18.01 Section, September 14, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Linear approximation, critical points, min/max problems
1. Find the critical points of x3 + 5x2 + 3x + 1.
2. What is the maximum area of a triangle such that the sum of the base and height is 1?
3. Use linear approximation to estimate the volume of a sphere of radius 1.01. (You shouldn’t
be cubing anything other than 1!)
4. I have an orange that weighs 0.5 pounds. I cut it open and measure the radius as 2 inches.
The next orange weighs 0.55 pounds. Approximately what is its radius? (Assume that
weight is roughly proportional to volume.)
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5. I need to make a closed box of volume 1 cubic foot. I require it to have a square base. Box
material has a fixed cost per square foot. What dimensions should I use if I’m trying to
minimize cost?
6. Find the tangent line to x3 + x2 + x + 1 at the point x = 1.
7. Use linear approximation to estimate
√
4.1.
8. Bonus question: We’ve talked about the linear approximation to a function f at a point
a: this is a linear equation `(x) such that `(a) = f (a) and `0 (a) = f 0 (a). Can you come up
with the quadratic approximation to f ?
Review
d
n
n−1
dx ax = anx
where f 0 (x) = 0
• Derivative of a polynomial:
• Critical points: points
• Linear approximation of f near a:
• Tangent line to f at a:
• Newton’s method:
(also works where n is a fraction)
f (x) ≈ f (a) + f 0 (a)(x − a)
f (a) + f 0 (a)(x − a)
xn+1 =
−f (xn )
+ xn
f 0 (xn )
2