Name:

Directions: Show your work! Answers without justification will likely result in few points. Your written work also allows me the option of giving you partial credit in the event of an incorrect final answer (but good reasoning). Indicate clearly your answer to each problem (e.g., put a box around it). Good luck!

Problem 1 (15 pts). Below is a list of properties of function:

Next to the following functions, indicate all that apply from the list above to the function by placing the correct letters adjacent to the function. Give the domains of the functions as well.

1. 

   Domain: ; Properties: C, R

2. 

   Domain: ; Properties: C, O, T

3. 

   Domain: ; Properties: C, E, W

Problem 2 (10 pts). Sketch the derivative of the function given by the following graph:

 

Problem 3 (10 pts). Carefully sketch the graph of a function which satisfies the following:

f is continuous, except for a jump discontinuity at x=1, and an infinite discontinuity at x=3. f is not differentiable at x=4.

Problem 4 (10 pts). Sketch the graph of the following function and use it to determine the values of a for which exists:

Problem 5 (10 pts). The size of your business has grown according to the following table:

Graph the data, and estimate the rate of change in size at 1994 in two different ways:

• using the slope of a secant line, and
• fitting a function to the data, sketching the tangent line, and estimating the slope.

Problem 6 (10 pts). Find the equation of the tangent line to the curve at the given point:

Problem 7 (10 pts). Evaluate the limit and justify each step by indicating the appropriate limit laws:

Problem 8 (25 pts). For the function f given by do the following:

1. Discuss the continuity and differentiability of f.
2. Draw the graph of f on the interval [-2,4]. What is the shape of the graph of f?
3. Use the definition of the derivative as a limit to find .
4. Carefully draw in the tangent line to the graph of f at x=2.
5. Sketch in the derivative function of f based on the graph of f (carefully distinguish it from the graph of f).

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