Single Variable Calculus - Problem Set3 - Mathematics, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

18.01 Problem Set 3

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

Part I (10 points)

Lecture 11. Fri. Sept. 29. Maximum-minimum problems. Read: 4.3, 4.4 Work: 2C-1, 2, 5, 11, 13.

Lecture 12. Tue. Oct. 3 Related rate problems. Read: 4.5 Work: 2E-2, 3, 5, 7

Lecture 13. Thu. Oct. 5 Newton’s method. Read: 4.6, (4.7 is optional)

Lecture 14. Fri. Oct. 6 Mean-value theorem. Inequalities. Read: 2.6 to middle p. 77, Notes MVT Work: assigned on PS

Part II (31 points + 8 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).
2. (Friday, 6pts: 3 + 3) a) 4.3/28 (Use as variable the distance x from the foot of the ladder to the house. Check endpoints.)

b) 4.4/

2. (Tuesday, 2pts) Hypocycloid. Show that every tangent line to the curve x^2 /^3 + y 2 /^3 = 1 in first quadrant has the property that portion of the line in the first quadrant has length 1. (Use implicit differentiation; this is the same as problem 45 page 114 of text.)
3. (Tuesday, 7pts: 3 + 3 + 1) Sensitivity of measurement, revisited. a) Recall that in problem 2, PS1/Part II, L^2 + 20 , 0002 = h^2. Use implicit differentiation to calculate dL/dh. Compare the linear approximation dL/dh to the error ΔL/Δh computed in examples on PS1. Explain why ΔL/Δh ≤ dL/dh if the derivative is evaluated at the left endpoint of the interval of uncertainty (or, in other words, Δh > 0). In what range of values of h is it true that |Δ L | ≤ 2 |Δ h |?

b) Suppose that the Planet Quirk is a not only flat, but one-dimensional (a straight line). There are several satellites at height 20 , 000 kilometers and you get readings saying that satellite 1 is directly above the point x 1 ± 10 −^10 and is at a distance h 1 = 21 , 000 ± 10 −^2 from you, satellite

ii) When the x’s and y’s are interchanged the formulas should be the same. What transformation of the plane does the exchange of x and y represent?

iii) It is impossible to find x 3 and y 3 if the lines are parallel, so the denominator in the formula must be zero when L 1 and L 2 have the same slope.

iv) Rescaling all variables by a factor c leaves the formula unchanged, so the numerator of the formula for x 3 and y 3 should have degree (in all variables) one greater than the denominator.

b) Write the equation involving x 2 and y 2 that expresses the property that ladder L 2 has length one. We will suppose that L 1 represents the ladder at a fixed position, and L 2 tends to L 1. Thus

x 2 = x 1 + Δx; y 2 = y 1 + Δy

Use implicit differentiation (related rates) to find

Δy lim Δx → 0 Δx

(Express the limit as a function of the fixed values x 1 and y 1 .)

c) Substitute x 2 = x 1 + Δx and y 2 = y 1 + Δy into the formula in part (a) for x 3 and use part (b) to compute X = (^) xlim x 3 = lim x 3 2 →x 1 Δx → 0

Simplify as much as possible. Deduce, by symmetry alone, the formula for

Y = (^) xlim y 3 2 →x 1

d) Show that X^2 /^3 +Y 2 /^3 = 1. (The limit point (X, Y ) that you found in part (c) is expressed as a function of x 1 and y 1. This is the unique point of the ladder L 1 that is also part of the boundary curve of the region swept out by the family of ladders.)