MATH 1823
FINAL EXAMINATION APRIL 1999 | TIME: 3 HOURS TOTAL POINTS = 100 |
INSTRUCTIONS: | |
(a) | Check that this examination contains seven pages, numbered 1 - 7. |
(b) | Answer all questions, and show the details of your solutions. Credit will be given to method and presentation of solutions. |
(c) | Calculators are not allowed. |
(d) | The reverse sides of the pages may be used for rough work and/or continuation of a solution. Indicate clearly if a solution is continued on the reverse side of a page. |
MARKS | |||
(24) | 1. | Find y' for the following functions: (DO NOT SIMPLIFY) | |
(a) | |||
(b) | y = (4-x^{2})^{5} | ||
(c) | y = x^{2} e^{x} | ||
(d) | |||
(e) | y = ln(1+x) | ||
(f) | |||
(12) | 2. | Evaluate the following limits: | |
(a) | |||
(b) | |||
(c) | |||
(6) | 3. | Are the following functions continuous at x = 0? Justify your answers. | |
(a) | y = 2x + 4 | ||
(b) | |||
(8) | 4. | Find the equations of the following lines: | |
(a) | the line through (-3,1) and (0,3); | ||
(b) | the line tangent to y = x^{3} + 3x^{2} + 1 when x = -1. | ||
(20) | 5. | Evaluate the integrals: | |
(a) | |||
(b) | |||
(c) | |||
(d) | |||
(e) | |||
(6) | 6. | Find the area between the curve y = x^{2} - 2x + 3 and the x-axis from x = 1 to x = 2. | |
(8) | 7. | A factory makes TV sets and the selling price is $1,000 each. If the manufacturing cost of ``x'' sets is $10,000 + 2x + .01x^{2}, how many sets should they make to maximize the profit? | |
(16) | 8. | Let f(x) = (x^{2} - 4)^{ 2}. | |
(a) | Compute f'(x) and f''(x). | ||
(b) | Determine the intervals in which the function is increasing/decreasing. | ||
(c) | Determine the intervals in which the function is concave up/concave down. | ||
(d) | Locate the local maximum and local minimum points. | ||
(e) | Locate the inflection points. | ||
(f) | Draw a neat sketch of f(x). |