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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 .

(2) The graph of a function is a straight line that is inclined at an angle of 30 degrees, as illustrated below.  Find .

g ' (x) = slope of the line =

(3) Sketch the graph of a function  f  for which , ,  , when , and   when .

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

When the well is 300 feet deep, it will cost an extra \$1000 to dig the next foot.
(Because rate of change of cost is \$1000 per foot when x = 300 feet.)

(5)

(6)

(7)

(8)

(9)

(10)  If , find .

(11)

(12)

(13)

(14)

(15)  Suppose f  is a function for which and .  If , find .

(16)  Find all points on the graph of    at which the tangent line passes through the point (0, 2).

A typical point on the graph will have form , and the line passing through this point and (0, 2) will have slope .
We are interested in when this line is a tangent line, i.e. when its slope is dy/dx = -2x.
Thus we need to solve the equation

Thus x must be 1 or -1, so the points on the graph are (1,f(1)) = (1, 0) and (-1,f(-1)) = (-1, 0).

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

The slope at x  is dy/dx = .
We are interested in the point where  , and the slope is then
A point on the tangent line is .

Using the point-slope formula,

(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 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 .

Thus

Rate of change is meters per radian

(19)  Use implicit differentiation to find :     .

(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 .)

Let V be the volume of the water in the tank.

Know

Want

By similar triangles, you get so .

By the volume formula,

feet per minute