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Calculus I                                                          Quiz #5                            March 16, 2004

Name____________________                   R.  Hammack                              Score ______
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(1) Use the limit definition of the derivative to find the derivative of the function  f(x) = x^(1/2).

f ' (x) = Underscript[lim , wx] (f(w) - f(x))/(w - x) = Underscript[lim , wx] ... x^(1/2))/(w^(1/2) + x^(1/2)) = Underscript[lim , wx] (w - x)/((w - x) (w^(1/2) + x^(1/2)))
Underscript[lim , wx] 1/(w^(1/2) + x^(1/2)) = 1/(x^(1/2) + x^(1/2)) = 1/(2x^(1/2))

Thus f ' (x) = 1/(2x^(1/2))


(2)  Use your answer from Question 1 to find the slope of the tangent line to the graph of the function f(x) = x^(1/2)
at the point where x = 9. (You do not need to use a limit to answer this question.)

f ' (9) = 1/(29^(1/2)) = 1/6


(3)   Use your answer from Question 2 to find the equation of the tangent line to the graph of the function f(x) = x^(1/2)
at the point where x = 9. Please put your final answer in slope-intercept form.

The line has slope 1/6.
The line passes through the point (9, f(9)) = (9, 9^(1/2)) = (9, 3).
Using the point-slope formula for a straight line, we get:
y - 3 = 1/6 (x - 9)

y - 3 = 1/6 (x - 9)

y - 3 = 1/6x - 3/2

y = 1/6x + 3/2