Definition of the Derivative

Recall that a function f ( x ) is differentiable at x  = a if

exists.

Project

Let m  denote the number of letters in your first name and n  denote the number of letters in your last name. If the number of letters in both of your names are the same, let . Define the function

Recall that the exponental function  is entered as exp(x) in Maple.

1. Is f (0) defined? Explain. Make a plot of f ( x ) on the interval [-2,2]. Does it look like f ( x ) has a limit as x  approaches 0? Explain.

2. Use the limit command from Maple to determine whether f ( x ) has a limit as x  approaches 0, and if it does, what the limit is.

3. Explain how to define f (0) in such a way that f ( x ) becomes continuous at x  = 0. Assume from here on that f ( x ) has been defined at x  = 0 so that f  is continuous at x  = 0.

4. From the plot of f ( x ) above, does it appear that the graph of f ( x ) is differentiable at x  = 0? Explain.

5. Use the definition of the derivative to verify that f ( x ) is indeed differentiable at x  = 0. Note that you will need to type in the explicit value of f (0) in your difference quotient since f (0) has no meaning in Maple at this point. Use the limit  command in Maple to compute the derivative at x  = 0.

6. Determine the equation of the line tangent to the graph of f ( x ) when x  = 0. Plot the graphs of both f and the tangent line when x  = 0 on the same set of axes.