Algebra-
1. You can rent a car for $100 plus 10 cents for each kilometer over 100, or $75 plus 13 cents for each kilometer over 150. How far would you have to drive the car for both options to cost the same amount?
2. Solve this system for x and y.
3x - 4y = 7
5x + 6y = 13
3. It takes 5 hours to put together 8 Ikea chairs and 7 Ikea bookcases. It takes 4 hours to put together 11 Ikea chairs and 3 Ikea bookcases. How long does it take to put together 5 Ikea chairs and 6 Ikea bookcases?
4. Solve this system for x, y, and z.
x - 2y + 3z = 4
2x + 3y - 4z = 9
3x + y + 5z = 15
5. Solve each equation for x.
(a) 2x2 + 3x - 6 = 0
(b) 11x2 + √5x - 1 = 0
(c) √(x +3) = x - 3
6. My pet square ran away, so I bought a pet rectangle of the same area. The length of the rectangle is five centimeters more than twice the side of the square; the height of the rectangle is six centimeters less than the side of the square. How long was the side of the square?
7. Evaluate log168.
8. Solve 7x = 11 for x, giving your answer in terms of logarithms.
9. Suppose log x = 43.7 and log y = 69.2. Find log (x2/ √y).
Calculus-
10. The average cost a(q) of producing q items is defined as the total cost C(q) divided by the number of items, q, so a(q) = C(q)/q. Explain why the average cost is minimized when the marginal cost is equal to the average cost in:
(a) mathematical terms;
(b) economic terms.
11. The average cost per item to produce q items is given by a(q) = 0.01q2 - 0.6q + 13, for q > 0.
(a) What is the total cost, C(q), of producing q goods?
(b) What is the minimum marginal cost? What is the practical interpretation of this result?
(c) At what production level is the average cost a minimum? What is the lowest average cost?
(d) Compute the marginal cost at q = 30. How does this relate to your answer to part (c)? Explain this relationship both analytically and in words.
12. Consider the function f(x) = x3 - 3x2.
(a) Find the equation of the derivative f'(x).
(b) Find the equation of the tangent to the graph of f at x = 2.
(c) Find the equation of the normal at x = 2. (The normal is the line perpendicular to the tangent).
(d) Sketch f(x), f'(x), the tangent at x = 2, and the normal at x = 2. Label any critical points of f.
13. Find the derivatives of the following functions using the various rules for computing derivatives:
(a) f(x) = x3 - 3x2 + 7x - 6
(b) g(x) = x + √(x2+(1/x2))
(c) r(t) = (t2 - t + 1)(5 + 2t - t3)
(d) q(s) = (2s - 7)5(3s + 1)6 (Factor the answer as much as possible.)
(e) w(z) = z2 + 8z +1/z2 - 1
14. Find the global maxima and minima of the following functions over the stated interval.
(a) f (x) = 2x2 + 6x + 2 on [-1, 3].
(b) g(x) = x3 - x2 + 12x - 7 on [-4, 4].
15. (a) A cruise line offers a trip for $2000 per passenger. If at least 100 passengers sign up, the price is reduced for all the passengers by $10 for every additional passenger (beyond 100) who goes on the trip. The boat can accommodate 250 passengers. What number of passengers maximizes the cruise lines total revenue? What price does each passenger pay then?
(b) The cost to the cruise line for n passengers is 80,000 + 400n. What is the maximum profit that the cruise line can make on one trip? How many passengers must sign up for the maximum to be reached and what price will each pay?