18.01 Section, October 5, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
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Quadratic approximations, concavity, and the second derivative test
Bonus question: What is a good definition of “cubic approximation”? Explain why your
definition is a good one. (And what about quartic? etc.)
1. (Done by Eva) Find a quadratic approximation for f (x) =
√
x2 + 1 at x = 1.
2. (Done by Eva) Find local min and max for f (x) = x3 − 2x, and sketch the function. Use
the second derivative test to find an interval on which the local maximum you found is
guaranteed to be a maximum.
3. Find quadratic approximations for:
(a)
1
1+x
at x = 2
(b) ln x at x = 1
4. When is the linear approximation equal to the quadratic approximation?
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5. Suppose f (x) has quadratic approximation f (x) ≈ 1 + 2x + 3x2 at the point x = 0. What
is the quadratic approximation for (f (x))3 at x = 0?
6. f (x) = x4 − 2x3 − 3x2 + 8x − 4 has a critical point at −1.1861. What kind of critical point
is this?
7. Find a function that is concave down everywhere but increasing everywhere.
8. If f (x) and g(x) are concave up everywhere, is f (g(x)) concave up everywhere?
Review
• Quadratic approximation of f (x) at x = a:
f (x) ≈ f (a) + f 0 (a)(x − a) +
f 00 (a)
(x − a)2
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• Second derivative test:
f 00 (x) > 0 in [a, b] =⇒ convex (“concave up”) in [a, b]
f 00 (x) < 0 in [a, b] =⇒ concave (“concave down”) in [a, b]
• Critical point is a local maximum if f 00 (a) < 0, local minimum if f 00 (a) > 0.
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