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Calculus I                                                  Test #2                                      April 9, 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) = 1/x^2.






(2) The graph of a function g(x) is shown below.  Using the same coordinate axis, sketch a graph of  y = g ' (x).

[Graphics:HTMLFiles/T2S04C_6.gif]



(3) Suppose f and g are functions for which f(3) = 10, g(3) = 8, f ' (8) = -2, and g ' (3) = 3.
Suppose also that h(x) = f(g(x)).  Find h ' (3).




(4) State two things that the derivative of a function f(x) tells you.  Be specific.

    



(5) d/dx[ x^5 + x^3 - 3] =



(6) d/dx[ 1/x^(1/2)] =  



(7)  d/dw[ w/(w + 1)] =



(8) d/dx[ x^2tan^2(x)    + sin(x) ] =



(9) d/dx[   (x^5 + x^3 - 3 )^(-1)] =



(10)  If     y = 1/x + 5x^2,    find (d^2y)/dx^2.




(11) d/dx[ sec^10(x)] =


(12) d/dx[ sec(x^10) ] =


(13)  d/dx[ (x + x^(1/2))^(1/2)] =

(14) d/dx[ tan(sin(cos(x))) ] =


(15)  Find all values of x for which the tangent line to   y = 2 sin(x)  at  (x, f(x)) has a slope of 1.



(16)  Find the slope of the tangent line to the graph of the equation  y/x = x y^2  at the point (1, 1).

 




(17)  Find the equation of the tangent line to the graph of y = x^3 at the point where x = 1.

 


(18)  A 10-foot ladder leans against a wall at an angle θ with the horizontal. The top of the ladder is y feet above the ground. If the bottom of the ladder is pushed toward the wall, find the rate y changes with respect to  θ when θ = π/3.

[Graphics:HTMLFiles/T2S04C_63.gif]




(19)  A 10-foot ladder leans against a wall.  If the bottom of the ladder is pushed toward the wall at a rate of 2 feet per second, how quickly is the top of the ladder moving up the wall when it's 8 feet above the floor?
[Graphics:HTMLFiles/T2S04C_68.gif]