### DEPARTMENT OF MATHEMATICS & STATISTICS 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) = x3 + 3x2 + 9x - 5 is one-to-one. (c) 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 x3 - 2y3 + 3x2y - xy2 = 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. 7. (a) Show that the function, y = aex + be-2x + ce3x 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 ? 8. 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.