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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 .

(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 .