hsiang-ping huang math 2210-90, spring 2008 5. webwork … · 2008-08-20 · hsiang-ping huang math...

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 1 due 01/17/2008 at 10:59pmMSTVectors Geometry, Dot and Cross ProductsThis assignment will cover the material from Chapters 11.1 –11.4.

1. (1 pt) set1/p1-1.pg

A child walks due east on the deck of a ship at 5 miles perhour.The ship is moving north at a speed of 16 miles per hour.

Find the speed and direction of the child relative to the sur-face of the water.

Speed = mphThe angle of the direction from the north =

(radians)Correct Answers:

• 16.7630546142402101• 0.302884868374971406

2. (1 pt) set1/p1-2.pg

Finda· b if|a| = 7,|b| = 7,and the angle betweena andb is π

9 radians.a· b =Correct Answers:

• 46.0449384185095108

3. (1 pt) set1/p1-3.pg

Gandalf the Grey started in the Forest of Mirkwood at a pointwith coordinates (0, -3) and arrived in the Iron Hills at the pointwith coordinates (1, 2). If he began walking in the direction ofthe vectorv = 3I + 1J and changes direction only once, whenhe turns at a right angle, what are the coordinates of the pointwhere he makes the turn.

( , )Correct Answers:

• 2.4• -2.2

4. (1 pt) set1/p1-4.pg

An object is at rest on the plane. Three forces,V,W,X areacting on the object. If

V = 9I +6J,W = 7I +2J,

thenX must beI+ J

Correct Answers:

• -16• -8

5. (1 pt) set1/p1-7.pg

Let T be the triangle with vertices at(9,−1) ,(3,8) ,(−6,2) .The area ofT is

Hint: Use the projection formula to find the length of an alti-tude orthogonal to any chosen base.

Correct Answers:

• 58.5

6. (1 pt) set1/p1-8.pg

Find the vectorV which makes an angle of 10 degrees withthe vectorW = 3I +10J and which is of the same length asWand is counterclockwise toW.

I+ JCorrect Answers:

• 1.2179414821310183• 10.3690220631506276

7. (1 pt) set1/p2-1.pg

If a = 3I +8J−3K andb =−5I +7J+9K ,find a· b = .Correct Answers:

• 14

8. (1 pt) set1/p2-2.pg

What is the angle in radians between the vectorsa =−9I +4J+1K andb =−4I +2J+2K

Angle: (radians)Correct Answers:

• 0.322317020097383697

9. (1 pt) set1/p2-3.pg

Find aunit vector in the same direction asa= 1I +10J+3K .I+ J+

K(Note that, a unit vector is a vector whose length is 1. Mul-

tiplying any non-zero vectorV by 1/|V| produces a unit vector,and multiplying any unit vector by -1 shows that there are twounit vector which are multiples of any non-zero vector. )

Correct Answers:

• 0.0953462589245592315• 0.953462589245592315• 0.286038776773677695

10. (1 pt) set1/p2-8.pg

What is the distance from the point (9, 2, 1) to the xz-plane?Distance =

Correct Answers:

• 2

1

11. (1 pt) set1/p2-5.pg

Find a unit vector with positive first coordinate that is orthogo-nal to the plane through the points P = (-1, -5, -1), Q = (3, -1, 3),and R = (3, -1, 8).

I+ J+ JCorrect Answers:

• 0.707106781186547524• -0.707106781186547524• 0

12. (1 pt) set1/p2-7.pg

Given vectorsu andv such thatu×v =−4I −5J−2K ,find:a)v×u = I+ J+ Kb) (u×v) · (u×v) = .c) (u×v)× (u×v) = I+ J+ KCorrect Answers:

• 4• 5• 2• 45• 0• 0• 0

13. (1 pt) set1/p2-10.pg

Consider the planes 2x+4y+1z= 1 and 2x+1z= 0.(A) Find the unique point P on the y-axis which is on both

planes. ( , , )(B) Find a unit vectoru with positive first coordinate that is

parallel to both planes.I + J + K

(C) Use the vectors found in parts (A) and (B) to find a vectorequation for the line of intersection of the two planes,r(t) =

I + J + KCorrect Answers:

• 0

• 0.25• 0• 0.447213595499957939• 0• -0.894427190999915879• t*1/sqrt( 2**2 + 1**2 )• 1/4• t*(-2)/sqrt( 2**2 + 1**2 )

14. (1 pt) set1/p4-7.pg

The vectors

U = cosθI +sinθJ,V =−sinθI +cosθJ,

for any θ, form an orthonormal basis for the plane; thatis, they are orthogonal vectors of length 1. Letθ = 2π

3 , andX = 8I +4J. Then we can write

X = uU+vV

withu =

,v =

Correct Answers:

• -0.535898384862245413• -8.92820323027550917

15. (1 pt) set1/p5-5.pg

The axis of a light in a lighthouse is tilted. When the light pointseast, it is inclined upward at 3 degree(s). When it points north,it is inclined upward at 7 degree(s). What is its maximum angleof elevation? (Hint: The maximum angle of elevation of planeof the beam above the horizontal plane is the same as the anglebetween the normal to the plane of the beam and the normal tothe horizontal plane.)

degreesCorrect Answers:

• 7.60410445369259154

Generated by the WeBWorK systemc©WeBWorK Team, Department of Mathematics, University of Rochester

2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 2 due 01/25/2008 at 10:59pmMSTLines, Curves, Velocity, Acceleration, Curvature, Tan-gents and NormalsThis assignment will cover the material from Chapters 11.5 –11.7.

1. (1 pt) set2/p1-5.pg

The distance between the two parallel linesL1 : 6x− 9y =−5,L2 : 6x−9y = 10 is

Correct Answers:

• 1.38675049056307281

2. (1 pt) set2/p1-6.pg

Find the distance of the point(9,8) from the line through(9,−8) which points in the direction of 1I −3J.

Correct Answers:

• 5.05964425626940693

3. (1 pt) set2/p2-6.pg

The equationX (t) = A + tL is the parametric equation of aline through the pointP : (2,−3,1) . The parametert representsdistance from the pointP, directed so that theI component ofL is positive. We know that the line is orthogonal to the planewith equation 5x−1y−8z=−6. ThenA= I+ J+ KL= I+ J+ K

Correct Answers:

• 2• -3• 1• 0.527046276694729889• -0.105409255338945978• -0.843274042711567822

4. (1 pt) set2/p2-9.pg

Enter T or F depending on whether the statement is true or false.(You must enter T or F – True and False will not work.)Note, all questions assume that you are inR3!

