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Math 1950

Fall, 2016

Final Exam

S. Kuhn

Name ______________________________________________

You may use a graphing calculator on this test but not one with symbolic capabilities, such as the TI-89.

Your name on this test indicates your pledge to abide strictly by the UTC Honor Code. Except where told

otherwise you must show all your work, including intermediate details of your calculations.

1. [4 points each] Give the value of each of the following limits; if the limit does not exist, write DNE. Briefly

justify your answers. Identify and label all indeterminate forms and label each use of L’Hopital’s Rule by writing

LH. You must do part a) without using L’Hopital’s Rule.

a)

=

b)

c)

=

=

d)

2. [2 points each] Assume that f is continuous at all values of x and that

. For each of the following

either: i) give its value, ii) say that it does not exist, or iii) say that there is not enough information to determine its

value.

a)

b)

3. [4 points each] Determine the equations of all vertical and horizontal asymptotes for the function

. If there are none, write NONE. Justify each of your answers by calculating an appropriate

limit.

a) The equation(s) of the vertical asymptote(s):

b) The equation(s) of the horizontal asymptote(s):

4. [ 6 points each] Compute each of the following. Perform only obvious simplifications.

a)

(

b)

(

c) if

)=

)=

then

=

d) if

5. [7 points] Determine

then

=

at the point (

) if

.

6. [8 points for a); 6 points for each of the others] Compute each of the following integrals. Completely simplify

your answer to part a), which should be exact.

a)

b)

c)

d)

7. [6 points] Determine the area of the region bounded by the graphs of x = 1, x = 3, y = 0, and

8. [7 points each] Consider the function

a) Use derivatives to determine the x- and y-coordinates at all relative maxima and minima (tell which is which).

Briefly explain how you draw the conclusions you draw, using either the First Derivative Test or the Second

Derivative Test; tell which test you use.

Consider the function

b) Use derivatives to determine all inflection points of f; give both x- and y-coordinates of these points. Explain

whow you know you have inflection points.

9. [3 points for a); 4 points for each of the others] An object moves along the x-axis and its position (in feet to the

right of the origin) at time t (in seconds) is given by the function

.

a) Determine the instantaneous velocity at time t = 2. Give appropriate units. In what direction is the object

moving at time t = 2 sec?

b) Determine the average velocity of the object on the time interval [1, 5]. Give appropriate units.

c) Determine a time in the time interval [1, 5] when the instantaneous velocity equals the average velocity over the

interval [1, 5]. Round your answer to two decimal places.

d) Determine all the time intervals on which velocity is decreasing.

10. [8 points] A conical tank has height 3 m and radius 2 m at the top. Water flows in at a rate of 1.7

. How

fast is the water level rising when it is 2.1 m from the bottom of the tank? (Round your answer to one decimal

place.) Include appropriate units in your answer.

11. [8 points] A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $9

per foot and on the other three sides by a metal fence costing $3 per foot. If the area of the garden is to be 800 sq.

ft., find the dimensions of the garden minimizing the cost. Verify that you are, in fact, minimizing the cost.

…

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