MATH 1053

FINAL EXAMINATION | |

DECEMBER 1996 | TIME: HOURS |

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 near x (zero).
0 | ||||

(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 (see Appendix):
y | ||||||

where depends on the material.
c | ||||||||

Suppose and .
c = 1 | ||||||||

(a) | Compute the bending moment. | |||||||

(b) | Locate the point where is largest.
|M| | |||||||

(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 . Find the components of the
velocity vector at time .
t | |||||||

(c) | Sketch together with the tangent vector obtained in part (b). | |||||||

(d) | What should the -interval in (1) be changed to in order to
generate the other branch of the hyperbola?
t | |||||||

(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 -values constitutes
a cycle (periodic solution) of period 3 for the system (2).
x | |||||||

(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. |

1. | . |

2. | Standard Cartesian equations for conic sections: |

3. | Curvature of a smooth curve : |