1. Two planes parallel to a line are parallel.2. Two planes either intersect or are parallel.3. Two lines either intersect or are parallel.4. Two lines perpendicular to a third line are parallel.5. Two planes perpendicular to a third plane are parallel.6. Two planes perpendicular to a line are parallel.7. Two lines perpendicular to a plane are parallel8. A plane and a line either intersect or are parallel.9. Two lines parallel to a third line are parallel.

10. Two lines parallel to a plane are parallel.11. Two planes parallel to a third plane are parallel.

Correct Answers:

• F• T• F• F• F• T• T• T• T• F• T

5. (1 pt) set2/p3-2.pg

The position of a particle in motion in the plane at timet isX(t) = exp(8.4t)I +exp(1t)J.At time t = 0, determine the following:(a) The speed of the particle is:(b) Find the unit tangent vector toX(t): I+

J(c) The tangential acceleration:(d) The normal acceleration:Correct Answers:

• 8.45931439302264022• 0.992988274194943794• 0.118212889785112356• 70.1834655169803464• 7.34811322904258407

6. (1 pt) set2/p3-4.pg

The position of a particle in motion in the plane at timet isX(t) = tI + ln(cos(t))J.At time t = (0.1π)/2, determine the following:(a) the unit tangent vector:

I+ J(b) the unit normal vector toX(t): I+

J(c) the acceleration vector:

I+ J(d) the curvature:

Correct Answers:

• 0.987688340595137752• -0.156434465040230709• -0.156434465040230709• -0.987688340595137752• 0• -1.02508563093691655• 0.987688340595137752

1

7. (1 pt) set2/p3-5.pg

Consider the vector functions

X(t) = 3I +cos(10t)J, Y (t) = sin(2t)J+8K .

LetZ (t) = X (t)×Y (t) .

ThendZdt

(t)= I+ J+ K .

Correct Answers:

• -8*10*sin(10*t)• 0.0• 3*2*cos(2*t)

8. (1 pt) set2/p3-6.pg

Given that the acceleration vector isa(t) = (−16cos(4t)) I +(−16sin(4t))J+(5t)K , the initial velocity isv(0) = I +K , andthe initial position vector isX (0) = I +J+K , compute:

A. The velocity vectorv(t) = I+ J+K

B. The position vectorX (t) = I+ J+K

Note: the coefficients in your answers must be entered in theform of expressions in the variablet; e.g. “5 cos(2t)”

Correct Answers:

• - 4 * sin( 4 * t ) + 1• 4 * cos( 4 * t ) - 4• ( 5 * t**2 ) / 2 + 1• cos( 4 * t ) + t• sin( 4 * t ) - 4 * t + 1• ( 5 * t**3 ) / 6 + t + 1

9. (1 pt) set2/p3-7.pg

If X(t) = cos(−1t)I +sin(−1t)J−10tK , compute:A. The velocity vectorv(t) = I+ J+

KB. The acceleration vectora(t) = I+ J+

KNote: the coefficients in your answers must be entered in the

form of expressions in the variablet; e.g. “5 cos(2t)”Correct Answers:

• - -1 * sin( -1 * t )• -1 * cos( -1 * t )• -10• - (-1)**2 * cos( -1 * t )• - (-1)**2 * sin( -1 * t )• 0

10. (1 pt) set2/p3-8.pg

Consider the helixX(t) = (cos(−4t),sin(−4t),2t). Compute,at t = π

6 :A. The unit tangent vectorT = I+ J+ KB. The unit normal vectorN = I+ J+ KCorrect Answers:

• -0.774596669241483377• 0.447213595499957939

• 0.447213595499957939• 0.5• 0.866025403784438647• 0

11. (1 pt) set2/p3-9.pg

Find the curvatureκ(t) of the curve X(t) = (5sint) I +(5sint) j +(1cost)K

Hint, to computeκ, you can use the formula listed on page 599of the text.

Correct Answers:

• (2**(1/2))*abs(5*1)/( 2*(5*cos(t))**2+(1*sin(t))**2 )ˆ(3/2)

12. (1 pt) set2/p3-10.pg

(A) Find the parametric equations for the line through the pointP = (4, -4, -5) that is perpendicular to the plane−5x+1y−4z= 1.Use ”t” as your variable, t = 0 should correspond to P, and thevelocity vector of the line should be the same as the normal vec-tor to the plane found directly from its equation.

x =y =z =

(B) At what point Q does this line intersect the yz-plane?Q = ( , , )Correct Answers:

• 4 + t * -5• -4 + t * 1• -5 + t * -4• 0• -3.2• -8.2

13. (1 pt) set2/p2-4.pg

Find a unit vector orthogonal to 6I−6J+10K and 0I +1J+0K :I+

J+K

Correct Answers:

• -0.857492925712544187• 0• 0.514495755427526512

14. (1 pt) set2/p3-1.pg

While a planetP rotates in a circle about its sun, a moonM rotates in a circle about the planet, and both motions are ina plane. Let’s call the distance betweenM and P one lunarunit. Suppose the distance ofP from the sun is 4.6× 103 lu-nar units; the planet makes one revolution about the sun every 5years, and the moon makes one rotation about the planet every0.333333333333333333 years. Choosing coordinates centeredat the sun, so that, at timet = 0 the planet is at(4.6×103,0),and the moon is at(4.6×103,1), then the location of the moonat timet, wheret is measured in years, is(x(t),y(t)), where

x(t)=y(t)=

2

Correct Answers:

• 4.6*1E3*cos(2*3.14159265358979324/5*t)-sin(2*3.14159265358979324/0.333333333333333333*t)• 4.6*1E3*sin(2*3.14159265358979324/5*t)+cos(2*3.14159265358979324/0.333333333333333333*t)

15. (1 pt) set2/p3-3.pg

The position of a particle in motion in the plane at timet isX(t) =−8t I +sin(3t) J.At time anyt, determine the following:(a) the speed of the particle is:

(b) the unit tangent vector toX(t) is: I+J

(Note that, a unit vector is a vector whose length is 1. Mul-tiplying any non-zero vectorV by 1/|V| produces a unit vector,and multiplying any unit vector by -1 shows that there are twounit vector which are multiples of any non-zero vector. The unittangent vector toX(t) is defined asV(t)/|V(t)|. )

Correct Answers:

• sqrt( (-8*-8)+(3*3)*(cos(3*t)*cos(3*t)) )• -8/sqrt( (-8*-8)+(3*3)*cos(3*t)*cos(3*t) )• 3*cos(3*t)/sqrt( (-8*-8)+(3*3)*cos(3*t)*cos(3*t) )


3

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 3 due 02/01/2008 at 10:59pmMSTSurfaces, Cylindrical and Spherical CoordinatesThis assignment will cover the material from Chapters 11.8 –11.9.

