Hsiang-Ping Huang Math 2210-90, Fall 2008 5. WeBWorK Assignment … · 2009-01-07 · Hsiang-Ping Huang Math 2210-90, Fall 2008 WeBWorK Assignment 1 due 09/04/2008 at 10:59pm MDT

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 1 due 09/04/2008 at 10:59pmMDTVectors 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 11 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:

• 12.0830459735946• 0.426627493126876

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

Find a · b if|a| = 5,|b| = 9,and the angle between a and b is π

4 radians.a · b =Correct Answers:

• 31.8198051533946

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

Gandalf the Grey started in the Forest of Mirkwood at a pointwith coordinates (-2, 3) and arrived in the Iron Hills at the pointwith coordinates (0, 8). If he began walking in the direction ofthe vector v = 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:

• 1.3• 4.1

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 =−5I−6J,W = 1I−5J,

then X must beI+ J

Correct Answers:

• 4• 11

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

Let T be the triangle with vertices at (−10,−4) ,(10,−1) ,(−2,10) .The area of T is

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

Correct Answers:

• 128

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

Find the vector V which makes an angle of 100 degrees withthe vector W = 1I− 8J and which is of the same length as Wand is counterclockwise to W.

I+ JCorrect Answers:

• 7.70481384588972• 2.37399317610352

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

If a =−2I−6J+9K and b = 9I+4J+0K,find a · b = .Correct Answers:

• -42

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

What is the angle in radians between the vectorsa = 1I+3J−6K and b =−10I−2J+2K

Angle: (radians)Correct Answers:

• 1.97931795539545

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

Find a unit vector in the same direction as a = −6I−10J−10K.

I+ J+K

(Note that, a unit vector is a vector whose length is 1. Mul-tiplying any non-zero vector V 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:

• -0.390566732942472• -0.650944554904119• -0.650944554904119

1

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

What is the distance from the point (2, 1, 7) to the xz-plane?Distance =

Correct Answers:• 1

11. (1 pt) set1/p2-5.pgFind a unit vector with positive first coordinate that is orthogo-nal to the plane through the points P = (3, 5, 2), Q = (4, 6, 3),and R = (4, 6, 5).

I+ J+ KCorrect Answers:• 0.707106781186547• -0.707106781186547• 0

12. (1 pt) set1/p2-7.pgGiven vectors u and v such that

u×v =−2I−6J−1K,find:a) v×u = I+ J+ K

b) (u×v) · (u×v) = .c) (u×v)× (u×v) = I+ J+ K

Correct Answers:• 2• 6• 1• 41• 0• 0• 0

13. (1 pt) set1/p2-10.pgConsider the planes 4x+3y+1z = 1 and 4x+1z = 0.

(A) Find the unique point P on the y-axis which is on bothplanes. ( , , )

(B) Find a unit vector u with positive first coordinate that isparallel 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• 0.333333333333333• 0• 0.242535625036333• 0• -0.970142500145332• t*1/sqrt( 4**2 + 1**2 )• 1/3• t*(-4)/sqrt( 4**2 + 1**2 )

14. (1 pt) set1/p4-7.pgFor any θ, the vectors

U = cosθI+ sinθJ, V =−sinθI+ cosθJ,

form an orthonormal basis for the plane; that is, they are orthog-onal vectors of length 1.

Let θ = 2π

4 , and X = 9I+7J. Then we can write

X = uU+ vV

with u = , v =Correct Answers:• 7• -9

15. (1 pt) set1/p5-5.pgThe axis of a light in a lighthouse is tilted. When the light pointseast, it is inclined upward at 8 degree(s). When it points north,it is inclined upward at 2 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:• 8.23996716862849

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

2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 2 due 09/12/2008 at 10:59pmMDTLines, 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 lines L1 : −7x−3y =−8,L2 :−7x−3y = 3 is

Correct Answers:• 1.44437076145695

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

Find the distance of the point (1,1) from the line through(−5,−5) which points in the direction of 10I−9J.


