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Final Jeopardy 203
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Col. 1 100
Differentials 100
MaxMin 100
MaxMin 100
Curves 100
Col. 1 101
Differentials 200
PartialDeriv’s 200
MaxMin 200
Doub. Ints. 200
Col.1 102
Differentials 300
Gradients 300
MaxMin 300
Doub Ints300
Col 1. 103
Differentials 400
Double Integrals 400
Empty Empty
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If a sequence of level curves are all closed (i.e. form a closed loop) with each one inside the previous one. Then, some point inside the innermost loop is a________.
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Formulas 100 Answer
A local max or min.
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Formulas 101
Show that
does not exist.
2 2
( , ) (0,0) 2 2
sinlim
2x y
x y
x y
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Formulas 101 Answer
The limit along the x-axis is 1/2, but along the y-axis it is 1.
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Partial Derivatives 102
Verify Clairault’s Theorem for the function:
2
( , ) x yf x y xe
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Partial Derivatives 102 Answer
fxy and fyx
2 2 2 2
2 2 2 2
2 2
2
( 2 ) (1 2 ) (1 2 )
( 2 ) (1 2 )
x y x y x y x yx xy
x y x y x y x yy yx
f e xe x x e f x e
f xe f e xe x x e
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Partial Derivatives 103
Suppose . Find
1 2 3
1 1 1 1
R R R R
1
R
R
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Partial Derivatives 103 Answer
Ans: 2
1
R
R
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Differentials 100
Suppose z= .Find dz.
2 cos sinx y xy x
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Differentials Answer
Ans
2(2 cos sin cos ) ( sin sin )x y y x xy x dx x y x x dy
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Gradients 200
Find the equation of plane that is tangent to the surface z= at the point (3,-2,15)
2 2x xy y x
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Gradients 200 Answer
z=15 +8(x-3) - 9(y+2)
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Differentials 300
Three positive numbers, each less than 20 are rounded to the nearest natural number and then multiplied together. Estimate the maximum possible error that can occur from rounding?
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Differentials 300 Answer
The error estimate should be 600
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Differentials 400
Find dz/du and dz/dv
23 2 2 2cos( )
3
u vz x y x y x u v y
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Differentials 400 Answer
Ans:
2 2 33 cos( ) (2 ) 2 sin( ) (1/ 3)z z x z y
x y x y u x y x yu x u y u
2 2 33 cos( ) ( 2 ) 2 sin( ) (2 / 3)z z x z y
x y x y v x y x y vv x v y v
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Differentials 100
What is the definition of the derivative of f in the direction of unit vector u=<a,b,c> at the point (x0 ,y0 ,z0)
Answer with proper notation
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Differentials 100 Answer
Def: .
0 0 0 0 0 00 0 0 0
( , , ) ( , , )( , , ) limu h
f x ha y hb z hc f x y zD f x y z
h
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Partial Derivatives 200
Find the derivative of at the point (4,1,1) in the direction of the vector <1,2,3>
Answer
( , , )x
f x y zy z
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Partial Derivatives 200 Answer
9
2 14
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Gradients 300
At the point (1,1,1), what is the maximum rate of change of the following function and in which direction is it achieved?
2 3 4( , , )f x y z x y z
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Gradients 300 Answer
The maximum rate of change isin the <2,3,4> direction
Follow-up question: in what direction is the minimum rate of change?
29
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MaxMin 100
Find all the max’s and min’s of the following function on the given domain
2 2 2( , ) 4f x y x y x y
{( , ) :| | 1,| | 1}D x y x y
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MaxMin 100 Answer
There is a local min of 4 at (0,0). This is tied with the point (0,-1) for the absolute min . There is no local max. The absolute max of 7 is achieved at (1,1) and (1,-1)
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MaxMin 200
Find the size of the largest rectangular box with edges parallel to the coordinate axes that can fit inside
9x² +36y² +4z² = 36
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MaxMin 200 Answer
The box has corners at
and has volume 48
( 2, 1, 3)
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MaxMin 300
An aquarium is to be built from glass and slate. The volume V is given in advance, the builder is free to choose the dimensions. The builder wants to minimize cost. The bottom be made from slate, which costs 5 times as much as the glass that the sides will be made from. What dimensions should the builder choose?
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MaxMin 300 Answer
The base of the aquarium should be square with side length 2V/5. Then the height must be 25/4V
2
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MaxMin 100
GiFind the maximum and minimum values of in the domain( , ) xyf x y e 2 24 1x y
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MaxMin 100 Answer
The max’s and min’s occur where the lines y=x/2 and y=-x/2 meet the boundary of the ellipse .
( 2 / 2, 2 / 4)
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Double Integrals 200
Find the volume under the surface z=3xy
2
over the rectangular region with corners at (2,3) and (4,5).
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Double Integrals 200 Answer
588
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Double Integrals 300
Find the volume under above the regions determined by the curves
and
2 23z x y
y x2x y y
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Double Integrals 300 Answer
2
2 2 2
0
1443
35
y
y yx y dxdy
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Double Integrals 400
Evaluate:
1 / 2 2
0 arcsincos 1 cosy
x xdxdy
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Double Integrals 400 Answer
1