1. (1 pt) set3/p4-1.pg

Match the function with the description of its level sets (thesetsz=constant).

A. a collection of ellipses.B. a collection of hyperbolas.C. a collection of parallel lines.D. a collection of circles centered at the origin.E. a collection of parabolas.

1. z=√

15−x2−y2

2. xy+z2 = 13. z=

√1+x+y

4. z= 5x−3y5. z= (x+y−1)−1

6. z=(x2 +2y2

)−2

7. z= x2−2xy+y2 +x+yCorrect Answers:

• D• B• E• C• C• A• E

2. (1 pt) set3/p4-3.pg

Match the surfaces with the appropriate descriptions.

1. z= 42. z= y2−2x2

3. x2 +2y2 +3z2 = 14. z= x2

5. z= 2x+3y6. z= 2x2 +3y2

7. x2 +y2 = 5

A. nonhorizontal planeB. elliptic paraboloidC. circular cylinderD. hyperbolic paraboloidE. parabolic cylinderF. horizontal planeG. ellipsoid

Correct Answers:

• F• D• G• E• A

• B• C

3. (1 pt) set3/p4-4.pg

Match the given equation with the verbal description of thesurface:

A. Half planeB. Elliptic or Circular ParaboloidC. PlaneD. SphereE. Circular CylinderF. Cone

1. φ = π3

2. r = 43. ρcos(φ) = 44. ρ = 2cos(φ)5. r = 2cos(θ)6. ρ = 47. r2 +z2 = 168. z= r2

9. θ = π3

Correct Answers:

• F• E• C• D• E• D• D• B• A

4. (1 pt) set3/p4-5.pg

Let P be the point(−9,−5,4) in cartesian coordinates.A. The cylindrical coordinates ofP are r = , θ =

, z= .B. The spherical coordinates ofP are ρ = , θ =

, φ = .Correct Answers:

• 10.2956301409870003• 0.507098504392336958• 4• 11.0453610171872608• 0.507098504392336958• 1.20023042026060199

5. (1 pt) set3/p4-6.pg

Let P be the point with the spherical coordinatesρ = 7, φ =π/2, θ = π/6.

A. The cylindrical coordinates ofP are r = , θ =, z= .

B. The cartesian coordinates ofP are x = , y =, z= .

Correct Answers:

• 7• 0.523598775598298873

1

• -1.89735380184963276E-19• 6.06217782649107053• 3.5• -1.89735380184963276E-19

6. (1 pt) set3/p4-8.pg

Let L be the liney = 8, x = 4z. If we rotateL around thex-axis,we get a surface whose equation isAx2 +By2 +Cz2 = 1, where

A= , B = , C = .Correct Answers:

• -0.0009765625• 0.015625• 0.015625

7. (1 pt) set3/p4-9.pg

Match the curves with the appropriate descriptions.

A. Two parallel lines.B. ellipse.C. parabola.D. hyperbola.

1. x2−2xy+y2 +x+y = 02. 2x2−3xy−2y2 +x+y = 123. x2−2xy= 164. x2−2xy+2y2 = 165. x2−2xy+y2 = 16

Correct Answers:

• C• D• D• B• A


2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 4 due 02/14/2008 at 10:59pmMSTFunctions of Two or More Variables, Partial Deriva-tives, Limits, Continuity, DifferentiabilityThis assignment will cover the material from Chapters 12.1 –12.4.

1. (1 pt) set4/p5-1.pg

Find the first partial derivatives off (x,y) = sin(x−y) at thepoint (8, 8).

A. fx(8,8) =B. fy(8,8) =Correct Answers:

• 1• -1

2. (1 pt) set4/URVC 5 15.pg

The level curves of a functionf (x,y) consist of a collectionof hyperbolas and two lines. If the lines intersect at a point P,what are the possibilities for P? Type the letters of all possibili-ties, with no punctuation, in alphabetical order.

A. P is a local maximum, that is,f (P)≥ f (Q) for all Q nearP.

B. P is a local minimum, that is,f (P)≤ f (Q) for all Q nearP.

C. P is neither a local maximum nor a local minimum.

Correct Answers:

• ABC

3. (1 pt) set4/URVC 5 F.pg

On a map showing the grave of George Mallory, the contourlines are:

• A. closely spaced• B. far apart

Correct Answers:

• A


Find the limit, if it exists, or type N if it does not exist.

lim(x,y)→(5,−4)

e√

4x2+1y2=

Correct Answers:

• eˆ(sqrt(4*(5)ˆ2 + 1*(-4)ˆ2))



lim(x,y)→(0,0)

2x2

4x2 +4y2 =

Correct Answers:

• N



lim(x,y)→(0,0)

(x+25y)2

x2 +252y2=

Correct Answers:

• N


Find the limit, if it exists, or type N if it does not exist.(Hint: use polar coordinates.)

lim(x,y)→(0,0)

3x3 +4y3

x2 +y2 =

Correct Answers:

• 0



lim(x,y,z)→(5,4,4)

5zex2+y2

5x2 +4y2 +4z2 =

Correct Answers:

• 50580513322533988.1



lim(x,y,z)→(0,0,0)

5xy+3yz+5xz25x2 +9y2 +25z2 =

Correct Answers:

• N

10. (1 pt) set4/URVC 5 10.pg

Find the first partial derivatives off (x,y) = 1x−2y1x+2y at the point

(x,y) = (4, 1).∂ f∂x (4,1) =∂ f∂y (4,1) =Correct Answers:

• 0.111111111111111111• -0.444444444444444444

1

11. (1 pt) set4/URVC 5 12.pg

Find the first partial derivatives off (x,y) = sin(x−y) at thepoint (8, 8).

A. fx(8,8) =B. fy(8,8) =Correct Answers:

• 1• -1

12. (1 pt) set4/URVC 5 13.pg

If sin(−3x−1y+ z) = 0, find the first partial derivatives∂z∂x

and ∂z∂y at the point (0, 0, 0).

A. ∂z∂x(0,0,0) =

B. ∂z∂y(0,0,0) =

Correct Answers:

• 3• 1


2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 5 due 02/21/2008 at 10:59pmMSTDirectional Derivatives, the Gradient, Tangent Planes,the Chain Rule, Implicit Differentiation, Approxima-tion with DifferentialsThis assignment will cover the material from Chapters 12.5 –12.7.

1. (1 pt) set5/p5-2.pg

If f (x,y) = −4x2 + 3y2, find the value of the directional de-rivative at the point(−3,2) in the direction given by the angleθ = 2π

3 .