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

The equation X (t) = A + tL is the parametric equation of aline through the point P : (2,−3,1) . The parameter t representsdistance from the point P, directed so that the I component ofL is positive. We know that the line is orthogonal to the planewith equation 10x−3y+2z = 10. ThenA= I+ J+ KL= I+ J+ K

Correct Answers:• 2• -3• 1• 0.940720868383597• -0.282216260515079• 0.188144173676719

4. (1 pt) set2/p2-9.pgEnter 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 in R3!

1. Two lines parallel to a plane are parallel.2. Two planes perpendicular to a line are parallel.3. Two lines perpendicular to a plane are parallel4. Two planes parallel to a third plane are parallel.5. Two lines either intersect or are parallel.6. Two planes either intersect or are parallel.7. Two lines parallel to a third line are parallel.8. Two planes parallel to a line are parallel.9. Two lines perpendicular to a third line are parallel.

10. A plane and a line either intersect or are parallel.11. Two planes perpendicular to a third plane are parallel.

Correct Answers:

• F• T• T• T• F• T• T• F• F• T• F

5. (1 pt) set2/p3-2.pgThe position of a particle in motion in the plane at time t is

X(t) = exp(8.9t)I+ exp(1t)J.

At time t = 0, determine the following:

(a) The speed of the particle is:

(b) Find the unit tangent vector to X(t): I+J

(c) The tangential acceleration:

(d) The normal acceleration:

Correct Answers:

• 8.95600357302296• 0.993746812116997• 0.111656945181685• 78.826341932969• 7.85059981572428


X(t) = tI+ ln(cos(t))J.

At time t = (−0.7π)/2, determine the following:

(a) the unit tangent vector:I+ J

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

(c) the acceleration vector:I+ J

(d) the curvature:

Correct Answers:

• 0.453990499739548• 0.891006524188367

1

• 0.891006524188367• -0.453990499739548• 0• -4.85183999631916• 0.453990499739548

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

Consider the vector functions

X(t) = 9I+ cos(6t)J, Y(t) = sin(3t)J+7K.

LetZ(t) = X(t)×Y(t) .

Then

dZdt

(t)= I+ J+ K.Correct Answers:

• -7*6*sin(6*t)• 0.0• 9*3*cos(3*t)

8. (1 pt) set2/p3-6.pgGiven that the acceleration vector is a(t) = (−25cos(−5t))I+(−25sin(−5t))J+(−1t)K, the initial velocity is v(0) = I+K,and the initial position vector is X(0) = I+J+K, compute:

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

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

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

Correct Answers:

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

9. (1 pt) set2/p3-7.pgIf X(t) = cos(−4t)I+ sin(−4t)J+4tK, compute:

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

B. The acceleration vector a(t) = I+ J+K

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

Correct Answers:

• - -4 * sin( -4 * t )• -4 * cos( -4 * t )• 4• - (-4)**2 * cos( -4 * t )• - (-4)**2 * sin( -4 * t )• 0

10. (1 pt) set2/p3-8.pgConsider the helix X(t) = (cos(−1t),sin(−1t),3t). Compute,at t = π

6 :A. The unit tangent vector T = I+ J+ KB. The unit normal vector N = I+ J+ KCorrect Answers:

• -0.158113883008419• -0.273861278752583• 0.948683298050514• -0.866025403784439• 0.5• 0

11. (1 pt) set2/p3-9.pgFind the curvature κ(t) of the curve X(t) = (−5sin t)I +(−5sin t) j+(4cos t)K

Hint, to compute κ, you can use the formula listed on page599 of the text.

Correct Answers:

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

12. (1 pt) set2/p3-10.pg(A) Find the parametric equations for the line through the pointP = (2, -1, -1) that is perpendicular to the plane−1x−3y+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:

• 2 + t * -1• -1 + t * -3• -1 + t * 4• 0• -7• 7

13. (1 pt) set2/p2-4.pgFind a unit vector orthogonal to −7I + 5J + 1K and 0I + 1J +0K:

I+J+K

Correct Answers:

