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Calculus I                                                                      Test #2                                             October 29, 2003

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^2.





(2) The graph of a function g(x) is a straight line that is inclined at an angle of 30 degrees, as illustrated below.  Find g ' (0).
[Graphics:HTMLFiles/T2F03Dsol_5.gif]


(3) Sketch the graph of a function  f  for which f(0) = -1, f ' (0) = 0,  f ' (2) = 0, f ' (x) ≥0 when x≥0, and  f ' (x) ≤0 when x≤0.
[Graphics:HTMLFiles/T2F03Dsol_14.gif]


(4) Suppose that the cost of drilling x feet for an oil well is C = f(x) dollars, and suppose that  f ' (300) = 1000. Explain, in non-mathematical terms, what the statement  f ' (300) = 1000 means.


(5) d/dx[ 3x^4 + x + 2] =

(6) d/dx[ -3/x^8 + 2x^(1/2)] =


(7)  d/ds[ 3/(2s + 1)] =


(8) d/dx[ sec(x) tan(x) ] =


(9) d/dx[sin(x) /x^2] =


(10)  If y = cos(x), find (d^2y)/dx^2.





(11) d/dx[ (x^3 + 3x)^10] =


(12) d/dx[ sin(x^3) ] =


(13)  d/dx[ sin(x^3)^(1/2)] =


(14) d/dx[ x^5sec(1/x)] =


(15)  Suppose f  is a function for which f(4) = 3 and f ' (4) = -5.  If g(x) = x f(x), find g ' (4).





(16)  Find all points on the graph of  y = 1 - x^2  at which the tangent line passes through the point (0, 2).

 



(17)  Find the equation of the tangent line to the graph of y = tan(x) at the point where x = π/4.

 





(18)  A search light is trained on a tall building. As the light rotates, the spot it illuminates moves up  the side of the building. That is, the distance D between the ground and the illuminated spot is a function of the angle θ formed by the light beam and the horizontal ground. If the search light is located 50 meters from the building, find the function giving the rate of change of D with respect to θ.

 

 

 


(19)  Use implicit differentiation to find dy/dx:     cos(x y) = 1/2.

 




(20)  A conical water tank has a height of 24 feet and a radius of 12 feet, as illustrated. If water flows into the tank at a constant rate of 20 cubic feet per minute, how fast is the depth h of the water changing when h = 2?
(Hint: You may be interested to know that the volume V of a cone of height h and radius r is V = π/3r^2h.)