Correct Answers:

• -1.60769515458673624

2. (1 pt) set5/p5-3.pg

Supposef (x,y) = 3x2+3xy−2y2, P= (1,0), andu =(−5

13 , 1213

).

A. Compute the gradient of f.∇ f = I+ JNote: Your answers should be expressions of x and y; e.g. “3x -4y”

B. Evaluate the gradient at the point P.(∇ f )(1,0) = I+ JNote: Your answers should be numbers

C. Compute the directional derivative of f at P in the directionu .(Du f )(P) =Note: Your answer should be a number

Correct Answers:

• 2*3*x + 3*y• 3*x + 2*-2*y• 6• 3• 0.461538461538461538

3. (1 pt) set5/p5-4.pg

Supposef (x,y) = xy, P = (4,−2) andv = 1I +3J.

A. Find the gradient of f.∇ f = I+ JNote: Your answers should be expressions of x and y; e.g. “3x -4y”

B. Find the gradient of f at the point P.(∇ f )(P) = I+ JNote: Your answers should be numbers

C. Find the directional derivative of f at P in the direction ofv, where the directionu of a vectorv is the unit vector obtainedby normalizing that vector, i.e.,u = v

||v|| .Du f =Note: Your answer should be a number

D. Find the maximum rate of change of f at P.

Note: Your answer should be a number

E. Find the (unit) direction vector in which the maximum rateof change occurs at P.u = I+ JNote: Your answers should be numbers

Correct Answers:

• 1/y• -x / y**2• -0.5• -1• -1.10679718105893277• 1.11803398874989485• -0.447213595499957939• -0.894427190999915879

4. (1 pt) set5/p5-6.pg

Consider the equationxz2− 4yz+ 4lnz = −9 as definingz

implicitly as a function ofx andy. The values of∂z∂x

and∂z∂y

at

(−5,1,1) are and .(This problem used to have ”log” instead of ”ln”, but the answerwas the same, because in webwork ”log” means the natural log-arithm.

Correct Answers:

• 0.1• -0.4

5. (1 pt) set5/p6-3.pg

Let w = 4xy−10x+ 9y, x = r + s+ t, y = r + s, andz= s+ t.Find the partial derivatives ofw with respect tor, s andt at thepoint r = 2, s= 1, t =−2.

∂w∂r

= .

∂w∂s

= .

∂w∂t

= .

Correct Answers:

• 15• 15• 2

6. (1 pt) set5/ur vc 6 5.pg

The dimensions of a closed rectangular box are measured as 100centimeters, 60 centimeters, and 80 centimeters, respectively,with the error in each measurement at most .2 centimeters. Usedifferentials to estimate the maximum error in calculating thesurface area of the box.

square centimetersCorrect Answers:

• 192

1

7. (1 pt) set5/p5-7.pg

Find the equation of the tangent plane to the surfacez= 1y2−4x2 at the point(4,0,−64).

z =Note: Your answer should be an expression of x and y; e.g.

“3x - 4y + 6”Correct Answers:

• ( -2 * 4 * 4 * x ) + (2 * 1 * 0 * y ) + ( 4* 16 - 1 * 0 )

8. (1 pt) set5/p5-8.pg

The intensity of light at a distancer from a source is givenby L = Ir−2, whereI is the illumination at the source. Startingwith the valuesI = 70,r = 10, suppose we increase the distanceby 4 and the illumination by 3. By (approximately) how muchdoes the intensity of light change?

dL =Correct Answers:

• -0.53

9. (1 pt) set5/p6-2.pg

Consider the surface 1x2 + 1y2 + 25z2 = 27 and the pointP =(1,1,1) on this surface.

a) The outward unit normal at the pointP is I +J + K .

b) The equation of the tangent plane at the pointP is

z= x+ y+ .Correct Answers:

• 0.0399361531915435866• 0.0399361531915435866• 0.998403829788589666• -0.04• -0.04• 1.08

10. (1 pt) set5/URVC 5 14.pg

Find all the first and second order partial derivatives off (x,y) = 8sin(2x+y)+8cos(x−y).

A. ∂ f∂x = fx =

B. ∂ f∂y = fy =

C. ∂2 f∂x2 = fxx =

D. ∂2 f∂y2 = fyy =

E. ∂2 f∂x∂y = fyx =

F. ∂2 f∂y∂x = fxy =

Correct Answers:

• 2*8*cos(2*x+y) - 8*sin(x-y)• 8*cos(2*x+y) + 8*sin(x-y)• -4*(8 * sin(2*x + y)) - 8*cos(x-y)• - 8*sin(2*x+y) - 8*cos(x-y)• -2*(8*sin(2*x+y)) + 8*cos(x-y)• -2*(8*sin(2*x+y)) + 8*cos(x-y)


2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 6 due 02/28/2008 at 10:59pmMSTMaxima and Minima, Lagrange’s MethodThis assignment will cover the material from Chapters 12.8 –12.9.

1. (1 pt) set6/p6-4.pg

Supposef (x,y) = x2 +y2−10x−8y+4(A) How many critical points doesf have inR2?

(Note, R2 is the set of all pairs of real numbers, or the(x,y)-plane.)

(B) If there is a local minimum, what is the value of the dis-criminant D at that point? If there is none, type N.

(C) If there is a local maximum, what is the value of the dis-criminant D at that point? If there is none, type N.

(D) If there is a saddle point, what is the value of the discrim-inant D at that point? If there is none, type N.

(E) What is the maximum value off onR2? If there is none,type N.

(F) What is the minimum value off on R2? If there is none,type N.

Correct Answers:

• 1• 4• N• N• N• 4 - 5**2 - 4**2

2. (1 pt) set6/p6-5.pg

Supposef (x,y) = xy(1−5x−8y).f (x,y) has 4 critical points. List them in increasing lexographicorder. By that we mean that (x, y) comes before (z, w) ifx < zor if x= zandy< w. Also, describe the type of critical point bytyping MA if it is a local maximum, MI if it is a local minimim,and S if it is a saddle point.

First point ( , ) of typeSecond point ( , ) of typeThird point ( , ) of typeFourth point ( , ) of type

Correct Answers:

• 0• 0• S• 0• 0.125• S• 0.0666666666666666667

• 0.0416666666666666667• MA• 0.2• 0• S

3. (1 pt) set6/p6-6.pg

You are to manufacture a rectangular box with 3 dimensions x,y and z, and volumev= 8. Find the dimensions which minimizethe surface area of this box.

x =y =z =

Correct Answers:

• 2• 2• 2

4. (1 pt) set6/p6-7.pg

Find the coordinates of the point (x, y, z) on the plane z = 2 x +1 y + 1 which is closest to the origin.x =y =z =

Correct Answers:

• -0.333333333333333333• -0.166666666666666667• 0.166666666666666667

5. (1 pt) set6/p6-8.pg

Find the maximum and minimum values off (x,y) = 7x+ y onthe ellipsex2 +25y2 = 1maximum value:minimum value:

Correct Answers:

• 7.00285656000463856• -7.00285656000463856


Supposef (x,y) = xy−ax−by.(A) How many local minimum points doesf have inR2?