• -0.14142135623731• 0• -0.989949493661166

2

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

While a planet P rotates in a circle about its sun, a moon Mrotates in a circle about the planet, and both motions are in aplane. Let’s call the distance between M and P one lunar unit.Suppose the distance of P from the sun is 2.8×103 lunar units;the planet makes one revolution about the sun every 3 years, andthe moon makes one rotation about the planet every 0.1 years.Choosing coordinates centered at the sun, so that, at time t = 0the planet is at (2.8×103,0), and the moon is at (2.8×103,1),then the location of the moon at time t, where t is measured inyears, is (x(t),y(t)), where

x(t)=y(t)=

Correct Answers:

• 2.8*1E3*cos(2*3.14159265358979/3*t)-sin(2*3.14159265358979/0.1*t)• 2.8*1E3*sin(2*3.14159265358979/3*t)+cos(2*3.14159265358979/0.1*t)


X(t) = 9t I+ sin(−3t) J.

At time any t, determine the following:

(a) the speed of the particle is:(b) the unit tangent vector to X(t) is: I+

J

(Note that, a unit vector is a vector whose length is 1. Mul-tiplying any non-zero vector V 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 to X(t) is defined as V (t)/|V (t)|. )

Correct Answers:• sqrt( (9*9)+(-3*-3)*(cos(-3*t)*cos(-3*t)) )• 9/sqrt( (9*9)+(-3*-3)*cos(-3*t)*cos(-3*t) )• -3*cos(-3*t)/sqrt( (9*9)+(-3*-3)*cos(-3*t)*cos(-3*t) )


3

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 3 due 09/18/2008 at 10:59pmMDTSurfaces, 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 (thesets z =constant).

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

1. z =√

15− x2− y2

2. z = (x+ y−1)−1

3. xy+ z2 = 14. z =

√1+ x+ y

5. z = x2−2xy+ y2 + x+ y6. z =

(x2 +2y2

)−2

7. z = 5x−3yCorrect Answers:• D• B• A• E• E• C• B

2. (1 pt) set3/p4-3.pgMatch the surfaces with the appropriate descriptions.

1. x2 + y2 = 52. z = 43. z = 2x2 +3y2

4. z = 2x+3y5. x2 +2y2 +3z2 = 16. z = y2−2x2

7. z = x2

A. parabolic cylinderB. circular cylinderC. elliptic paraboloidD. hyperbolic paraboloidE. nonhorizontal planeF. horizontal planeG. ellipsoid

Correct Answers:• B• F• C• E• G

• D• A

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

Match the given equation with the verbal description of thesurface:

A. ConeB. PlaneC. Half planeD. Circular CylinderE. Elliptic or Circular ParaboloidF. Sphere1. θ = π

32. ρ = 2cos(φ)3. φ = π

34. z = r2

5. r = 46. ρ = 47. r = 2cos(θ)8. ρcos(φ) = 49. r2 + z2 = 16

Correct Answers:• C• F• A• E• D• F• D• B• F

4. (1 pt) set3/p4-5.pgLet P be the point (1,−9,6) in cartesian coordinates.

A. The cylindrical coordinates of P are r = , θ =, z = .

B. The spherical coordinates of P are ρ = , θ =, φ = .

Correct Answers:• 9.05538513813742• -1.460139105621• 6• 10.8627804912002• -1.460139105621• 0.985621929890572

5. (1 pt) set3/p4-6.pgLet P be the point with the spherical coordinates ρ = 8, φ =π/4, θ = π/5.

A. The cylindrical coordinates of P are r = , θ =, z = .

B. The cartesian coordinates of P are x = , y =, z = .

Correct Answers:• 5.65685424949238• 0.628318530717959

1

• 5.65685424949238• 4.57649122254147• 3.32501550221963• 5.65685424949238

6. (1 pt) set3/p4-8.pgLet L be the line y = 8, x = 5z. If we rotate L around the x-axis,we get a surface whose equation is Ax2 +By2 +Cz2 = 1, where

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

• -0.000625• 0.015625• 0.015625

7. (1 pt) set3/p4-9.pgMatch the curves with the appropriate descriptions.

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

1. x2−2xy = 162. x2−2xy+2y2 = 163. 2x2−3xy−2y2 + x+ y = 124. x2−2xy+ y2 = 165. x2−2xy+ y2 + x+ y = 0

Correct Answers:

• D• B• D• A• C


2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 4 due 10/02/2008 at 10:59pmMDTFunctions 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 of f (x,y) = sin(x− y) at thepoint (-6, -6).

