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Derivatives and Systems of Linear Equations

Derivatives
Use the definitions of the derivative to find the derivative of each of the given functions.
y = x^3 + 5

Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.
y = 8 - x^2; at x = 1

Find the derivative of the given function.
G(t) = (t^3+ t-2)/〖(2t-1)〗^5

The total energy consumption (in quadrillion Btu's) for the United States can be approximated by the function.
f(x) = -.000144 x^3+ .014151x^2 + .1388x + 23.35
where x = 0 corresponds in the year 1970
Find the energy consumption for 1990, 2000, and 208.
Find on the average rate of change in energy consumption between 2000 and 2008.
At what rate was energy consumption changing in 2008?

The net revenue for Bank of America (in billions of dollars) can be approximated by the function
g(x) = -.033741x^2 + 1.62176x^3 - 28.4297x^2 + 216.603x - 599.806
(9 ≤ x ≤ 18).
Where x = 9 corresponds to the year 1999.
Find the revenue in 2006.
Find the revenue in 2008.
Find the rate of change of revenue in 2007.

Applications of Derivatives
Find the absolute extrema of each function on the given interval
f(x) = x^4 - 18x^2 + 1; [-4, 4]

Solve the problem
A restaurant has an annual demand for 900 bottles of a California wine. It costs $1 to store one bottle for one year, and it costs $5 to place a reorder. Find the number of orders that should be placed annually.

Systems of Two Linear Equations in Two Variables
Determine whether the given ordered list of numbers is a solution of the system of equations.
(2, -.5)
0.5x + 8y = -3
x + 5y = -5

Use substitution to solve each system.
x + y = 7
x-2y = -5

Use elimination to solve each system.
2x + 5y - 8
6x + 15y = 18

Solve word problem.
A 200-seat theater charges $8 for adults and $5 for children. If all seats were filled and the total ticket income was $1435, how many adults and how many children were in the audience?

An apparel shop sells skirts for $45 and blouses for $35. Its entire stock is worth $51, 750, but sales are slow and only half the skirts and two-thirds of the blouses are sold, for a total of $30, 600. How many skirts and blouses are left in the store?

Systems of Linear Equations and Matrices
Solve the system by any method.
3x + y - z = 0
2x - y + 3z = -7

The owner of a small business borrows money on three separate credit cards: x dollars on his Master card, y dollars on his Visa, and z dollars on his American Express card. These amounts satisfy the following equations:
1.18 + 1.15y + 1.09z = 11,244.25
3.54x - .55y + .27z = 3,732.75
.06x + .05y + .03z = 414.75

Applications of Systems of Linear Equations
Use systems of equations to work these applied problems.
Use this example to answer question:

Ellen McGillicuddy plans to invest a total of $100,000 in a money market account, a bond fund, an international stock fund, and a domestic stock fund. She wants 60%of her investment to be conservative (money market and bonds). She wants the amount in international stocks to be one-fourth of the amount in domestic stocks. Finally she needs an annual return of $4000. Assuming she gets annual returns of 2.5% on the money market account, 3.5% on the bond fund, 5% on the international stock fund, and 6% on the domestic fund. How much should she put in each investment?

Suppose that Ellen McGillicuddy in the example finds that her annual return on the international stock fund will be only 4%. Now how much should she put in each investment?

Shipping charges at an online bookstore are $4 for one book, $6 for two books, and $7 for three to five books. Last week, there were 6400 orders of five or fewer books and total shipping charges for these orders were $33,600. The number of shipments with $7 charges was 1000 less than the number with $6 charges. How many shipments were made in each category (one book, two books, and three-to-five books)?

Work the following problems by writing and solving a system of equations.
1. Kate borrows $10,000. Some is from her friend at 8% annual interest, twice as much as that from her bank at 9%, and the remainder from her insurance company at 5%. She pays a total of $830 in interest for the first year. How much did she borrow from each source.

2. An auto manufacturer sends cars from two plants, I and II, to dealerships A and B, located in a mid-western city. Plant I has a total of 28 cars to send, and plant II has 8. Dealer A needs 20 cars, and dealer B needs 16. Transportation costs based on the distance of each dealership from each plant are $220 from I to A, $300 from I to B, $400 from II to A, and $180 from II to B. The manufacturer wants to limit transportation costs to $10,640. How many cars should be sent from each plant to each of the two dealerships?

3. An electronics company produces transistors, resistors, and computer chips. Each transistor requires 3 units of copper, 1 unit of zinc, and 2 units of glass. Each resistor requires 3, 2, 1 unit of the three materials, and each computer chip requires 2, 1, and 2 units of these materials, respectively. How many of each product can be made with the following amounts of materials?
a. 810 units of copper, 410 of zinc, and 490 of glass
b. 765 units of copper, 385 of zinc, and 470 of glass
c. 1010 units of copper, 500 of zinc, and 610 of glass

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