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Calculus  I                                                     Final Exam                                    December 13, 2001

Name____________________                R.  Hammack                                            Score ______
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(1) Calculate the limits.

(a)

(b)

(c)

(2) Find the derivatives of the following functions. You may use any applicable rule. You do not need to simplify your answer.

(a)    f(x) =      f ' (x) =

(b)   f(x) =      f ' (x) =

(c)  y =

(d)

(e)

(a)

(b) 0

(c)  infinity

(d) State the horizontal asymptotes of f.    Line y = 0

(e) State the vertical asymptotes of f.   Line x = 3

(f) Find the inverse of f.  (You may assume that f is one-to-one.)

(g) Find the derivative of f.

(h) Find the equation of the line tangent to the graph of  y = f(x) at the point x = 2.

Point:
Slope:

By Point-Slope Formula:

(4) This problem concerns the function  .

(a)  2x + 2

(b) Find all critical points of f.
x = -1

(c) Find the absolute maximum and minimum values of f on the interval [].

Absolute Maximum is 8, which happens at x = 2.
Absolute Minimum is -1, which happens at x = -1.

(5) This problem concerns the function .

(a)

(b) Find all critical points of f.
-1, 0, 1

(c) State the intervals on which f is increasing and those on which it is decreasing.
-1      0      1
----+-----+----+-----
- - -| + + | - - -| + + + f  '(x)

Increases on and
Decreases on and

(d) List the locations of all relative maxima of f.   x = 0

(e) List the locations of all relative minima of f.   x = -1, x = 1.
(6)  Use the limit definition of the derivative to find the derivative of .

(7) Find all values of x for which the slope of the tangent line to the graph of  y = cot(x)  at  x has slope

slope =

x = , where n is an integer.

(8) Consider the equation

(a) Find .

(b) Find the slope of the tangent line to the graph of at the point  ().

Plugging this point into the above expression gives

(9) Suppose you have 600 feet of fencing material to enclose two rectangular regions, as illustrated. What dimensions x and y would enclose the greatest area?
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Call the horizontal length x and the vertical length y
Then 2x + 3y = 600,   so y = (600 - 2x) / 3

We want to maximize area = x y = x(600 - 2x) / 3 = (600x - 2) / 3 on the interval [0, 300].
Let A(x) = (600x - 2) / 3
So A ' (x) = (600 - 4x) / 3  = 0
The critical number is then x = 150.
Since A ''(x) = -4 < 0, the second derivative test says this must be a relative maximum,
and since there's only one critical number, this must be an absolute maximum.

Therefore, area is maximized when x = 150 feet and y = 100.
Thus the fenced area should be 150 by 100 feet.

(10) The height of a conical bucket is 3 feet and its radius is 1 foot. Water is poured into the bucket at a rate of 1 cubic foot per minute. How quickly is the depth h of the water increasing when h  = 2?

Let V be the volume of the water in the bucket.
We know: dV / dt = 1
We want: dh / dt

Formula for the volume of a cone of radius r and height h is
By similar triangles, , thus

feet per minute
(11)  An object, moving on a horizontal line, is a distance of     feet from a point A at time t . (t is in seconds.)

(a) At what time(s) t is the object a distance of 1 foot from the point A?
When  1 = s(t)

ANSWER: At times t = 0 and t = 3

(b) Find the function giving the object's velocity at time t.

(c) Find the function giving the object's acceleration at time t.

(d) At what time(s)  t does the object have a velocity of 9 feet per second?
When  9 = v(t)

ANSWER: At times t = 0 and t = 4

(e) At what time(s)  t does the object change direction?
This will happen when the velocity   changes sign.

1     3
-----+----+-----
+ + | - - -| + + v(t)
ANSWER: Object changes direction at times t = 0 and t = 4

(f) When is the object's acceleration 0 feet per second per second?
When  0 = a(t)

ANSWER: At time t = 2

(g) Find the times at which the object is speeding up.
This will happen when velocity and acceleration have the same sign.
Here is a chart for the sign of acceleration.

2
---+-------
- - | + + +

From this chart and the one for velocity in part e, we see that the object is speeding up on the time intervals [1,2] and

(h) When does the object attain its minimum velocity? (Remember, that velocity could be negative.)
To find the minimum of v(t), we must look at the critical numbers of v(t).
Since , the only critical number is t = 2.
The chart from part g shows that there is a relative minimum there, so that must be an absolute minimum (since there is just one critical number).
ANSWER: At time t = 2