A. fx(−6,−6) =B. fy(−6,−6) =Correct Answers:• 1• -1

2. (1 pt) set4/UR VC 5 15.pg

The level curves of a function f (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/UR VC 5 F.pg

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

• A. far apart• B. closely spaced

Correct Answers:• B

4. (1 pt) set4/UR VC 5 3.pg

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

lim(x,y)→(−4,3)

e√

5x2+5y2=

Correct Answers:• eˆ(sqrt(5*(-4)ˆ2 + 5*(3)ˆ2))



lim(x,y)→(0,0)

5x2

5x2 +4y2 =

Correct Answers:• N



lim(x,y)→(0,0)

(x+15y)2

x2 +152y2=

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)

6x3 +2y3

x2 + y2 =

Correct Answers:

• 0



lim(x,y,z)→(1,1,5)

5zex2+y2

1x2 +1y2 +5z2 =

Correct Answers:

• 1.4545386021517



lim(x,y,z)→(0,0,0)

4xy+3yz+5xz16x2 +9y2 +25z2 =

Correct Answers:

• N

10. (1 pt) set4/UR VC 5 10.pg

Find the first partial derivatives of f (x,y) = 4x−3y4x+3y at the point

(x,y) = (1, 2).∂ f∂x (1,2) =∂ f∂y (1,2) =Correct Answers:

• 0.48• -0.24

11. (1 pt) set4/UR VC 5 13.pg

If sin(−5x + 3y + 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:

• 5• -3

1


2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 5 due 10/09/2008 at 10:59pmMDTDirectional 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.pgIf f (x,y) = −3x2 + 3y2, find the value of the directional de-rivative at the point (−2,3) in the direction given by the angleθ = 2π

6 .

Correct Answers:

• 21.5884572681199

2. (1 pt) set5/p5-3.pgSuppose f (x,y) = −1x2 + 1xy + 3y2, P = (−2,1), and u =( 8

10 , 610

).

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 )(−2,1) = I+ JNote: Your answers should be numbers

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

Correct Answers:

• 2*-1*x + 1*y• 1*x + 2*3*y• 5• 4• 6.4

3. (1 pt) set5/p5-4.pgSuppose f (x,y) = x

y , P = (1,2) and v =−2I+4J.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 direction u of a vector v 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• -0.25• -0.447213595499958• 0.559016994374947• 0.894427190999916• -0.447213595499958

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

Consider the equation xz2 + 4yz + 2lnz = −2 as defining z

implicitly as a function of x and y. The values of∂z∂x

and∂z∂y

at

(−6,1,1) 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:

• 0.166666666666667• 0.666666666666667

5. (1 pt) set5/p6-3.pgLet w = −3xy + 7x− 7y, x = r + s + t, y = r + s, and z = s + t.Find the partial derivatives of w with respect to r, s and t at thepoint r =−1, s =−2, t = 5.

∂w∂r

= .

∂w∂s

= .

∂w∂t

= .

Correct Answers:

• 3• 3• 16

6. (1 pt) set5/ur vc 6 5.pgThe dimensions of a closed rectangular box are measured as 50centimeters, 100 centimeters, and 100 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:

• 200

1

7. (1 pt) set5/p5-7.pgFind the equation of the tangent plane to the surface z = 9y2−4x2 at the point (−1,0,−4).

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

“3x - 4y + 6”Correct Answers:• ( -2 * 4 * -1 * x ) + (2 * 9 * 0 * y ) + ( 4

* 1 - 9 * 0 )

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

The intensity of light at a distance r from a source is given byL = Ir−2, where I is the illumination at the source. Starting withthe values I = 100, r = 30, suppose we increase the distance by2 and the illumination by 3. By (approximately) how much doesthe intensity of light change?

dL =Correct Answers:• -0.0114814814814815

9. (1 pt) set5/p6-2.pgConsider the surface 16x2 + 16y2 + 1z2 = 33 and the pointP = (1,1,1) on this surface.

