MIT 18.01 Single Variable Calculus: Problem Set 5, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

18.01 Problem Set 5

Due Friday 10/27/06, 1:55 pm

Part I (20 points)

Lecture 19. Fri. Oct. 20 First fundamental theorem. Properties of integrals.

Read: 6.6, 6.7 (The second fundamental theorem, discussed in Lecture 20, is stated in

the text as (13) at the bottom of page 215 and also as Step 1, page 207, of the proof of the first

fundamental theorem.)

Work: 3C-1, 2a, 3a, 5a; 3E-6bc; 4J-

Lecture 20. Tues. Oct. 24 Second Fundamental Theorem. Def’n of ln x.

Read: Notes PI, p.2 [eqn.(7) and example]; Notes FT.

Work: 3E-1, 3a; 3D-1, 4bc, 5, 8a; 3E-2ac

Lecture 21. Thur. Oct. 26 Areas between curves. Volumes by slicing.

Read: 7.1, 7.2, 7.3 Work 1 : 4A-1b, 2, 4; 4B-1de, 6, 7

Lecture 22. Fri. Oct. 27 Volumes by disks and shells.

Read: 7.4 Work on PS 6

Part II (30 points + 5 extra credit)

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; 3 pts) 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).

1. (Lec 19, 6 pts: 3 + 3) (+ 5 extra for part (c))

a) Suppose that at the beginning of day 0, some time last summer, the temperature in Boston

was y(0) = 65 ◦ Fahrenheit and that over a 50-day period, the temperature increased according to

the rule y � (t) = y(t)/100, with time t measured in days. Find the formula for y, and draw a graph

of temperature on days 3 and 4, 3 ≤ t ≤ 5, and label with the correct day and shade in the regions

whose areas represent the average temperature each of the two days. 2

1 A more colorful way of expressing 4B-6 is in terms of the volume of a tent, as in the textbook problem 7.3/7.

Unfortunately, the problem is ill-posed and can’t be done without pretending that the cross-sections are triangles.

In real life, the canvas would have creases. In general, the shapes formed by stretching canvas or nylon over various

arrays of tent poles are quite hard to compute. 2 The continuous average of a function is � (^) b 1 f (x)dx b − a (^) a

In this case b − a = 1, so the average is the same as integral. For more, see Notes, AV, and Lecture 23.