DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1823

FINAL EXAMINATION
APRIL 1999
TIME: 3 HOURS
TOTAL POINTS = 100

INSTRUCTIONS:

(a) Check that this examination contains seven pages, numbered 1 - 7.
(b) Answer all questions, and show the details of your solutions. Credit will be given to method and presentation of solutions.
(c) Calculators are not allowed.
(d) The reverse sides of the pages may be used for rough work and/or continuation of a solution. Indicate clearly if a solution is continued on the reverse side of a page.

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MARKS
(24)1. Find y' for the following functions: (DO NOT SIMPLIFY)
(a) ${\displaystyle y = 3x^{\textstyle 2} + 2 \sqrt{x} + 4 + \frac{1}{x}}$
(b) y = (4-x2)5
(c) y = x2 ex
(d) ${\displaystyle y = \frac{x^{\textstyle 2} + 1}{x-1}}$
(e) y = ln(1+x)
(f) $y = (e^{\textstyle x} + \sqrt{x + 1})^{\textstyle 1/2}$
(12)2. Evaluate the following limits:
(a) ${\displaystyle \lim_{x \rightarrow 1}\;\frac{x^{\textstyle 2} +
x - 2}{x-1}}$
(b) ${\displaystyle \lim_{x \rightarrow 0}\;\frac{x^{\textstyle 2} +
1}{x^{\textstyle 3} - x}}$
(c) ${\displaystyle \lim_{x \rightarrow
\infty}\;\frac{5x^{\textstyle 3} - 2x^{\textstyle 2} +
1}{2x^{\textstyle 3} - x}}$
(6)3. Are the following functions continuous at x = 0? Justify your answers.
(a) y = 2x + 4
(b) ${\displaystyle y = \left\{ \begin{array}{ccl}
x & \mbox{if} & x > 0\\
1 & \mbox{if} & x \leq 0
\end{array} \right. . }$
(8)4. Find the equations of the following lines:
(a) the line through (-3,1) and (0,3);
(b) the line tangent to y = x3 + 3x2 + 1 when x = -1.
(20)5. Evaluate the integrals:
(a) ${\displaystyle \int \left( \frac{x^{\textstyle 3}}{2} -
\sqrt{x} + \frac{1}{x} \right)\;dx}$
(b) ${\displaystyle \int_0^4\;x\sqrt{x^{\textstyle 2} + 9}\;dx}$
(c) ${\displaystyle \int\;\frac{1}{4x+1}\;dx}$
(d) ${\displaystyle \int\;xe^{\textstyle x}\;dx}$
(e) ${\displaystyle \int_0^{\infty}\;\frac{1}{(2x + 1)^{\textstyle
2}}\;dx}$
(6)6. Find the area between the curve y = x2 - 2x + 3 and the x-axis from x = 1 to x = 2.
(8)7. A factory makes TV sets and the selling price is $1,000 each. If the manufacturing cost of ``x'' sets is $10,000 + 2x + .01x2, how many sets should they make to maximize the profit?
(16)8. Let f(x) = (x2 - 4) 2.
(a) Compute f'(x) and f''(x).
(b) Determine the intervals in which the function is increasing/decreasing.
(c) Determine the intervals in which the function is concave up/concave down.
(d) Locate the local maximum and local minimum points.
(e) Locate the inflection points.
(f) Draw a neat sketch of f(x).