(The answer is an integer).

(B) How many local maximum points doesf have inR2?

(C) How many saddle points doesf have inR2?

Correct Answers:

• 0• 0• 1

1


Consider the functionf (x,y) = xsin(y).In the following questions, enter an integer value or type INFfor infinity.

(A) How many local minima doesf have inR2?

(B) How many local maxima doesf have inR2?

(C) How many saddle points doesf have inR2?

Correct Answers:

• 0• 0• INF


Each of the following functions has at most one critical point.Graph a few level curves and a few gradiants and, on this ba-sis alone, decide whether the critical point is a local maximum(MA), a local minimum (MI), or a saddle point (S). Enter theappropriate abbreviation for each question, or N if there is nocritical point.

(A) f (x,y) = e−2x2−4y2

Type of critical point:(B) f (x,y) = e2x2−4y2

Type of critical point:(C) f (x,y) = 2x2 +4y2 +4

Type of critical point:(D) f (x,y) = 2x+4y+4

Type of critical point:Correct Answers:

• MA• S• MI• N


Find the maximum and minimum values off (x,y) = 8x2 + 9y2

on the disk D:x2 +y2 ≤ 1.maximum value:minimum value:

Correct Answers:

• 9• 0

10. (1 pt) set6/ur vc 7 10.pg

For each of the following functions, find the maximum andmimimum values of the function on the circular disk:x2 +y2 ≤1. Do this by looking at the level curves and gradients.

(A) f (x,y) = x+y+4:maximum value =minimum value =

(B) f (x,y) = 4x2 +5y2:maximum value =minimum value =

(C) f (x,y) = 4x2−5y2:maximum value =minimum value =

Correct Answers:

• 5.41421356237309505• 2.58578643762690495• 5• 0• 4• -5

11. (1 pt) set6/ur vc 7 11.pg

For each of the following functions, find the maximum and min-imum values of the function on the rectangular region:−4≤ x≤4,−5≤ y≤ 5.Do this by looking at level curves and gradients.

(A) f (x,y) = x+y+2:maximum value =minimum value =

(B) f (x,y) = 2x2 +3y2:maximum value =minimum value =

(C) = f (x,y) = (5)2x2− (4)2y2:maximum value =minimum value =

Correct Answers:

• 11• -7• 107• 0• 400• -400

12. (1 pt) set6/ur vc 7 12.pg

Find the maximum and minimum values off (x,y,z) = 3x+1y+5z on the spherex2 +y2 +z2 = 1.maximum value =minimum value =

Correct Answers:

• 5.91607978309961604• -5.91607978309961604

13. (1 pt) set6/ur vc 7 13.pg

Find the maximum and minimum values off (x,y) = xy on theellipse 2x2 +y2 = 3.maximum value =minimum value =

Correct Answers:

• 1.06066017177982129• -1.06066017177982129

2

14. (1 pt) set6/ur vc 7 14.pg

You are hiking the Inca Trail on the way to Machu Piecho.When you arrive at the hightest point on the trail, which of thefollowing are possibilities? In alphabetical order without punc-tuation or spacing, list the letters which indicate possibilities.

(A) The path passes through the center of a set of concentriccontour lines.

(B) The path is tangent to a contour line.(C) The path follows a contour line.(D) The path crosses a contour line.possibilities:

Correct Answers:

• ABCD


3

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 7 due 03/13/2008 at 10:59pmMDTDouble Integrals and ApplicationsThis assignment will cover the material from Chapters 13.1 –13.5.

1. (1 pt) set7/p7-1.pg

Evaluate the iterated integralR 2

0

R 10 12x2y3 dxdy

Correct Answers:

• 16

2. (1 pt) set7/p7-2.pg

Calculate the double integralR R

R(4x+10y+40)dAwhereR isthe region: 0≤ x≤ 5,0≤ y≤ 2.

Correct Answers:

• 600

3. (1 pt) set7/p7-3.pg

Calculate the volume under the elliptic paraboloidz= 2x2 +3y2

and over the rectangleR= [−2,2]× [−4,4].

Please note, the notation [-2,2] x [-4,4] refers to the Cartesianproduct of these two closed intervals, that is, all pairs (x,y) withx in [-2,2] and y in [-4,4] which is a rectangle with corners at(-2,-4) (2,-4) (-2,4) and (2,4).

Correct Answers:

• 597.333333333333333

4. (1 pt) set7/p7-4.pg

Find the volume of the solid bounded by the planes x = 0, y = 0,z = 0, and x + y + z = 3.

Correct Answers:

• 4.5

5. (1 pt) set7/p7-5.pg

Match the following integrals with the verbal descriptions of thesolids whose volumes they give. Put the letter of the verbal de-scription to the left of the corresponding integral.

1.R 1−1

R√1−x2

−√

1−x21−x2−y2 dydx

2.R 1√

30

R 12

√1−3y2

0

√1−4x2−3y2 dxdy

3.R 1

0

R√y

y2 4x2 +3y2 dxdy

4.R 2

0

R 2−2

√4−y2 dydx

5.R 2−2

R 4+√

4−x2

4 4x+3y dydx

A. Solid under an elliptic paraboloid and over a planar re-gion bounded by two parabolas.

B. Solid under a plane and over one half of a circular disk.C. One eighth of an ellipsoid.D. One half of a cylindrical rod.E. Solid bounded by a circular paraboloid and a plane.

Correct Answers:

• E• C• A• D• B

6. (1 pt) set7/p7-6.pg

Using polar coordinates, evaluate the integralRR

Rsin(x2 +y2)dAwhere R is the region 1≤ x2 +y2 ≤ 25.

Correct Answers:

• -1.41654571713486144

7. (1 pt) set7/p7-7.pg

A lamina occupies the part of the diskx2+y2≤ 25 in the firstquadrant and the density at each point is given by the functionρ(x,y) = 3(x2 +y2).

A. What is the total mass?B. What is the moment about the x-axis,

Rxρ(x,y)dxdy?

C. What is the moment about the y-axis,R

yρ(x,y)dxdy?

