18.01 Section, October 13, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Taylor polynomials and piecewise functions
1. A car starts from rest and accelerates at a rate of 10 m/s2 for 5 seconds before suddenly
hitting the brakes. Throughout, (kinetic) friction causes a decceleration of 1 m/s2 (but this
is included in the 10 m/s2 acceleration). How long does it take before the car stops?
2. (a) Write an equation for a piecewise function with continuous derivative consisting of a
line glued to a quadratic equation. (You’re not allowed to use a horizontal line, and you’re
not allowed to use a linear function as the “quadratic”.)
(b) Is it possible to do this so that the second derivative is continous? If so, write an
equation for one; if not, explain why.
(c) What if you replace the quadratic with a cubic?
1
3. Determine all the cubic equations f (x) that go through (0, 0) and (1, 2), and satisfy f 00 (1) =
0.
4. Draw a function f such that all Taylor polynomials (at x = 0) are a “really bad approximation” for f outside the interval [−1, 1]. (As the question is deliberately vague, I’m asking
for a sketch, not an equation.)
5. Bonus question: (a) What is the fourth-order Taylor polynomial for f (x) = x5 at x = 0?
(b) Your friend says the following: “It should be the polynomial of degree ≤ 4 that is the
best approximation to f , and zero is a terrible approximation! x3 , say, is much better: just
try some points, e.g. x = 1, x = −1, x = 2, . . . ”. How do you respond?
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
+ ...