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Exam (elaborations) mathematics 1
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Basic integrals and functions exercises
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Worksheet 3 (17-18). Optimization-STEPHANIE REINA WedOct/25/2017, 16004Student: _____________________ Instructor: STEPHANIE REINA Assignment: Worksheet 3 (17-18).
Date: _____________________ Course: MATHEMATICS I Optimization
1. Find the extreme values (absolute and local) of the following function and where they occur.
2
f(x) = x − 81
Select the correct choice below and fill in the answer box within the choice.
(Use a comma to separate answers as needed.)
A. The given function has minimum value 0 at x = .
B. The given function has maximum value 0 at x = .
C. The given function has minimum value − 1 at x = .
D. The given function has maximum value − 1 at x = .
2. Find the extreme values of the function and where they occur.
12x
y=
2
x + 36
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The maximum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no maxima.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The minimum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no minima.
3. ex
Find the absolute minimum value on (0,∞) for f(x) = .
3
x
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The absolute minimum is f ≈ .
(Round to two decimal places as needed.)
B. There is no absolute minimum.
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,Worksheet 3 (17-18). Optimization-STEPHANIE REINA WedOct/25/2017, 16004
4. Find the absolute maximum value on (0, ∞) for f(x) = 7 ln 2x e − x .
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The absolute maximum is f ≈ .
(Round to two decimal places as needed.)
B. There is no absolute maximum.
5. Identify the absolute extrema of the function and the x-values where they occur.
81
f(x) = 6x + + 3, x > 0
2
x
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The absolute minimum is and occurs at the x-value .
(Type an integer or decimal rounded to the nearest thousandth as needed.)
B. There is no solution.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The absolute maximum is and occurs at the x-value .
(Type an integer or decimal rounded to the nearest thousandth as needed.)
B. There is no solution.
6. Find the absolute extrema of the function, if they exist, as well as all values of x where they occur.
ln x
f(x) =
10
x
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The absolute minimum is at the x-value .
(Type an integer or decimal rounded to four decimal places as needed.)
B. There is no solution.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The absolute maximum is at the x-value .
(Type an integer or decimal rounded to four decimal places as needed.)
B. There is no solution.
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, Worksheet 3 (17-18). Optimization-STEPHANIE REINA WedOct/25/2017, 16004
7. The total profit P(x) (in thousands of dollars) from the sale of x hundred thousand pillows is approximated by
3 2
P(x) = − x + 12x + 144x − 400, x ≥ 5.
Find the number of hundred thousands of pillows that must be sold to maximize profit. Find the maximum profit.
The maximum profit is $ .
The maximum profit will occur when pillows are sold.
8. If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where
x
p(x) = 150 − .
10
a. Find an expression for the total revenue from the sale of x thousand candy bars.
b. Find the value of x that leads to maximum revenue.
c. Find the maximum revenue.
a. R(x) =
b. The x-value that leads to the maximum revenue is .
c. The maximum revenue is $ .
9. For the cost and price functions below, find a) the number, q, of units that produces maximum profit; b) the price, p,
per unit that produces maximum profit; and c) the maximum profit, P.
C(q) = 90 + 11q; p = 75 − 2q
a) The number, q, of units that produces maximum profit is q = .
b) The price, p, per unit that produces maximum profit is p = $ .
c) The maximum profit is P = $ .
10. Suppose that the cost function for a product is given by C(x) = 0.003x3 + 6x + 10,895. Find the production level
(i.e., value of x) that will produce the minimum average cost per unit C(x).
The production level that produces the minimum average cost per unit is x = .
(Round to the nearest whole number as needed.)
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