D. Where is the center of mass? ( , )E. What is the moment of inertia about the origin,

R(x2 +

y2)ρ(x,y)dxdy?Correct Answers:

• 736.31077818510779• 1875• 1875• 2.54647908947032537• 2.54647908947032537• 12271.8463030851298

8. (1 pt) set7/p7-8.pg

findR R

Rx√

ydA, whereR is the region in the first quadrantbounded above by the curvey = 25−x2.

Correct Answers:

• 416.666666666666667

9. (1 pt) set7/p8-2.pg

A sprinkler distributes water in a circular pattern, supplyingwater to a depth ofe−r feet per hour at a distance of r feet fromthe sprinkler.

A. What is the total amount of water supplied per hour insideof a circle of radius 11?

f t3perhourB. What is the total amount of water that goes throught the

sprinkler per hour?f t3perhour

Correct Answers:

• 6.2819260286074643• 6.28318530717958648

1

10. (1 pt) set7/p8-4.pg

Electric charge is distributed over the diskx2 + y2 ≤ 14 so that the charge density at (x,y) isσ(x,y) =13+x2 +y2 coulombs per square meter.Find the total charge on the disk.

Correct Answers:

• 879.645943005142107

11. (1 pt) set7/p8-5.pg

Using polar coordinates, evaluate the integral which givesthe area which lies in the first quadrant between the circlesx2 +y2 = 16 andx2−4x+y2 = 0.

Correct Answers:

• 6.28318530717958648

12. (1 pt) set7/p8-7.pg

Use single variable calculus methods to find the area of theregion in the first quadrant bounded by the curvesy2 = 3x,,y2 = 4x, x2 = 8y, x2 = 9y. In the next set, we will do the sameproblem using multivariable calculus methods.

Correct Answers:

• 0.333333333333333333


2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 8 due 04/03/2008 at 10:59pmMDTSurface Area, Triple IntegralsThis assignment will cover the material from Chapters 13.6 –13.8.

1. (1 pt) set8/p8-1.pg

A cylindrical drill with radius 4 is used to bore a holethrought the center of a sphere of radius 8. Find the volumeof the ring shaped solid that remains.

Correct Answers:

• 1392.99791173187931

2. (1 pt) set8/p8-3.pg

You are getting married and your dearest relative has bakedyou a cake which fills the volume between the two planes,z= 0andz = 9x+ 8y+ c, and inside the cylinderx2 + y2 = 1. Youare to cut it in half by making two vertical slices from the centeroutward. Suppose one of the slices is atθ = 0 and the other is atθ = ψ.

What is the limit, limc→∞

ψ?

Correct Answers:

• 3.14159265358979324

3. (1 pt) set8/p8-7.pg

Use the multivariable calculus change of variables method tofind the area of the region in the first quadrant bounded by thecurvesy2 = 2x,, y2 = 3x, x2 = 2y, x2 = 3y by mapping the regionto a rectangle by a transformation whose Jacobian is constant.You may use (but it is also good to check!) that the Jacobian ofthe inverse transformation is the inverse of the Jacobian of theforward transformation.

Correct Answers:

• 0.333333333333333333

4. (1 pt) set8/p9-1.pg

Find the surface area of the part of the plane 2x+4y+z= 4 thatlies inside the cylinderx2 +y2 = 25.

Correct Answers:

• 359.914653444810207

5. (1 pt) set8/p9-2.pg

The vector equationr (u,v) = ucosvI + usinvJ + vK , 0≤ v≤5π, 0≤ u≤ 1, describes a helicoid (spiral ramp). What is thesurface area?

(See the formulas regarding the surface area element for para-metrically defined surfaces around Example 9, on pages 760-2of the text.)

Correct Answers:

• 18.0294993105176174

6. (1 pt) set8/p9-3.pg

Find the mass of the region (in cylindrical coordinates)r3≤ z≤5, where the density function isρ(r,θ,z) = 8z.

Answer: .Correct Answers:

• 688.955448490142634

7. (1 pt) set8/p9-4.pg

Evaluate the triple integralZ Z ZE

xyzdV

whereE is the solid: 0≤ z≤ 1, 0≤ y≤ z, 0≤ x≤ y.

Correct Answers:

• 0.0208333333333333333

8. (1 pt) set8/p9-5.pg

Find the average value of the functionf (x,y,z) = x2 + y2 + z2

over the rectangular prism 0≤ x≤ 3, 0≤ y≤ 2, 0≤ z≤ 1

Correct Answers:

• 4.66666666666666667

9. (1 pt) set8/p9-6.pg

Use cylindrical coordinates to evaluate the triple integralRRRE

√x2 +y2dV, whereE is the solid bounded by the circular

paraboloidz= 9−4(x2 +y2

)and the xy-plane.

Correct Answers:

• 25.4469004940773252

10. (1 pt) set8/p9-7.pg

Use spherical coordinates to evaluate the triple integralRRRE x2 +y2 +z2dV, whereE is the ball:x2 +y2 +z2 ≤ 81.

Correct Answers:

• 148406.323681458961

11. (1 pt) set8/p9-8.pg

Match the integrals with the type of coordinates which makethem the easiest to do. Put the letter of the coordinate system tothe left of the number of the integral.

1.R 1

0

R y2

01x dx dy

2.RR

D1

x2+y2 dAwhere D is:x2 +y2 ≤ 4

3.RRR

E z2 dV where E is:−2≤ z≤ 2,1≤ x2 +y2 ≤ 24.

RRRE dV where E is:x2+y2+z2≤ 4,x≥ 0,y≥ 0,z≥ 0

5.RRR

E z dV where E is: 1≤ x≤ 2,3≤ y≤ 4,5≤ z≤ 6

A. spherical coordinatesB. polar coordinates

1

C. cylindrical coordinatesD. cartesian coordinates

Correct Answers:

• D

• B• C• A• D


2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 9 due 04/10/2008 at 10:59pmMDTVector Fields and Line IntegralsThis assignment will cover the material from Chapters 14.1 –14.2.

1. (1 pt) set9/p10-1.pg

Compute the gradient vector fields of the following func-tions:

A. f (x,y) = 6x2 +5y2

∇ f (x,y) = I+ JB. f (x,y) = x4y7,

∇ f (x,y) = I+ JC. f (x,y) = 6x+5y

∇ f (x,y) = I+ JD. f (x,y,z) = 6x+5y+4z

∇ f (x,y) = I+ J+ KE. f (x,y,z) = 6x2 +5y2 +4z2

∇ f (x,y,z) = I+ J+ KCorrect Answers:

• 2*6*x• 2*5*y• (4*xˆ(4 -1))*yˆ(7)• (7*yˆ(7 -1))*xˆ(4)• 6• 5• 6• 5• 4• 2*6*x• 2*5*y• 2*4*z

2. (1 pt) set9/p10-2.pg

Match the following vector fields with the verbal descriptionsof the level curves or level surfaces to which they are perpendic-ular by putting the letter of the verbal description to the left ofthe number of the vector field.

