18.01 Section, October 26, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Limits, continuity, and intermediate value theorem
1. The linear approximation for
√
4.1 is 2.025. Give error bounds for this estimate.
2. Redo the following midterm problem using Taylor’s theorem: how many decimal places is
the quadratic approximation cos(0.1) ≈ 1 − 21 (0.1)2 = 0.995 good to?
3. How many terms in the Taylor expansion do I need to sum in order to calculate e to within
an error of 0.01?
Actually, 1 + 1 +
1
2!
+
1
3!
+
1
4!
= 2.7083 and e − 2.7083 ≈ 0.009945. What happened?
1
4. Suppose f has derivatives of all orders, and f 00 (x) = c everywhere. Use Taylor’s theorem
to show that f is a quadratic.
5. Bonus question: Use Taylor’s theorem to re-derive the following fact:
If f 00 (x) > 0 in [a, b], then f (x) ≥ the linear approximation in [a, b].
Review
• Mean Value Theorem:
f (b)−f (a)
b−a
= f 0 (x) for some x in the interval [a, b].
• Taylor’s Theorem (with approximation):
f (x) = f (a)+f 0 (a)(x−a)+
1 00
1
1
f (a)(x−a)2 +· · ·+ f (n) (a)(x−a)n +
f (n+1) (ξ)(x − a)n+1
2!
n!
(n + 1)!
for some ξ between x and a. (The boxed term is the error term.)
• Taylor series for ex at 0 is 1 + x +
1 2
2! x
• Taylor series for sin x at 0 is x −
x3
3!
• Taylor series for cos x at 0 is 1 −
x2
2!
+
+
x5
5!
+
x4
4!
1 3
3! x
+
1 4
4! x
−
x7
7!
+ ...
−
x6
+ ...
6!
2
+ ...