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, ${\displaystyle y =
\frac{x^{\textstyle 2} + 1}{x}}$.
(b) Use the definition of a one-to-one function to show that ${\displaystyle f(x) = \frac{3x + 1}{2x+5}}$ is one-to-one.
(c) Find the inverse of the function, ${\displaystyle
y = \log_7(4x-1),\;x > \frac{1}{4}}$.
2. Evaluate the following limits:
(a) ${\displaystyle \lim_{x \rightarrow \infty}\;\frac{1 + x -
2x^{\textstyle 2}}{x \sqrt{1 + x^{\textstyle 2}}}}$; (b) ${\displaystyle \lim_{x \rightarrow
3}\;\frac{x - 3}{x - \sqrt{4x-3}}}$;
(c) ${\displaystyle \lim_{x \rightarrow 2-}\;\frac{x-3}{x^{\textstyle
2} + x - 6}}$; (d) ${\displaystyle \lim_{x \rightarrow \pi/2}\;\frac{\sin 2x}{2x - \pi}}$;
(e) ${\displaystyle \lim_{x \rightarrow -1}\;\frac{2x^{\textstyle 3}
+ x^{\textstyle 2} + 1}{x^{\textstyle 3} + 2x^{\textstyle 2} - 1}}$; (f) ${\displaystyle \lim_{x \rightarrow 1}\;
\frac{x - \vert x\vert}{x - 1}}$.
3. (a) State the $\varepsilon$-definition of a limit, ${\displaystyle \lim_{x \rightarrow a}\;f(x) = L}$; and use this definition to prove that:
\begin{displaymath}\lim_{x \rightarrow 2}\;\frac{x^{\textstyle 2} + 1}{x - 1} = 5.
\end{displaymath}
(b) Use the limit definition of the derivative of a function to calculate the derivative of the function, ${\displaystyle \frac{1}{\sqrt{x}},\;x
\neq 0}$.
4. Determine f'(x) for each of the following functions, f(x):
(a) $f(x) = 2^{\textstyle x^{\textstyle 2}+1} \cdot \sqrt{x^{\textstyle 2}
+ 2x + 3}$; (b) $f(x) = \sin(\ln (1 + \sqrt{x}))$;
(c) ${\displaystyle f(x) = \frac{\cos(5x+1)}{x + \tan (2x-3)}}$; (d) ${\displaystyle f(x) = \frac{1}{\tan^{\textstyle -1}x}}$;
(e) ${\displaystyle f(x) = \log_{10} \sqrt{\frac{x(x^{\textstyle
2}+1)}{4x-1}}}$; (f) $f(x) = x^{\textstyle x}$.
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, $f(x) = x \sin \pi
x$, to determine an approximation for the functional value, f(0.53).
7. For the function, ${\displaystyle f(x) = \frac{1-x^{\textstyle
2}}{x^{\textstyle 3}}}$:
(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.