1. F = yI +xJ2. F = xI +yJ3. F = xI −yJ4. F = xI +yJ−zK5. F = xI +yJ+zK6. F =−yI +xJ7. F = 2I +J8. F = 2xI +yJ+zK9. F = 2I +J+K

10. F = 2xI +yJ11. F = xI +yJ−K

A. ellipsesB. circlesC. paraboloids

D. ellipsoidsE. linesF. hyperboloidsG. planesH. spheresI. hyperbolas

Correct Answers:

• I• B• I• F• H• E• E• D• G• A• C

3. (1 pt) set9/p10-3.pg

Let F be the radial force fieldF = xI + yJ. Find the workdone by this force along the following two curves, both whichgo from (0, 0) to (5, 25). (Compare your answers! Compute theline integrals directly from the definition. In the next set, youwill do the same integrals using the Fundamental Theorem forLine Integrals.)

A. If C1 is the parabola:x = t, y = t2, 0 ≤ t ≤ 5, thenRC1

F · dX =B. If C2 is the straight line segment:x = 5t2, y = 25t, 0≤

t ≤ 1, thenRC2

F · dX =Correct Answers:

• 325• 325

4. (1 pt) set9/p10-4.pg

Let C be the counter-clockwise planar circle with center atthe origin and radius r> 0. Without computing them, deter-mine for the following vector fieldsF whether the line integralsRC F · dX are positive, negative, or zero and type P, N, or Z as

appropriate.A. F = the radial vector field =xI +yJ:B. F = the circulating vector field =−yI +xJ:C. F = the circulating vector field =yI −xJ:D. F = the constant vector field =I +J:Correct Answers:

• Z• P• N• Z

1

5. (1 pt) set9/p10-5.pg

If C is the curve given by X (t) = (1+5sint) I +(1+5sin2 t

)J +

(1+1sin3 t

)K , 0≤ t ≤ π

2 and F is the ra-dial vector fieldF(x,y,z) = xI + yJ + zK , compute the workdone byF on a particle moving along C.

Correct Answers:

• 36.5

6. (1 pt) set9/p10-6.pg

Let R be the rectangle with vertices (0,0), (8,0), (0,3), (8,3), andlet C be the boundary of R traversed counterclockwise. For thevector field

F(x,y) = 4yI +xJ,

find ZC

F ·dX.

Correct Answers:

• -72

7. (1 pt) set9/p10-8.pg

Suppose C is any curve from(0,0,0) to (1,1,1) andF(x,y,z) =(3z+2y) I + (1z+2x)J + (1y+3x)K . Compute the line inte-gral

RC F ·dX.

Correct Answers:

• 6

8. (1 pt) set9/p10-9.pg

Find the work done by the force fieldF(x,y,z) = 2xI +2yJ+1K on a particle that moves along the helixX(t) = 5cos(t)I +5sin(t)J+5tK ,0≤ t ≤ 2π.

Correct Answers:

• 31.4159265358979324

9. (1 pt) set9/p10-10.pg

Calculate the divergence and curl of these vector fields:A. F (X) =

(x3−3xy2

)I +

(−3x2y+y3

)J

curl (F)= I+ J+ Kdiv (F)=B. G(X) = x3yI +x2zJ+yz3Kcurl (F)= I+ J+ Kdiv (G)=

Correct Answers:

• 0• 0• 0• 0• zˆ3-xˆ2• 0• 2*x*z-xˆ3• 3*xˆ2*y+3*zˆ2*y


2

Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 10 due 04/17/2008 at 10:59pmMDTIndependence of Path, Green’s TheoremThis assignment will cover the material from Chapters 14.3 –14.4.

1. (1 pt) set10/p10-7.pg

For each of the following vector fieldsF , decide whether it isconservative or not by computing curlF . Type in a potentialfunction f (that is,∇ f = F). If it is not conservative, type N.

A. F(x,y) = (−4x+5y) I +(5x+4y)Jf (x,y) =

B. F(x,y) =−2yI −1xJf (x,y) =

C. F(x,y,z) =−2xI −1yJ+Kf (x,y,z) =

D. F(x,y) = (−2siny) I +(10y−2xcosy)Jf (x,y) =

E. F(x,y,z) =−2x2I +5y2J+2z2Kf (x,y,z) =

Note: Your answers should be either expressions of x, y andz (e.g. “3xy + 2yz”), or the letter “N”

Correct Answers:

• -2*x**2 + 5*x*y + 2*y**2• N• -2*x**2/2 + (-2+1)*y**2/2 + z• -2*x*sin(y) + 5*y**2• (1/3)*(-2*x**3 + 5*y**3 + 2*z**3)

2. (1 pt) set10/p11-1.pg

Let C be the positively oriented circlex2 +y2 = 1. Use Green’sTheorem to evaluate the line integral

RC 19ydx+7xdy.

Correct Answers:

• -37.6991118430775189

3. (1 pt) set10/p11-2.pg

Let F =−5yI +5xJ. Use the tangential vector form of Green’sTheorem to compute the circulation integral

RC F ·dX where C

is the positively oriented circlex2 +y2 = 25.

Correct Answers:

• 785.39816339744831

4. (1 pt) set10/p11-3.pg

Let F = 4xI +4yJ and letn be the outward unit normal vectorto the positively oriented circlex2 + y2 = 4. Compute the fluxintegral

RC F ·nds.

Correct Answers:

• 100.530964914873384

5. (1 pt) set10/p11-4.pg

Let F be the radial force fieldF = xI + yJ. Find the workdone by this force along the following two curves, both whichgo from (0, 0) to (10, 100). (Use the Fundamental Theorem forLine Integrals instead of computing the line integral from thedefinition, as you did in the previous set. This way shows whythe answers to the two parts must be the same - independence ofpath!)

A. If C1 is the parabola:x = t, y = t2, 0 ≤ t ≤ 10, thenZC1

F · dX =

B. If C2 is the straight line segment:x= 10t2, y= 100t2, 0≤t ≤ 1, then

ZC2

F · dX =

Correct Answers:

• 5050• 5050

6. (1 pt) set10/p11-5.pg

Let F(x,y) = −yI+xJx2+y2 and let C be the circleX (t) = (cost) I +

(sint)J, 0≤ t ≤ 2π.A. Compute

RC F ·dX

Note: Your answer should be a numberB. Is F conservative? Type Y if yes, type N if no.