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

b) The equation of the tangent plane at the point P is

z = x+ y+ .Correct Answers:

• 0.706417257101357• 0.706417257101357• 0.0441510785688348• -16• -16• 33

10. (1 pt) set5/UR VC 5 14.pg

Find all the first and second order partial derivatives off (x,y) =−9sin(2x+ y)+6cos(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*-9*cos(2*x+y) - 6*sin(x-y)• -9*cos(2*x+y) + 6*sin(x-y)• -4*(-9 * sin(2*x + y)) - 6*cos(x-y)• - -9*sin(2*x+y) - 6*cos(x-y)• -2*(-9*sin(2*x+y)) + 6*cos(x-y)• -2*(-9*sin(2*x+y)) + 6*cos(x-y)


2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 6 due 10/23/2008 at 10:59pmMDTMaxima and Minima, Lagrange’s MethodThis assignment will cover the material from Chapters 12.8 –12.9.

1. (1 pt) set6/p6-4.pgSuppose f (x,y) = x2 + y2−4x−8y+1

(A) How many critical points does f have in R2?

(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 of f on R2? If there is none,type N.

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

Correct Answers:

• 1• 4• N• N• N• 1 - 2**2 - 4**2

2. (1 pt) set6/p6-5.pgSuppose f (x,y) = xy(1−1x−9y).f (x,y) has 4 critical points. List them in increasing lexographicorder. By that we mean that (x, y) comes before (z, w) if x < zor if x = z and y < 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• 0.111111111111111• S

• 0.333333333333333• 0.037037037037037• MA• 1• 0• S

3. (1 pt) set6/p6-6.pgYou are to manufacture a rectangular box with 3 dimensionsx, y and z, and volume v = 343. Find the dimensions whichminimize the surface area of this box.

x =y =z =

Correct Answers:

• 7• 7• 7

4. (1 pt) set6/p6-7.pgFind 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:

• -0.333333333333333• -0.166666666666667• 0.166666666666667

5. (1 pt) set6/p6-8.pgFind the maximum and minimum values of f (x,y) = 5x + y onthe ellipse x2 +4y2 = 1maximum value:minimum value:

Correct Answers:

• 5.02493781056044• -5.02493781056044

6. (1 pt) set6/ur vc 7 2.pgSuppose f (x,y) = xy−ax−by.

(A) How many local minimum points does f have in R2?(The answer is an integer).

(B) How many local maximum points does f have in R2?

(C) How many saddle points does f have in R2?

Correct Answers:

• 0• 0• 1

1

7. (1 pt) set6/ur vc 7 3.pgConsider the function f (x,y) = xsin(y).In the following questions, enter an integer value or type INFfor infinity.

(A) How many local minima does f have in R2?

(B) How many local maxima does f have in R2?

(C) How many saddle points does f have in R2?

Correct Answers:

• 0• 0• INF

8. (1 pt) set6/ur vc 7 5.pgEach 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−3x2−3y2

Type of critical point:(B) f (x,y) = e3x2−3y2

Type of critical point:(C) f (x,y) = 3x2 +3y2 +1

Type of critical point:(D) f (x,y) = 3x+3y+1

Type of critical point:Correct Answers:

• MA• S• MI• N

9. (1 pt) set6/ur vc 7 8.pgFind the maximum and minimum values of f (x,y) = 20x2 +21y2

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

Correct Answers:

• 21• 0

10. (1 pt) set6/ur vc 7 10.pgFor 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+2:maximum value =minimum value =

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

(C) f (x,y) = 2x2−3y2:maximum value =minimum value =

Correct Answers:

• 3.41421356237309• 0.585786437626905• 3• 0• 2• -3

11. (1 pt) set6/ur vc 7 11.pgFor each of the following functions, find the maximum and min-imum values of the function on the rectangular region: −1≤ x≤1,−2≤ y≤ 2.Do this by looking at level curves and gradients.

