MATH 1823
INSTRUCTIONS: | |
(a) | Answer each question in your answer booklets. All your work must be shown. |
(b) | For most questions calculators are not required; on such questions the calculator may only be used to check accuracy. |
MARKS | |||
(15) | 1. | Find the derivative of each of these functions: | |
(a) | |||
(b) | |||
(c) | |||
(d) | |||
(e) | |||
(12) | 2. | Determine these limits: | |
(a) | |||
(b) | |||
(c) | |||
(d) | |||
(13) | 3. | (a) | Find the equation of the tangent line to the graph of when x = 1. |
(b) | Simplify: | ||
(c) | Recall that an investment of $P compounded continuously for x years at (100r)% p.a. is worth . If the annual interest rate is 5%, how long does it take for $1000 to double in value? | ||
(d) | Set up (but do not evaluate) the limit required when computing the derivative of from the definition. | ||
(8) | 4. | A concert promoter finds from a consumer survey that she can sell 500-5x tickets, if the price is set at x dollars each. On the other hand, concert costs run at $10,000 plus an additional $20 for each ticket. | |
(a) | Give the revenue function (as a function of x). | ||
(b) | Give the cost function. | ||
(c) | Give the profit function. | ||
(d) | What ticket price should the promoter charge in order to maximize her profit? Be sure to justify your answer using the methods of calculus. | ||
(14) | 5. | For the function : | |
(a) | Determine the intervals where is increasing and where is decreasing. | ||
(b) | Find the relative maxima and minima. | ||
(c) | Determine the intervals where is concave up and where is concave down. | ||
(d) | Sketch the graph of . | ||
(5) | 6. | If the marginal cost function is given by and the fixed cost (i.e. at the 0 production level) is 500, determine the cost function. | |
(14) | 7. | Evaluate the following integrals: | |
(a) | |||
(b) | |||
(c) | |||
(d) | |||
(6) | 8. | Find the area enclosed by the curves | |
and y = 2 - x |