### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1053

 FINAL EXAMINATION DECEMBER 1996 TIME: HOURS
CALCULATORS PERMITTED
ATTEMPT QUESTIONS 1, 2, 3, 4 AND AT LEAST TWO OTHERS.
(Note the formulas in the Appendix.)
 MARKS (20) 1. (a) Find the first derivative of each of the following functions: (b) Evaluate the following limits: (i) (ii) (iii) (c) Find for (Hint: Apply the definition of derivative and then substitute .) (20) 2. For each of the following functions defined for all , find (i) the stationary point(s); (ii) asymptote(s) (if any); (iii) points where there is no derivative (if any); (iv) the maximum value of the function. (v) Also, sketch the graph. (a) ; (b) . (20) 3. (a) (i) Obtain the third-order Taylor polynomial approximation to for x near 0 (zero). (ii) Compute from the result of part (i). (b) (i) State the Mean Value Theorem. (ii) Show that for all x > 1. (iii) Show that for all . (Hint: Consider the derivative of each function.) (10) 4. In theories allowing for large deflections of elastic beams or columns, the bending moment is proportional to the curvature , which is a nonlinear function of y (see Appendix): where c depends on the material. Suppose c = 1 and . (a) Compute the bending moment. (b) Locate the point where |M| is largest. (15) 5. Consider the curve defined parametrically by (a) Show that must be (one branch of) a hyperbola. (b) Suppose that and in (1) are the components of the position vector of a particle at time t. Find the components of the velocity vector at time . (c) Sketch together with the tangent vector obtained in part (b). (d) What should the t-interval in (1) be changed to in order to generate the other branch of the hyperbola? (15) 6. Consider the recurrence system with . (a) Sketch the graph of . (b) Locate the fixed points of f. (c) Determine whether the fixed points are attracting or repelling. (d) Show that the set of three x-values constitutes a cycle (periodic solution) of period 3 for the system (2). (15) 7. Consider the curve defined by (a) Explain why there are no points on the graph for and 0 < x < 1. (b) Obtain a formula for the tangent slope . (c) Locate the point(s) on , with , where the tangent is vertical and the point(s) where it is horizontal. (d) Sketch the curve for . (e) Sketch the complete curve.

### APPENDIX

 1. . 2. Standard Cartesian equations for conic sections: 3. Curvature of a smooth curve :