Correct Answers:

• 6.28318530717958648• N

7. (1 pt) set10/p11-7.pg

Let C be the positively oriented square with vertices(0,0),(1,0), (1,1), (0,1). Use Green’s Theorem to evaluate the lineintegral

RC 5y2xdx+4x2ydy.

Correct Answers:

• -0.5

8. (1 pt) set10/p11-8.pg

Find a parametrization of the curvex2/3 +y2/3 = 1 and use it tocompute the area of the interior.

Correct Answers:

• 1.17809724509617246

9. (1 pt) set10/p11-10.pg

Let F = 1x3I +5y3J and letn be the outward unit normal vectorto the positively oriented circlex2 + y2 = 9. Compute the fluxintegral

RC F ·nds.

Correct Answers:

• 1145.11052223347964


1

WeBWorK demonstration assignment

The main purpose of this WeBWorK set is to familiarizeyourself with WeBWorK.

Here are some hints on how to use WeBWorK effectively:

• After first logging into WeBWorK change your pass-word.

• Find out how to print a hard copy on the computer sys-tem that you are going to use. Print a hard copy of thisassignment.

• Get to work on this set right away and answer thesequestions well before the deadline. Not only will thisgive you the chance to figure out what’s wrong if an an-swer is not accepted, you also will avoid the likely rushand congestion prior to the deadline.

• The primary purpose of the WeBWorK assignments inthis class is to give you the opportunity to learn by hav-ing instant feedback on your active solution of relevantproblems. Make the best of it!

1. (1 pt) setDemo/demopr1.pg

Evaluate the expression1(2−3) = .

Correct Answers:

• -1


Evaluate the expression2/(3+1) = .Enter you answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Correct Answers:

• 0.5


Let r = 9.Evaluate 4/π∗ r = .

Next, enter the expression 4/(π∗ r) = and let WeB-WorK compute the result.

Correct Answers:

• 11.4591559026164642• 0.141471060526129187


Enter here the expression1a + 1b.

Enter here the expression1a+b.

Correct Answers:

• 1/a+1/b• 1/(a+b)


Enter here the expression

a+12+b


a+bc+d

If WeBWorK rejects your answer use the preview button tosee what it thinks you are trying to tell it.

Correct Answers:

• (a+1)/(2+b)• (a+b)/(c+d)


Enter here the expression√

a+b

Enter here the expressiona√

a+bEnter here the expression

a+b√a+b

Correct Answers:

• sqrt(a+b)• a/sqrt(a+b)• (a+b)/sqrt(a+b)


Enter here the expression√x2 +y2


x√

x2 +y2

Enter here the expressionx+y√x2 +y2

Correct Answers:

• sqrt(x**2+y**2)• x*sqrt(x**2+y**2)• (x+y)/sqrt(x**2+y**2)



−b+√

b2−4ac2a

Note: this is an expression that gives the solution of aquadraticequationby thequadratic formula.

Correct Answers:

• (-b+sqrt(b**2-4*a*c))/(2a)

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Hsiang-Ping HuangMath 2210-90, Spring 2008WeBWorK Assignment 11 due 04/24/2008 at 10:59pmMDTSurface Integrals, Divergence and Stokes’ TheoremsThis assignment will cover the material from Chapters 14.5 –14.7.

1. (1 pt) set11/p12-1.pg

Let F = 2xI +4yJ+10zK . Compute the divergence and the curl.A. div F =B. curl F = I+ J+ KCorrect Answers:

• 16• 0• 0• 0

2. (1 pt) set11/p12-2.pg

Let F = (1yz) I +(1xz)J+(7xy)K . Compute the following:A. div F =B. curl F = I+ J+ KC. div curlF =Note: Your answers should be expressions of x, y and/or z;

e.g. ”3xy” or ”z” or ”5”Correct Answers:

• 0• (7 - 1)*x• (1 - 7)*y• (1 - 1)*z• 0

3. (1 pt) set11/p12-3.pg

A fluid has density 5 and velocity fieldv =−yI +xJ+2zK .Find the rate of flow outward through the spherex2+y2+z2 = 9

Correct Answers:

• 1130.9724

4. (1 pt) set11/p12-4.pg

Use Stokes’ theorem to evaluateZ Z

ScurlF · dS where

F(x,y,z) =−19yzI +19xzJ+9(x2+y2)zK and S is the part of theparaboloidz= x2 + y2 that lies inside the cylinderx2 + y2 = 1,oriented upward.

Correct Answers:

• 119.380520836412143

5. (1 pt) set11/p12-5.pg

Use Stokes’ Theorem to evaluateZ

CF · dr whereF(x,y,z) =

xI + yJ + 3(x2 + y2)K andC is the boundary of the part of theparaboloid wherez= 4−x2−y2 which lies above the xy-planeandC is oriented counterclockwise when viewed from above.

Correct Answers:

• 0

6. (1 pt) set11/p12-6.pg

SupposeF = F(x,y,z) is a gradient field withF = ∇ f , S is alevel surface of f, and C is a curve on S. What is the value of theline integral

RC F ·dr?

Correct Answers:

• 0

7. (1 pt) set11/p12-7.pg

EvaluateZ Z

S

√1+x2 +y2 dSwhereS is the helicoid:r(u,v) =

ucos(v)I +usin(v)J+vK , with 0≤ u≤ 2,0≤ v≤ 2π

(See the formulas regarding the surface area element for para-metrically defined surfaces around Example 9, on pages 760-2of the text.)

Correct Answers:

• 29.3215314335047369

8. (1 pt) set11/p12-8.pg

Find the surface area of the part of the spherex2 + y2 + z2 = 1that lies above the conez=

√x2 +y2

Correct Answers:

• 1.84030236902122023

9. (1 pt) set11/p12-10.pg

Let S be the part of the plane 2x+ 2y+ z= 4 which lies in thefirst octant, oriented upward. Find the flux of the vector fieldF = 2I +2J+3K across the surface S.

Correct Answers:

• 22

10. (1 pt) set11/p6-1.pg

Find an equation of the tangent plane to the parametric surface

x = 3r cosθ, y = −4r sinθ, z= r at the point(

3√

2,−4√

2,2)

whenr = 2, θ = π/4.z =Note: A normal to a surface defined parametrically by

x(r,θ),y(r,θ),z(r,θ) is given by< xr ,yr ,zr >×< xθ,yθ,zθ > .Your answer should be an expression of x and y; e.g. “3x -

4y”Correct Answers:

• 2 + 0.235702260395515841 * (x- ((3)*sqrt(2))) + -0.176776695296636881 * (y- ((-4)*sqrt(2)))


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hsiang-ping huang math 2210-90, spring 2008 5. webwork … · 2008-08-20 · hsiang-ping huang math...

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