(A) f (x,y) = x+ y+5:maximum value =minimum value =

(B) f (x,y) = 5x2 +6y2:maximum value =minimum value =

(C) = f (x,y) = (2)2x2− (1)2y2:maximum value =minimum value =

Correct Answers:

• 8• 2• 29• 0• 4• -4

12. (1 pt) set6/ur vc 7 12.pgFind the maximum and minimum values of f (x,y,z) = 5x+5y+2z on the sphere x2 + y2 + z2 = 1.maximum value =minimum value =

Correct Answers:

• 7.34846922834953• -7.34846922834953

13. (1 pt) set6/ur vc 7 13.pgFind the maximum and minimum values of f (x,y) = xy on theellipse 2x2 + y2 = 7.maximum value =minimum value =

Correct Answers:

• 2.47487373415292• -2.47487373415292

2

14. (1 pt) set6/ur vc 7 14.pgYou 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, Fall 2008WeBWorK Assignment 7 due 11/07/2008 at 10:59pmMSTDouble Integrals and ApplicationsThis assignment will cover the material from Chapters 13.1 –13.5.

1. (1 pt) set7/p7-1.pgEvaluate the iterated integral

R 10

R 20 6x2y3 dxdy


2. (1 pt) set7/p7-2.pgCalculate the double integral

R RR(2x+4y+8)dA where R is the

region: 0≤ x≤ 2,0≤ y≤ 1.


3. (1 pt) set7/p7-3.pgCalculate the volume under the elliptic paraboloid z = 4x2 +5y2

and over the rectangle R = [−2,2]× [−1,1].

Please note, the notation [-2,2] x [-1,1] refers to the Carte-sian product of these two closed intervals, that is, all pairs (x,y)with x in [-2,2] and y in [-1,1] which is a rectangle with cornersat (-2,-1) (2,-1) (-2,1) and (2,1).


4. (1 pt) set7/p7-4.pgFind the volume of the solid bounded by the planes x = 0, y = 0,z = 0, and x + y + z = 5.


5. (1 pt) set7/p7-5.pgMatch the following integrals with the verbal descriptions ofthe solids whose volumes they give. Put the letter of the verbaldescription to the left of the corresponding integral.

1.R 1

0R√y

y2 4x2 +3y2 dxdy

2.R 2

0R 2−2

√4− y2 dydx

3.R 2−2

R 4+√

4−x2

4 4x+3y dydx

4.R 1−1

R√1−x2

−√

1−x21− x2− y2 dydx

5.R 1√

30

R 12

√1−3y2

0

√1−4x2−3y2 dxdy

A. Solid under a plane and over one half of a circular disk.B. One half of a cylindrical rod.C. One eighth of an ellipsoid.D. Solid under an elliptic paraboloid and over a planar re-

gion bounded by two parabolas.

E. Solid bounded by a circular paraboloid and a plane.Correct Answers:• D• B• A• E• C

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

Using polar coordinates, evaluate the integralRR

R sin(x2 +y2)dA where R is the region 9≤ x2 + y2 ≤ 49.

Correct Answers:• -3.80673946434912

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

A lamina occupies the part of the disk x2 + y2 ≤ 1 in the firstquadrant and the density at each point is given by the functionρ(x,y) = 1(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:• 0.392699081698724• 0.2• 0.2• 0.509295817894065• 0.509295817894065• 0.261799387799149

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

R RR x√

ydA, where R is the region in the first quadrantbounded above by the curve y = 36− x2.


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

A sprinkler distributes water in a circular pattern, supplyingwater to a depth of e−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 10?

f t3 perhourB. What is the total amount of water that goes throught the

sprinkler per hour?f t3 perhour


1

• 6.28318530717959

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

Electric charge is distributed over the diskx2 + y2 ≤ 6 so that the charge density at (x,y) is σ(x,y) =17+ x2 + y2 coulombs per square meter.Find the total charge on the disk.

Correct Answers:

• 376.991118430775

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 = 4 and x2−2x+ y2 = 0.


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 curves y2 = 2x,,y2 = 3x, x2 = 4y, x2 = 5y. In the next set, we will do the sameproblem using multivariable calculus methods.



2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 8 due 11/21/2008 at 10:59pmMSTSurface 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 1 is used to bore a holethrought the center of a sphere of radius 5. Find the volumeof the ring shaped solid that remains.

