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

 

-6

 

 

 

 

 

-4

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 


2. [5 pts]  Refer to the above graph to answer the following questions.

 

a)     List all of the x-values where the following occur:

 

a.      Corner points

 

 

 

b.     Vertical tangents

 

 

 

c.      Points of Discontinuity

 

 

 

 

b)     What do all of these points have in common?

 

 

 

 

 

 

 

 

 

 

3.  [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.

 

 

 

 

 

 

 

 

 

 

 

 

 

4.  [10 pts] Choose 2 of the following 4 functions and find their derivatives.  Circle the 2 you want graded.

 

(a)  

 

 

 

 

 

 

(b) 

 

 

 

 

 

 

(c)  

 

 

 

 

 

 

(d) 
5.  [5 pts]   Find the line tangent to at (4,10).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.  [10 pts] An object dropped from a 100 foot building has a velocity which is represented by  where t is in seconds and v is in .

a)     How far has the object fallen after 2 seconds? 

 

 

 

 

 

 

 

 

 

b)     What is the acceleration of the object at t = 2 seconds?

 

 

 

 

 

 

 

 

c)     What conclusions about the acceleration can you draw from your answer to part b)?

 

 

 

 

7.  [10 pts]  A company has a revenue function given by  and a cost function given by  where x is the number of units and R(x) and C(x) are measured in dollars.

 

a)     Find the Profit Function.

 

 

 

 

 

 

b)     Find the Marginal Profit

 

 

 

 

 

 

 

c)     For what value of x is the Profit maximized?

 

 

 

 

 

 

 

 

 

 

 

d)     What is the maximum profit?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.  [5 pts]  Find the area between  and .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9.  [5 pts]  Determine the following antiderivative using integration by parts.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.  [10 pts]  Given find the following partial derivatives:

 

 

a) 

 

 

 

 

b) 

 

 

 

 

 

c) 

 

 

 

 

 

d) 

 

 

 

 

 

e) 

 

 

 

 

 

 

f) 

 

 

 

 

 

 

 

 

11) [10 pts]  Consider the function

a)  Find all of the critical points and inflection points.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)  Find the largest intervals for which the function is increasing, decreasing, concave up and concave down. 

 

 

 

 

 

 

 

 

 

 

 

c)  List the value and location of each relative max and min, and list the location of each inflection point. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12.  [10 pts]  Evaluate the following integral  by:

 

a)     Using the area of geometric figures (Hint: Draw a sketch).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)     Using the Fundamental Theorem of Calculus.