1. [15 pts] In the following table, each row corresponds to x = a, where ‘a’ is some value of x for the function f(x) graphed above. Fill in the table accordingly. For any value that does not exist, write DNE. The first row is given as an example.

x = a					Continuous? Yes or No
-2	-1	1	DNE	1	No
-5	-1	-1	-1	DNE	No
3	-2	0	DNE	-3	No
0	3	3	3	3	Yes

2. [5 pts] Find the following limit:

= = = 4 + 6 = 10

3. [5 pts] Determine all values of x for which the following function is continuous.

Continuous on (0,7)

4. [5 pts] Consider the following piecewise function:

a) Find (if it exists), and

b) Determine if any discontinuities exist. If the function is not continuous for one or more x values, make one change to f(x) so that it is continuous for all real number.

(Hint: Make a sketch!)

a) = 4

b) f (2) = 1 ¹ 4 so not continuous at x = 2

is an example of a continuous function.

5. [10 pts] Answer the following questions using the function graphed above.

a) For what values of x is the function non-differentiable? List all of them.

x = -6, -3, 4, 6, 7

For each of the next four questions, there may be more than one correct answer, but you are to report only one answer for each item.

b) Give one interval of x for which the average rate of change of f is positive.

(-7,-5) and there are others

c) Give one value of x for which the instantaneous rate of change of f is negative.

x = -4 and there are others

d) Give one interval of x for which the rate of change of f is constant, but not 0.

(6, 7) or (7, 8)

e) Give one value of x for which the instantaneous rate of change of f is 0.

x = 0

6. [5 pts] Let . Find using the definition of the derivative. Be sure to sure show your steps. In a sentence explain why this answer makes sense considering the original function.

=

This makes sense since our original function always has a slope (rate of change) of -5. We can tell this since it is linear.

7. [20 pts] Find the derivatives of the following functions.

(a)

(b)

(c)

(d)

(e)

8. [5 pts] Find the line tangent to at x =0 (hint use 7e).

m = 2, point (0,2) so y - 2 = 2(x – 0) or y = 2x + 2

**Note for problems 9-11**

You are only required to do 2 of these 3 problems. Indicate by circling the numbers of the 2 problems you want graded. If none or all of the problems are circled, the 2 lowest scores will be chosen.

9. Suppose you drop a ball off the top of a 256 foot building. The height of the ball in feet, h, is given by the equation , where t is the time measured in seconds.

a) [5 pts] Find the average velocity between 1 second and the time the ball hits the ground. (Hint: First find the time when the ball hits the ground!)

Average Velocity =

b) [10 pts] How fast is it going at exactly 2 seconds after being released? Use the definition of the derivative for this calculation. (Show ALL steps!)

10. A group of biology researchers showed that the number of eggs (y) that a certain insect lays on a mungbean plant could be approximated by , where x is the age of the mungbean plant in years, for .

a) [5 pts] Find (Hint: DON’T use the definition).

b) [3 pts] Which is the best interpretation of ?

i) The number of eggs on the plant after x years

ii) The number of years it takes for the plant to accumulate y eggs

iii) The time expected for the insect to lay one more egg

iv) The additional number of eggs expected on the plant in year 7 compared to year 2.

v) The rate at which the eggs are accumulating or dropping off at a given point of time

c) [5 pts] Find at x = 4 years, and interpret this number with a sentence.

At 4 years the rate the egg production is increasing with respect to age is 12.6 eggs/year

d) [2 pts] Notice that the graph of is always above the x-axis for . What does this mean in terms of the age of the plant and the number of eggs?

Since the derivative is always positive for the interval the rate of the number of eggs produced is always increasing as the age increases.

11. If the demand equation for a certain product is given by, where x is the number of items sold and p is the price in dollars:

(a) [3 pts] Find the revenue equation.

R(x)=

(b) [6 pts] Find the marginal revenue equation from your answer to part (a).

(c) [6 pts] For what value(s) of x is the marginal revenue zero? Interpret what this means using a sentence.

This is where the revenue is going from either increasing to decreasing or vice versa.

Or

The revenue is neither increasing or decreasing at this point.