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