Calculus I,    B-Track Test #1 September  30, 2002 Name____________________ R.  Hammack Score ______

(1) (10 points)

(a)

(b) Find all the x values that are not in the domain of the function .
The denominator will be zero for those values of x for which sin(x) =1/2.
Therefore the values x = and  x = (where n is an integer) are not in the domain of f.

(2) (10 points)

(a) Sketch the graph of the equation    .
To find the x-intercept, set y = 0 and solve for x. You get x = -2.
To find the y-intercept, set x = 0 and solve for y. You get y  = 4.

(b) Find the equation of the line that is parallel to the graph of    (from Part a, above) and which passes through the point . Put your final answer in slope-intercept form, and simplify as much as possible.

Looking at the graph above, you see that the line has slope 2. We seek a line that is parallel to this one, so its slope will be 2 also. Moreover, it is to pass through (-2,-3), so the point-slope formula gives

(3) (10 points)  Consider the functions and

(a)

(b)

(4)  (20 points)  Evaluate the following limits.

(a)

(b)   0

(c)    (0)(1) = 0

(d)

(5) (35 points) This problem concerns the function  .

(a) State the domain of f.
Observe =(provided x is neither 0 nor 2)

Note that x = 0 or x = 2 will give a zero in the denominator, so these values are not in the domain. However, any other value of x will work. Therefore the domain of f is all real numbers except 0 and 2.

(b)     (2+2)/2 = 2

(c)     (Note highest power of x occcurs on both top and bottom, each with coefficient 1.)

(d)    (Note, top gets closer to 2, and bottom gets closer to 0, but positive.)

(e)

(f) List the vertical asymptote(s) of f  (if any).  (Feel free to use any relevant information from parts a-d above)
By part, (d), the line x=0 is a vertical asynptote,

(g) List the horizontal asymptote(s) of f  (if any).   (Feel free to use any relevant information from parts a-d above)
By part, (c), the line y=1 is a horizontal asynptote,

(6) (10 points)  This problem concerns the function
(a)
(b)
(c)
(d) D.N.E. because right- and left-hand limits do not agree.
(e) Is f continuous at x = 0? (Explain, using above information.)
NO, because does not exist, so it's impossible that

(f)
(g)
(h)
(i) 0, by parts g and h above
(j) Is f continuous at x = 1? (Explain, using above information.)
YES, because the above shows

(7) (5 points) Sketch the graph of a function f  satisfying the following properties:  ,   ,   ,  ,   and   ,  There are many correct answers.