Correct Answers:

• 492.499134782156

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 = 0and z = 2x + 4y + c, and inside the cylinder x2 + y2 = 1. Youare to cut it in half by making two vertical slices from the centeroutward. Suppose one of the slices is at θ = 0 and the other is atθ = ψ.

What is the limit, limc→∞

ψ?

Correct Answers:

• 3.14159265358979

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 thecurves y2 = 7x,, y2 = 8x, 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:

• 0.333333333333333

4. (1 pt) set8/p9-1.pgFind the surface area of the part of the plane 4x+4y+z = 3 thatlies inside the cylinder x2 + y2 = 9.

Correct Answers:

• 162.423682276052

5. (1 pt) set8/p9-2.pgThe vector equation r(u,v) = ucosvI + usinvJ + vK, 0 ≤ v ≤9π, 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.)


6. (1 pt) set8/p9-3.pgFind the mass of the region (in cylindrical coordinates) r3 ≤ z≤2, where the density function is ρ(r,θ,z) = 1z.

Answer: .Correct Answers:• 7.48045122572275

7. (1 pt) set8/p9-4.pgEvaluate the triple integralZ Z Z

ExyzdV

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

Correct Answers:• 0.0208333333333333

8. (1 pt) set8/p9-5.pgFind the average value of the function f (x,y,z) = x2 + y2 + z2

over the rectangular prism 0≤ x≤ 2, 0≤ y≤ 1, 0≤ z≤ 5


9. (1 pt) set8/p9-6.pgUse cylindrical coordinates to evaluate the triple integralRRR

E√

x2 + y2 dV , where E is the solid bounded by the circularparaboloid z = 4−4

(x2 + y2

)and the xy-plane.


10. (1 pt) set8/p9-7.pgUse spherical coordinates to evaluate the triple integralRRR

E x2 + y2 + z2 dV , where E is the ball: x2 + y2 + z2 ≤ 100.


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.RRR

E dV where E is: x2 +y2 +z2 ≤ 4,x≥ 0,y≥ 0,z≥ 02.

RRD

1x2+y2 dA where D is: x2 + y2 ≤ 4

3.R 1

0R y2

01x dx dy

4.RRR

E z2 dV where E is: −2≤ z≤ 2,1≤ x2 + y2 ≤ 25.

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

A. polar coordinates1

B. cylindrical coordinatesC. cartesian coordinatesD. spherical coordinates

Correct Answers:

• D• A• C• B• C


2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 9 due 11/29/2008 at 10:59pmMSTVector 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) = 10x2 +6y2

∇ f (x,y) = I+ JB. f (x,y) = x3y2,

∇ f (x,y) = I+ JC. f (x,y) = 10x+6y

∇ f (x,y) = I+ JD. f (x,y,z) = 10x+6y+3z

∇ f (x,y) = I+ J+ KE. f (x,y,z) = 10x2 +6y2 +3z2

∇ f (x,y,z) = I+ J+ KCorrect Answers:

• 2*10*x• 2*6*y• (3*xˆ(3 -1))*yˆ(2)• (2*yˆ(2 -1))*xˆ(3)• 10• 6• 10• 6• 3• 2*10*x• 2*6*y• 2*3*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 = xI+ yJ−K2. F = xI+ yJ3. F = 2I+J+K4. F = 2xI+ yJ+ zK5. F = xI+ yJ− zK6. F = xI− yJ7. F = 2xI+ yJ8. F = xI+ yJ+ zK9. F = yI+ xJ

10. F = 2I+J11. F =−yI+ xJA. ellipsoidsB. paraboloidsC. circles

D. ellipsesE. hyperboloidsF. hyperbolasG. linesH. spheresI. planes

Correct Answers:• B• C• I• A• E• F• D• H• F• G• G

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

Let F be the radial force field F = xI + yJ. Find the workdone by this force along the following two curves, both whichgo from (0, 0) to (10, 100). (Compare your answers! Computethe line integrals directly from the definition. In the next set,you will do the same integrals using the Fundamental Theoremfor Line Integrals.)

A. If C1 is the parabola: x = t, y = t2, 0 ≤ t ≤ 10, thenRC1

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

t ≤ 1, thenR

C2F · dX =

Correct Answers:• 5050• 5050

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 fields F whether the line integralsR

C F · dX are positive, negative, or zero and type P, N, or Z asappropriate.

