MATB41 Relevant Tutorial Notes

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MATB41 Tutorial Notes
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;,IWKHSODWH¶VGHQVLW\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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