**Unformatted text preview: **Math 21C
Kouba Discussion Sheet 3 1.) Compute zw and 2,, for each of the following functions.
a.) z :x3y+y4—2$+5 b.) z=f(:c)+g(y) c.) z: f(m3)+g(4y)
d.) z : f(:1:2 + 313) + g(:vy2) e.) y2 + .22 + sin(xz) = 4
f.) z : f(u,v) where u = ln(a: -— y) and ”u : ewy 2.) Find %— and (2—1: ifw = f(4[t2 — 3s) and f’(:c) : lnzzt. 3.) Assume that f is differentiable function of one variable with z = a: f (my) Show that
55.2,, — yzy : z . 4.) Assume that f and g are twice differentiable functions of one variable. Show that 8% 8%
u : f(ac + at) + g(a: — at) satisﬁes a25—5 2 (at—2 , where a is a constant.
:3 5.) Find and classify critical points as determining relative maximums, relative minimums,
or saddle points. a.) 223332 —6;cy+y2+12:c— 16y+1 b.) 2 = $23] —— £132 — 2y2 c.) z : 1:2 — 8ln(:cy) + y2 d.) z : 3m2y — 6m2 + y3 — 6312 6.) Determine the absolute extrema for each function on the indicated region.
a.) f(a:,y) : 2m + 4y +12 on
i.) the triangle with vertices (0,0), (0,3), and (3,0).
ii.) the circle 2:2 + y2 = 4.
b.) f($, y) = my — a: — 3y on the triangle with vertices (0,0), (0, 4), and (5, 0).
C.) f(a:, y) = 11:2 — 3y2 — 2x + 63/ on the square with vertices (0, 0), (0, 2),( , 0) and
(2, 2). 7.) Find the point on the plane a: + 2y + 32 = 6 nearest the origin. 8.) Determine the minimum surface area of a closed rectangular box with volume 8 ft.3
THE FOLLOWING PROBLEM IS FOR RECREATIONAL PURPOSES ONLY. 9.) Divide the ﬁgure into four equal parts, each one of the same size and shape. 1 ...

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