### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1053

 FINAL EXAMINATION DECEMBER 1998 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 range of the function, . (b) Use the definition of a one-to-one function to show that 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 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) = x3 + 3x2 + 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 2x3 - y3 - 3xy2 + x2y + 2y = 3 at the point (-1,1). 6. (a) Use the 2nd MVT to determine a quadratic approximation for the function, (2x+1)ex, near x = 1. (b) Use a linear approximation of the function, , to determine an approximation for the functional value, f(0.53). 7. For the function, : (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 - x2. (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.