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Calculus I                                                                           Test #1                                              October  6, 2003

Name____________________                                   R.  Hammack                                              Score ______
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(1) (12 points)

(a)

(b) 1

(c) Find all solutions of  the equation    .
This is equivalent to  . Looking at the unit circle, we see that and  make this equation true. But a multiple of could be added to these numbers, and the equation would still hold. Thus the solutions are all numbers of form or  .

(2) (8 points)

(a) Sketch the graph of the equation    .

(b) Find the equation of the line that is perpendicular to the graph of    (from Part a, above) and which has an x-intercept of 1.  Put your final answer in slope-intercept form, and simplify as much as possible.

The above line has slope 1/2, so the slope of the line we seek is the negative reciprocal of this, namely -2.
Since the line we seek has x-intercept 1, it passes through the point (1, 0).
Using the point-slope formula,

(3) (40 points)  The problems on this page concern the functions and

(a)

(b)

(c) State the domain of Note: the denominator factors as 3x(x - 3).
Domain: All real numbers except 0 and 3.

(d)  List the values of x at which f is not continuous. 0 and 3

(e)

(f)

(g)      (because top approaches 3, bottom approaches 0, and is positive)

(h)

(i) List the vertical asymptote(s) of f  (if any).  (Feel free to use any relevant information from parts a-d above)
By part (g) above, the line x = 3 is a vertical asymptote.

(j) List the horizontal asymptote(s) of f  (if any).   (Feel free to use any relevant information from parts a-d above)
By part (h) above, the line y = 1/3 is a vertical asymptote.
(4)  (16 points)  Evaluate the following limits.

(a)

(b)

(c)    1

(d)

(5) (20 points)  This problem concerns the function
(a) 0

(b)

(c)

(d) Sketch a graph of the function f.

(e)
Is the function f continuous at x = 1?  Yes, because by the above work.

(6) (4 points)

(Note that x is negative in this problem, since it's approaching negative infinity. Therefore we must multiply the 1/x terms by -1 to convert them to positive quantities that can be brought into the radical.)