This

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Professor: E. Moore

Typed by: Daniel Addams (2010-2011 President, AMACSS)

Projections

Ex.: A 50-m pole at the equator loans /18 rad from the vertical. On the first day of fall, at noon,

the pole casts a shadow on the ground. How long is this shadow?

Sol.:

àL

6F

5< L8

=.

Ü-.Lx

Ü5L!!:s;:vè{;Lwr :vè{;

? the shadow is 50 cos(4/9) m (~8.68m) long.

Arc Length

Ex.: A flat metal plate has shape determined by the area under the graph of B:T;L5

5>ëâTÐ

:ráw;,IWKHSODWH¶VGHQVLW\x units from the y-axis) is given by x2 (g/cm2), what is its total mass?

Sol.:

Height: 5

5>ë Linear density: ë.

5>ë

õILë.|ë

5>ëIrmr_j LìI

9

4Lìë.

5>ë

9

4TL®L59

6E x

Critical Points

Bounded Surface

n critical pts. Global Extremum

Unbounded Surface

1 critical pt. Global Extremum

>1 critical pt. Local Extrema

H1f

H2f

H3f

Result

:T4áU4áV4;

+

+

+

loc. min.

:T5áU5áV5;

±

+

±

loc. max.

:T6áU6áV6;

Any pattern different from the above two

saddle pt.

Hj = 0 degenerate pt.

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