_____________________________________________________________________________________

Calculus I Test
#2 October
29, 2003

Name____________________ R. Hammack Score
______

_____________________________________________________________________________________

(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