We are given the following information about p and its derivatives at x = 0:

Our objective is to determine the coefficients a₀, a₁, ..., a₄ from this information. In your worksheet you will find a "generic" definition of a polynomial q(x) with all the coefficients initially set to 0. As you find (or guess) each coefficent, you will plot your approximating function so far and describe the extent to which q(x) approximates p(x).

1. How is a₀ related to p(0)? Use the known value of p(0) to set the constant term of q, and plot the resulting function q(x). Describe the ways in which q does and does not approximate p. [The graph of a constant function doesn't look much like the graph of p, of course. But this q must have something in common with p. What is it?]

2. How is a₁ related to p'(0)? [Hint: Differentiate p(x) symbolically, and set x to 0.] Use the known value of p'(0) to set the first-degree coefficient of q, and plot the resulting function q(x). Describe the ways in which q does and does not approximate p. [The new q should have something more in common with p than in the preceding step.]

3. How is a₂ related to p''(0)? [Hint: Differentiate symbolically again, and set x to 0. Be careful -- the answer is a little different this time.] Set the second-degree coefficient of q, and plot the resulting function q(x). Describe the ways in which q does and does not approximate p.

4. Now you should know how to find a₃. Set the cubic coefficient of q, and plot the resulting function q(x). Describe the ways in which q does and does not approximate p.

5. Finally, find a₄. Set the quartic coefficient of q, and plot the resulting function q(x). If all the coefficients have been correctly computed, q should now be an exact match for p, whose graph we repeat here. If the match is not exact, review your calculations.