Final Jeopardy 203
Dec 19, 2015
Final Jeopardy 203
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
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________.
Formulas 100 Answer
A local max or min.
Formulas 101
Show that
does not exist.
2 2
( , ) (0,0) 2 2
sinlim
2x y
x y
x y
Formulas 101 Answer
The limit along the x-axis is 1/2, but along the y-axis it is 1.
Partial Derivatives 102
Verify Clairault’s Theorem for the function:
2
( , ) x yf x y xe
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
Partial Derivatives 103
Suppose . Find
1 2 3
1 1 1 1
R R R R
1
R
R
Partial Derivatives 103 Answer
Ans: 2
1
R
R
Differentials 100
Suppose z= .Find dz.
2 cos sinx y xy x
Differentials Answer
Ans
2(2 cos sin cos ) ( sin sin )x y y x xy x dx x y x x dy
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
Gradients 200 Answer
z=15 +8(x-3) - 9(y+2)
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?
Differentials 300 Answer
The error estimate should be 600
Differentials 400
Find dz/du and dz/dv
23 2 2 2cos( )
3
u vz x y x y x u v y
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
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
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
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
Partial Derivatives 200 Answer
9
2 14
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
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
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
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)
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
MaxMin 200 Answer
The box has corners at
and has volume 48
( 2, 1, 3)
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?
MaxMin 300 Answer
The base of the aquarium should be square with side length 2V/5. Then the height must be 25/4V
2
MaxMin 100
GiFind the maximum and minimum values of in the domain( , ) xyf x y e 2 24 1x y
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)
Double Integrals 200
Find the volume under the surface z=3xy
2
over the rectangular region with corners at (2,3) and (4,5).
Double Integrals 200 Answer
588
Double Integrals 300
Find the volume under above the regions determined by the curves
and
2 23z x y
y x2x y y
Double Integrals 300 Answer
2
2 2 2
0
1443
35
y
y yx y dxdy
Double Integrals 400
Evaluate:
1 / 2 2
0 arcsincos 1 cosy
x xdxdy
Double Integrals 400 Answer
1