MASSACHUSETTS INSTITUTE OF TECHNOLOGYINTERPHASE CALCULUS III WORKSHOP

26 JUNE 2018

Problem 1

Find a formula for the distance from the point (x, y, z) to the plane x = �1.

Solution

|x + 1|

Problem 2

Find an equation whose solution set is the set of points whose distance to (�3, 2, 5) is equal to 4. What is the radiusof the intersection of this sphere with the yz-plane?

Solution

(x + 3)2 + (y � 2)2 + (z � 5)2 = 16

sowhydoes this make sense

well do the Y or Z coordinate matter No

The only coordinate that matters isX interns of distance to the planets 1

So I Xtl l is the distance if Xs 1the distance should be zero which isperf

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16 0 32tCy 2 I Cz g2

z plane 8 0 16 32 Cy254 12

bydefinition 7 9 2 8 5 2

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Problem 3

A rope of length 12º units is partially wrapped around a tree of radius 12 units, as shown in the figure below. Thepart of the rope not touching the tree is pulled tight. Find the coordinates of the end of the rope, labeled E.

60�

E

12

Solution

Length of rope on tree: 4ºPoint of tangency: (6, 6

p3)

E = (6 � 4p

3º, 6p

3 + 4º)how much rope is on treeuse Arc length S res 121

Point ontree 6,605 4Thc 12 451 817 left on a tangentLine Then

941

4103

Es 6 4 BE 6ft 4

Problem 4

Find linear transformations with the following geometric descriptions.(a) reflect across the x-axis(b) rotate 180 degrees and double the distance from the origin(c) halve the distance from the origin while preserving the angle between (x, y), the origin, and the positive x-axis(d) rotate 90 degrees counterclockwise(e) project a point in R3 onto the xy-plane

Solution

(a) f (x, y) = (x,�y)(b) f (x, y) = (�2x,�2y)(c) f (x, y) = ( 1

2 x, 12 y)

(d) f (x, y) = (�y, x)(e) f (x, y, z) = (x, y, 0)

a

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and DoublyC 2k 24

4 moves it along a ragfromthe origin but hater

distance Fitz

µ e all the Z components

get crushed toZeroontuxy planeCx y o