MIT 18.01 Single Variable Calculus Problem Set 2B, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

18.01 Problem Set 2B

Due Friday 9/29/06, 1:55 pm

2B is the second half of Problem Set 2, all of which is due along with the first half 2A.

Part I (10 points)

Lecture 9. Tues. Sept. 26. Linear and quadratic approximations. Read: Notes A Work: 2A-2, 3, 7, 11, 12ade

Lecture 10. Thurs. Sept. 28. Curve-sketching. Read: 4.1, 4.2 Work: 2B-1,2: a,e,h; 2B-4, 6ab, 7ab

Lecture 11. Fri. Sept. 29. Maximum-minimum problems. Read: 4.3, 4.4 Work: assigned on PS

Part II (16 points + 3 extra)

Directions: Attempt to solve each part of each problem yourself. If you collaborate, solutions must be written up independently. It is illegal to consult materials from previous semesters. With each problem is the day it can be done.

1. (not until due date; 2 points) Write the names of all the people you consulted or with whom you collaborated and the resources you used, or say “none” or “no consultation”. (See full explanation on PS1).
2. (10 points: 2 + 4 + 4; and 3 extra) Golf balls The area of a section of a sphere of radius R between two parallel planes that are a distance h apart is 1 area of a spherical section = 2 πhR

Slice the sphere of radius R by a horizontal plane. The portion of the plane inside the sphere is a disk of radius r ≤ R. The portion of the spherical surface above the plane is called a spherical cap. For example, if the plane passes through the center, then the disk has radius r = R, its circumference is the equator, and the spherical cap is the Northern Hemisphere. More generally, a spherical cap is the portion of surface of the Earth north of a latitude line. The formula above applies to regions between two latitude lines, and, in particular, to spherical caps.

a) Consider a spherical cap which is the portion of the surface of the sphere above horizontal plane that slices the sphere at or above its center. Find the area of the cap as a function of R and r. Do this by finding first the formula for the height h of the spherical cap in terms of r and R. (This height is the vertical distance from the horizontal slicing plane to the North Pole.) Then use your formula for h and the formula above for the area of spherical sections.

This formula will be derived in Unit 4. Two examples may convince you that it is reasonable. For h = R, it gives the area of the hemisphere, 2 πR^2. For h = 2R it gives 4 πR^2 , the area of the whole sphere.

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