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 :