MATH 1053

FINAL EXAMINATION DECEMBER 1997 | TIME: 3 HOURS |

**Answer all questions without the use of a
calculator.
Show all work; credit will be given for presentation and
methods of solution.**

1. | (a) | Determine the domain of the function, . | ||

(b) |
Use the definition of a one-to-one function to show that
g(x) = x^{3} + 3x^{2} + 9x - 5 is one-to-one.
| |||

Find the inverse of the function, . | ||||

2. | Evaluate the following limits: | |||

(a) | ; | (b) | ; | |

(c) | ; | (d) | ; | |

(e) | ; | (f) | . | |

3. | (a) | State the
definition of the limit, L of a
function, f (x), as x approaches a number, a. Use this
definition to prove that
.
| ||

(b) | Use the definition of the derivative of a function to determine the derivative of . | |||

4. |
Determine f '(x) for each of the following functions, f (x):
| |||

(a) | ; | (b) | ; | |

(c) | ; | (d) | ; | |

(e) | ; | (f) | . | |

5. | (a) | Find the equation of the tangent line at the point (1,-1) on the curve | ||

x^{3} - 2y^{3} + 3x^{2}y - xy^{2} = 2x - 3.
| ||||

(b) | Let
G(x) = g(x)^{2} where g is the inverse of a function f.
Calculate G'(3) given
.
| |||

(c) | Use logarithmic differentiation to determine the derivative of the function, | |||

6. | (a) | Use the 1st MVT to determine a linear approximation to the
function
near x = a. Hence deduce an
approximation for the number f (0.48).
| ||

(b) | A line segment, L, passes through the point (1,2) and has
its end points on the positive x- and positive y- axes. Calculate
the slope of the line for which L has minimum length.
| |||

Show that the function,
y = ae^{x} + be^{-2x} + ce^{3x}
satisfies the (differential) equation,
| ||||

for all values of the constants a, b
and c.
| ||||

(b) | The angle of elevation of the sun is decreasing at a rate of 0.25 rads/hour. How fast is the shadow cast by a building of height 400 feet increasing, when the angle of elevation of the sun is ? | |||

For the function : | ||||

(a) | locate the horizontal and vertical asymptotes; | |||

(b) | determine the intervals over which the function is increasing/decreasing, and locate all local extreme points; | |||

(c) | determine the intervals over which the function is concave up/concave down, and locate all inflection points; | |||

(d) | draw a neat sketch of the graph of the function. |