I constructed 2-hour lessons, including worksheets, diagrams, collaborative learning for 15+ students enrolled in my workshop.

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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