MATH 1053
FINAL EXAMINATION DECEMBER 1998 |
TIME: 3 HOURS |
1. | (a) |
| ||
| Use the definition of a one-to-one function to show that is one-to-one. | |||
(c) |
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2. | Evaluate the following limits: | |||
(a) | ; | (b) | ; | |
(c) | ; | (d) | ; | |
(e) | ; | (f) | . | |
3. | (a) | State the
-definition of a limit,
;
and use this
definition to prove that:
| ||
(b) | Use the limit definition of the derivative of a function to calculate the derivative of the function, . | |||
4. | Determine f'(x) for each of the following functions, f(x): | |||
(a) | ; | (b) | ; | |
(c) | ; | (d) | ; | |
(e) | ; | (f) | . | |
5. | (a) | Use the methods of differential calculus to show that f(x) = x^{3} + 3x^{2} + 4x - 6 is a one-to-one function and, hence, has an inverse. Compute the derivative of f^{-1}(x) at x = 2. | ||
(b) | Find the equation of the tangent to the curve 2x^{3} - y^{3} - 3xy^{2} + x^{2}y + 2y = 3 at the point (-1,1). | |||
6. | (a) | Use the 2nd MVT to determine a quadratic approximation for the function, (2x+1)e^{x}, near x = 1. | ||
(b) | Use a linear approximation of the function, , to determine an approximation for the functional value, f(0.53). | |||
7. |
| |||
(a) | locate the horizontal and vertical asymptotes; | |||
(b) | determine the behaviour of the graph of f near the horizontal asymptote; | |||
(c) | determine the intervals of x over which the function is increasing/decreasing, and locate all local extreme points of the graph of f; | |||
(d) | determine the intervals of x over which the function is concave up/concave down, and locate all inflection points; | |||
(e) | draw a neat sketch of the graph of f. | |||
8. | (a) | Sketch the parabola, y = 4x - x^{2}. | ||
(b) | Hence determine the dimensions of the largest rectangle that can be drawn in the first quadrant, given that two vertices must be on the x-axis and two vertices on the parabola. |