, D(x) = x + 4

3x^2-3x+5 +-21/x+4

If $15,000 is invested at an interest rate of 4% per year, compounded semiannually, find the value of the investment after the given number of years. 5 years 10 years 15 years

5: 18284.92 10:22289.21 15:27170.42

Find the inverse function of f. f(x) = 9x/x − 6

6x/x-9

Find all real solutions of the equation. (Enter your answers as a comma-separated list.) 2x + 9 + 3 = x

8

Solve the logarithmic equation for x. (Enter your answers as a comma-separated list.) log(9x + 8) = 2

92/9

A farmer has 1,230 acres of land on which he grows corn, wheat, and soybeans. It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow soybeans. Because of market demand the farmer will grow twice as many acres of wheat as of corn. He has allocated $65,550 for the cost of growing his crops. How many acres of each crop should he plant?

corn:270 wheat:540 soybeans:420

Consider the following. 1 + e^5x + 1 = 25

ln(24)/5−1/5 0.435611

Find the slant asymptote and the vertical asymptote. (If an answer does not exist, enter DNE.) r(x) = x^2/x − 9

slant: y= x+9 vertical: x=9

Find all horizontal and vertical asymptotes (if any). (If an answer does not exist, enter DNE. Enter your answers as a comma-separated list of equations.) s(x) = 6x^2+1/2x^2+7x-4

vertical: x=-4, x=1/2 horizontal: y=3

Find the intercepts and asymptotes. (If an answer does not exist, enter DNE. Enter your asymptotes as a comma-separated list of equations if necessary.) r(x) = 6x-6/x + 2

x-intercept: 1, 0 y-intercept: 0, -3 vertical: x=-2 horizontal: y=6 domain: (-infinity, -2) u (-2, infinity) range: (-infinity, 6) u (6, infinity)

Find the quotient and remainder using synthetic division. x5 + 4x3 − 13 x − 1

x^4+x^3+5x^2+5x+5