18.01 PROBLEM SET 7

Hand in to Room 2-108 by 1:00 pm Friday, May 3.

The due date and time are absolute. Late homeworks are accepted until the solutions are posted, but the score will be multiplied by 1/2. Everything included here may be used later. You do not need to

memorize the proofs, but you need to know the facts.

1. 181/15. 184/1. 187/11. 191/3, 10. 195/1, 10. AV.2/1, 2, 5.

(10 points)

2. Find the volume of a solid whose profile is x^{2} + y^{2}/2 = 1 and whose horizontal

cross-sections are squares. (5 points)

3. Compute the tangent line of f(x) = −2x^{−1/2} at x = 1. Conclude that

x − 3 ≥ −2x^{−1/2} for all x > 0

with equality only for x = 1. (5 points)

4. A hole of radius r is bored through the center of a sphere of radius a ≥ 1.

(a) Find the volume V and the surface area A of the remaining solid.

(b) Show that

48π^{2} · V = (\frac{A}{a})^{3} .

(c) Find the minimum of V − A. (10 points)

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