1.  Let f(x) = tanx.  Find the third degree Taylor polynomial for f about a = p/4.  (You must show some work; do not simply use the Taylor command on the TI-92; you can use the ``box'' to help you take a few derivatives however.)

2.  Consider the function f(x) = e^2x.  Estimate the error in approximating e^0.2 by using the Taylor polynomial P₁(0.1).  Your estimation should use the techniques of section 10.5 in the text.

3.  Use Taylor series methods to find:   lim_x®0[(1-e^2x)/( x)].

4.  There are zillions of trig identities.  One of them is  sin(2x) = 2sinxcosx.  Use Taylor series to give a proof of this identity.  (Well, not really a ``proof,'' but check the first 3 nonzero terms and see that they agree.)

5.  Find the radius of convergence and interval of convergence for the series  å_{n = 0}^¥[((x-4)ⁿ)/( (n+1)3^n-1)]      Radius:________           Interval:___________

6.  Show how to find a Taylor series for ln(1+x) by integrating an appropriate version of the geometric series.

7. a.  What is the sum of ³/₄+[3/ 16]+[3/ 64]+¼ ?

b.  Why won't the same trick work for the similar looking, but ``upside down,'' series  ⁴/₃+[16/ 3]+[64/ 3]+¼

8.  Suppose P₂(x) = a+bx+cx² is the second degree Taylor polynomial for the function f about a = 0.  Tell the signs of a, b, c if f has the graph given.  Briefly explain your answers.

sign of b: ________ ,

sign of c:_________ ,