DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1053

FINAL EXAMINATION
DECEMBER 1997
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 domain of the function, $f(x) = \ln(2x^2 + 5x -
3)$.
(b) Use the definition of a one-to-one function to show that g(x) = x3 + 3x2 + 9x - 5 is one-to-one.
(c) Find the inverse of the function, ${\displaystyle h(x) =
\frac{3x+1}{2x-5}}$.
2. Evaluate the following limits:
(a) ${\displaystyle \lim_{x \rightarrow -2}\;\frac{x+2}{x +
2\sqrt{x^2-3}}}$; (b) ${\displaystyle \lim_{x \rightarrow
\infty}\;\frac{x^2 - 3x + 7}{(2x + 1)^2}}$;
(c) ${\displaystyle \lim_{x \rightarrow 0}\; x \cot 3x}$; (d) ${\displaystyle \lim_{x \rightarrow 1-}\;\frac{x-2}{3x^2 - 2x -
1}}$;
(e) ${\displaystyle \lim_{x \rightarrow -\infty}\;\frac{x^3+1}{x
\sqrt{x^2 + 1}}}$; (f) ${\displaystyle \lim_{x \rightarrow
3+}\;\frac{\vert x-4\vert - \vert x-2\vert}{\vert x - 3\vert}}$.
3. (a) State the $\varepsilon-\delta$ definition of the limit, L of a function, f (x), as x approaches a number, a. Use this definition to prove that ${\displaystyle \lim_{x \rightarrow
2}\;\frac{2x + 1}{3x - 1} = 1}$.
(b) Use the definition of the derivative of a function to determine the derivative of ${\displaystyle f(x) = \frac{2x+1}{3x-1},\;\;x \neq
\frac{1}{3}}$.
4. Determine f '(x) for each of the following functions, f (x):
(a) $f(x) = 10^{3x+1} \cos 2x$; (b) $f(x) = \sec
\sqrt[3]{x^2 + 2}$;
(c) $f(x) = \tan^{-1} \sqrt{x^2 + 3x + 3}$; (d) $f(x) = \log_7(1 +
x \sinh x)$;
(e) $f(x) = \sqrt{x^2 + \sqrt{2x+1}}$; (f) ${\displaystyle f(x) = \frac{e^x}{3 \sin 4x + \ln(x^2 + 1)}}$.
5.(a) Find the equation of the tangent line at the point (1,-1) on the curve
x3 - 2y3 + 3x2y - xy2 = 2x - 3.
(b) Let G(x) = g(x)2 where g is the inverse of a function f. Calculate G'(3) given $f(-1) = 3,\;\;f'(-1) = 5$.
(c) Use logarithmic differentiation to determine the derivative of the function,
\begin{displaymath}y = \frac{e^x \tan x}{x \sqrt{x^2 + x + 1}}\;\;.
\end{displaymath}
6.(a) Use the 1st MVT to determine a linear approximation to the function $f(x) = x \cos \pi x$ near x = a. Hence deduce an approximation for the number f (0.48).
(b) A line segment, L, passes through the point (1,2) and has its end points on the positive x- and positive y- axes. Calculate the slope of the line for which L has minimum length.
7.(a) Show that the function, y = aex + be-2x + ce3x satisfies the (differential) equation,
\begin{displaymath}\frac{d^3y}{dx^3} - 2\;\frac{d^2y}{dx^2} - 5\;\frac{dy}{dx} + 6y = 0
\end{displaymath}
for all values of the constants a, b and c.
(b) The angle of elevation of the sun is decreasing at a rate of 0.25 rads/hour. How fast is the shadow cast by a building of height 400 feet increasing, when the angle of elevation of the sun is ${\displaystyle \frac{\pi}{6}}$?
8. For the function ${\displaystyle f(x) = \frac{8x}{(2x+1)^2}}$:
(a) locate the horizontal and vertical asymptotes;
(b) determine the intervals over which the function is increasing/decreasing, and locate all local extreme points;
(c) determine the intervals over which the function is concave up/concave down, and locate all inflection points;
(d) draw a neat sketch of the graph of the function.