18.01 Section, October 14, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
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More Taylor polynomials
1. Calculate the fifth order Taylor polynomial for ex . (Do this from scratch – don’t look at
your notes!) What about the nth order one?
2. (a) Calculate the 1st , 2nd , 3rd , 4th , and 5th order Taylor polynomials for f (x) = x3 at a = 0.
(b) Do the same at the point a = 1.
3. (a) Calculate the fourth order Taylor polynomial for f (x) = ln(x) at a = 1.
1
(b) Calculate f (k) (x) and f (k) (1) for arbitrary k, and give an expression for the N th order
Taylor polynomial for f at a = 1.
4. Suppose there is some point a such that the Taylor approximations for f at a stabilize (i.e.
TN (x) = TN +1 (x) = TN +2 (x) = . . . ). Does that mean f is a polynomial?
5. Bonus question: Normally, if you take the 27th order Taylor polynomial centered at some
point a, it’s different from the 27th order Taylor polynomial centered at some other point
a0 . Suppose I have a function such that these order-27 polynomials are the same no matter
what a they’re taken at. What can you tell me about the function? (Why?)
Hint: Do the problem first with 27 replaced by 1.
You should assume that the function has continuous derivatives of all orders.
Review
• k th order Taylor polynomial for f at a:
f (x) ≈ f (a) + f 0 (a)(x − a) +
f 000 (a)
f (k) (a)
f 00 (a)
(x − a)2 +
(x − a)3 + · · · +
(x − a)k
2!
3!
k!
• Taylor series for sin x at 0 is x −
x3
3!
+
x5
5!
−
x7
7!
+ ...
• Taylor series for cos x at 0 is 1 −
x2
2!
+
x4
4!
−
x6
6!
+ ...
• Taylor series for tan−1 (x) at 0 is x −
x3
3
+
x5
5
−
2
x7
7
+ ...