1 n x2 x3 x =1+x + + + When does a function equal its Taylor series? When does a function equal its Taylor series? For each N This is called the Taylor polynomial of order N for f (x). When does a function equal its Taylor series?

When one thing is linked to another. Self-selected survey is one in which the respondents themselves decide whether to be included. CONTROLLED EXPERIMENTS If a new drug is introduced its effectiveness needs to be tested. How does one do this?

Note that the integral is improper. X x + 1/2 s s + 1/2 3/2 dx = lim - | = lim - + 3 2 1 s s (x + 1) (x + 1) (s + 1)2 4 1 since the first term goes to zero. Is convergent. Explain your argument!

(e) (3 points) Explain how to use the comparison test to obtain your conclusion in part (d) in a different way.

1 6 = x 2 - 2 x + 2 ln x - - 2 ln x - 3 + c x 2 In the previous example there were actually two different ways of dealing with the x2 in the denominator. Which gives the following two terms in the decomposition, A B + x x2 We used the second way of thinking about it in our example. Notice...

A. Is the integrand a rational expression (i.e is the integrand a polynomial divided by a polynomial)? If so, then partial fractions may work on the integral. B. Is the integrand a polynomial times a trig function, exponential, or logarithm?

Now, let"s notice that we can use partial fractions on the series term to get, 1 1 1 1 = = - 2 i + 3i + 2 ( i + 2 )( i + 1) i + 1 i + 2 I"ll leave the details of the partial fractions to you. By now you should be fairly adept at this since we spent a fair amount of time doing...

Of examples. Such proofs are called exhaustive proofs, since these proofs proceed by exhausting all possibilities. Example1. Prove that (n+1)^3>=3^n if nN with n=<4.

2. If 23x-1 = 16, what is the value of x? A: 5 B: 1 C: 4 1 , what is f (x)?

2 C: x2 ln 5 x ln 5 2 8. If f (x) = 2ex + 2x + , what is f (x)?