The document outlines topics and exercises for a math final review session, focusing on double integrals, gradient estimation, tangent planes, and optimization problems involving geometric shapes. It includes specific tasks such as evaluating integrals, finding areas, and determining maximum dimensions of configurations. Key problems cover regions defined by inequalities, and the use of cylindrical coordinates for volume calculation.

1. Math 20C
Final Review Session(Week11)
”I’ve worked too hard and too long to let anything stand in the way of my goals.
I will not let my teammates down and I will not let myself down” –Mia Hamm
7) Set up, but do not evaluate the double integral of the shaded region below of the function f(x,y)=xy(e^xy))
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5) Let R be the region on the xy-plane that is outside the unit circle, but within the square whose vertices are the points
(0,0) (0,1) (1,1) (1,0)
a) Sketch R and express it using inequalities
b) Evaluate the double integral of (12x^3) using the region R.
c) Set up, but do not evaluate, a double integral that finds the area of R.
6) Let f(x,y) = 4+x-(x^2)-(y^3)
a) Find the equation of the tangent plane at (1,1,3)
b) Approximate f(0.9,1.2)
4) If f(x,y)= xz^2+2y(e^x)+(y^3)z,
a) Find the direction which f is decreasing the fastest at (1,2,3)
b) Calculate the directional derivative of f at (1,1,1) in the direction (1,2,3).
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2) For the graph,
a) Estimate the gradient at (-2,-2),(-1,-1),(0,0), & (1,1) using the graph
b) The function is f(x,y)=x^3+y^3-3x-3y. Calculate the gradient at
the above points.
c) Using (a)&(b), find if there are any saddle points, local mins, or
local maxes.
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2. 10) John happens to acquire 420 feet of fencing and decides to use it to start a kennel by building 5 identical adjacent
rectangular runs (see diagram below). Find the dimensions of each run that maximizes its area.
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9) Let W be the region enclosed by z=x+y+5, z=0, (x^2+y^2=4), & (x^2+y^2=9). Find the volume of W
using cylindrical coordinates.
8) W is the region bounded by x=0, y=0, z=0, and x+y+z=2. Solve the triple integral of the
region W of the function (x)dV
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