____________________________________________________________________

Calculus I Test
#2 April
9, 2004

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 shown below. Using the same coordinate axis, sketch a graph of .

(3) Suppose f
and g are functions for which ,
,
,
and .

Suppose also that . Find
.

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

1. f
'(x) equals the slope
of the tangent line to y = f
(x) at the point (x, f
(x)).

2. f
'(x) equals the rate
at which the quantity f
(x) is changing with respect to x,
at x.

3. f
'(x) equals the velocity
at time x of an object whose position
at time x is f
(x).

(5)

(6)

(7)

(8)

(9)

(10) If , find
.

(11)

(12)

(13)

(14)

(15) Find all values of x
for which the tangent line to at
has a slope of 1.

We seek those x for which , which
means cos(x) = 1/2

The set of all such values of x is

and
,
where n
is an integer.

(16) Find the slope of the
tangent line to the graph of the equation at
the point (1, 1).

Now plug in the point (x,y) = (1,1)
to get

. The tangent line has slope -2.

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

Point is and
slope is . Using
the point - slope form, we get

(18) A 10-foot ladder leans
against a wall at an angle
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
when .

Using trig, , so
rate of change = .

When ,
the rate of change is feet
per radian

(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?

We know
(negative because x is decreasing)

We want ?

By Pathagorean Theorem,

Since we know
and y = 8, let's plug those in now.

Now we just need to find x. Since y
= 8 at this instant,

we can use the Pythagorean Theorem to get .

ANSWER feet
per second