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Calculus I                                                                 Final Exam                                   December 10, 2002

Name____________________                             R.  Hammack                                        Score ______
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(1) Evaluate the following limits.

(a)

(b)

(c)

(d)

(e)

(f)

(2) Use the limit definition of the derivative to find the derivative of the function .

(3) Find the following derivatives.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(4) Simplify the following expressions as much as possible.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(5)   Sketch the graph of

(6)   Solve the equation      for x.

(a)

(b)

(c)   List the vertical asymptotes of f  (if any).

(d)   List  the horizontal asymptotes of f  (if any).

(e)   Find the inverse of f.

(f) State the domain of f.  All values of x except 2.

(g)   State the domain of.

(h)  State the range of  f

(i)   State the range of.

(j) Find the equation of the tangent line to the graph of at the point where .

(8) This problem concerns the function .

(a)

(b)

(c)  Find the interval(s) on which  f  is increasing and on which it is decreasing.

(d)  Find the interval(s) on which  f  is concave up and on which it is concave down.

(e)  List the x-coordinates of all inflection points of  f.

(f)  List all the critical numbers of f.

(g)  List the x-coordinates of the relative extrema of  f  (if any) and say whether there is a relative minima or a relative maxima.

(9) Given the equation   ,  find  .

(10) A rectangular box with with two square sides and an open top is to have a volume of 36 cubic feet. Find the dimensions of the container with minimum surface area.

(11) A spherical balloon is deflated in such a way that its radius decreases at a constant rate of 15 cm/min. At what rate is air escaping when the radius is 2 cm?

Hint: The volume of a sphere of radius r is .