MATH 1823

April 1998


(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 may be used for numerical calculations only. The use of graphing calculators is not permitted.
(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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(10) 1. (a) Determine the domains of the two functions:
(i) ${\displaystyle f(x) = \frac{2x+1}{x^2-x}}$ (ii) $g(x) = \sqrt{5x + 3}$
(b) Evaluate the following limits:
(i) ${\displaystyle \lim_{x \rightarrow -1}\;\frac{x^2-1}{2x^2 +
2x}}$ (ii) ${\displaystyle \lim_{x \rightarrow \infty}\;\frac{3x^2-x+4}{x(2x+1)}}$
(17)2. (a) For the function ${\displaystyle f(x) = \frac{2x-5}{x-2}}$, calculate the one-sided limits:
(i) ${\displaystyle \lim_{x \rightarrow 2^-}\;f(x)}$ (ii) ${\displaystyle \lim_{x \rightarrow 2^+}\;f(x)}$
(b) Use the limit definition to determine the derivative of the function,
f(x) = 3x2 + 6x - 7.
(c) State the derivatives of the following five functions:
\begin{displaymath}(2x+1)^{-0.7};\hspace*{0.5cm} e^{3x+1}; \hspace*{0.5cm} \ln(4x+1);
\hspace*{0.5cm} 10^{5x+1}; \hspace*{0.5cm} \log_{10}(6x+1).
(18) 3. Compute the derivatives of the following functions, but do not simplify your answers:
(a) ${\displaystyle y = 3x^2 - x \sqrt{x} + \frac{5}{\sqrt{x}}}$
(b) ${\displaystyle y = e^{-2x} \ln(x^2+1)}$
(c) ${\displaystyle y = \frac{2x^2 + 3}{\sqrt{6x-1}}}$
(d) ${\displaystyle y = \ln \left( x + \sqrt{x+1}\;\right)}$
(e) ${\displaystyle y = \log_2 \left( \frac{1}{4x+1}\right)}$
(f) ${\displaystyle y = 7^{\textstyle x^{\textstyle 2}+ 2x + 5}}$
(15) 4. Compute the following integrals:
(a) ${\displaystyle \int \left( 3x^2 - \frac{4}{x\sqrt{x}}\right)\;dx}$
(b) ${\displaystyle \int \left( e^{2x} + \frac{1}{5x}
(c) ${\displaystyle \int x^2 \ln x\;dx}$
(d) ${\displaystyle \int \frac{2x}{x^2 + 1}\;dx}$
(e) ${\displaystyle \int \sqrt{x}\;e^{x\sqrt{x}}\;dx}$
(f) ${\displaystyle \int_{-1}^2 (6x^2 - 4x + 5)\;dx}$
(16)5. Let ${\displaystyle f(x) = x^3 +
\frac{3}{2}\;x^2 - 6x + 4}$.
(a) Compute f '(x) and f ''(x).
(b) Determine the intervals of x over which the function is increasing/decreasing.
(c) Also, determine the intervals over which f (x) is concave up/concave down.
(d) Find the critical points of f (x).
(e) Use the 1st Derivative Test to determine whether the negative critical point in (d) is a local maximum or local minimum.
(f) Also, use the 2nd Derivative Test to determine whether the positive critical point in (d) is a local maximum or local minimum point.
(g) Draw a neat sketch of the graph of f (x).
(10)6. (a) Sketch the parabola, y = -x2 + 2x.
(b) Determine the equation of the tangent to the parabola of part (a) at the point (-1,-3).
(c) Compute the area of the finite region between the parabola of part (a) and the x-axis.
(14)7. (a) Find the absolute maximum and minimum values of the function
${\displaystyle f(x) = \frac{2x}{x^2+4}}$ defined over $1
\leq x \leq 4$.
(b) At any time t, the area of a rectangle (A) is increasing at a rate of 4 square units/minute. If the height (h) of the rectangle is always 2 units longer than the width (x), calculate the rate of increase of x when x = 3 units.
(c) A manufacturer's cost and revenue functions for the production of x units of an item are given, in dollars, by
\begin{displaymath}C(x) = 1000 - 50x + \frac{x^2}{10}\;\;, \;\;\;\;R(x) = 400 +
Calculate the profit earned from the production and sale of 25 units, and estimate the profit that may be earned from the production of a 26th unit.