### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1053

 FINAL EXAMINATION DECEMBER 1995 TIME: HOURS

#### CALCULATORS PERMITTED

##### ATTEMPT ALL QUESTIONS; EACH HAS THE SAME VALUE
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 1. (a) Compute the derivative for each of the following functions : (i) (ii) (iii) (iv) (b) Find the equation of the line tangent to at x = 0. (c) Show that for small |x|.
 2. For the function (a) find the critical points and classify them as local maxima, minima or degenerate; (b) find the inflection points; (c) find the absolute maximum value of ; (d) sketch the graph.
 3. (a) Compute the following limits: (i) (ii) (iii) (b) For the following function , defined for , sketch its graph and identify the points x where it is discontinuous, or not differentiable, or both.
 4. (a) State the mean value theorem. (b) Show that for all x > 1, where n is any integer . (c) Show that satisfies , for every pair of numbers . 5. Suppose that the equation represents a curve. [Note: You may assume that and ] (a) Find the points where this curve intersects the coordinate axes. (b) Explain why the shape of the curve in the first quadrant determines its shape in the other three quadrants. (c) Find the tangent slope for any point on this curve. (d) At what points on the curve is the tangent line horizontal? At what points is the tangent vertical? (e) Show that is on the curve and compute the tangent slope there. (f) From the information in parts (a) to (e), draw a rough sketch of the curve. (g) Equation (1) expresses conservation of energy during motion of a pendulum, with x as angular position variable and y as angular velocity. What is the amplitude of this motion (i.e., the maximum value of |x|)?
 6. (a) (i) Show that and 3 are fixed points of . (ii) Determine which, if any, of these fixed points is attracting? (b) Use Newton's method for solving : to find the positive root of accurate to two decimal places, starting from .