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

5. (1 pt) set9/p10-5.pgIf C is the curve given by X(t) = (1+4sin t)I +(1+1sin2 t

)J +

(1+4sin3 t

)K, 0 ≤ t ≤ π

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

1

Correct Answers:

• 25.5

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

Let R be the rectangle with vertices (0,0), (2,0), (0,2), (2,2),and let C be the boundary of R traversed counterclockwise. Forthe vector field

F(x,y) = 8yI + xJ,

find ZC

F ·dX .

Correct Answers:

• -28

7. (1 pt) set9/p10-8.pgSuppose C is any curve from (0,0,0) to (1,1,1) and F(x,y,z) =(4z+4y)I + (2z+4x)J + (2y+4x)K. Compute the line inte-gral

RC F ·dX.

Correct Answers:

• 10

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

Find the work done by the force field F(x,y,z) = 6xI +6yJ + 3K on a particle that moves along the helix X(t) =3cos(t)I+3sin(t)J+5tK,0≤ t ≤ 2π.


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, Fall 2008WeBWorK Assignment 10 due 12/05/2008 at 10:59pmMSTIndependence of Path, Green’s TheoremThis assignment will cover the material from Chapters 14.3 –14.4.

1. (1 pt) set10/p10-7.pgFor each of the following vector fields F , decide whether it isconservative or not by computing curl F . Type in a potentialfunction f (that is, ∇ f = F). If it is not conservative, type N.

A. F(x,y) = (2x+3y)I+(3x+14y)Jf (x,y) =

B. F(x,y) = 1yI+2xJf (x,y) =

C. F(x,y,z) = 1xI+2yJ+Kf (x,y,z) =

D. F(x,y) = (1siny)I+(6y+1xcosy)Jf (x,y) =

E. F(x,y,z) = 1x2I+3y2J+7z2Kf (x,y,z) =

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

Correct Answers:• 1*x**2 + 3*x*y + 7*y**2• N• 1*x**2/2 + (1+1)*y**2/2 + z• 1*x*sin(y) + 3*y**2• (1/3)*(1*x**3 + 3*y**3 + 7*z**3)

2. (1 pt) set10/p11-1.pgLet C be the positively oriented circle x2 + y2 = 1. Use Green’sTheorem to evaluate the line integral

RC 1ydx+14xdy.


3. (1 pt) set10/p11-2.pgLet F =−4yI +5xJ. Use the tangential vector form of Green’sTheorem to compute the circulation integral

RC F · dX where C

is the positively oriented circle x2 + y2 = 16.


4. (1 pt) set10/p11-3.pgLet F = 2xI + 2yJ and let n be the outward unit normal vectorto the positively oriented circle x2 + y2 = 16. Compute the fluxintegral

RC F ·nds.


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

Let F be the radial force field F = 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.pgLet F(x,y) = −yI+xJ

x2+y2 and let C be the circle X(t) = (cos t)I +(sin t)J, 0≤ t ≤ 2π.

A. ComputeR

C F ·dX

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

Correct Answers:• 6.28318530717959• N

7. (1 pt) set10/p11-7.pgLet C be the positively oriented square with vertices (0,0),(3,0), (3,3), (0,3). Use Green’s Theorem to evaluate the lineintegral

RC 2y2xdx+7x2ydy.


8. (1 pt) set10/p11-8.pgFind a parametrization of the curve x2/3 + y2/3 = 1 and use it tocompute the area of the interior.


9. (1 pt) set10/p11-10.pgLet F = 4x3I+2y3J and let n be the outward unit normal vectorto the positively oriented circle x2 + y2 = 1. Compute the fluxintegral

RC F ·nds.



1

WeBWorK demonstration assignmentThe main purpose of this WeBWorK set is to familiarize

yourself 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/demo pr1.pg

Evaluate the expression8(2−8) = .

Correct Answers:

• -48


Evaluate the expression8/(5+5) = .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:

• 0.8

3. (1 pt) setDemo/demo pr3.pgLet r = 7.

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

Correct Answers:

• 8.91267681314614• 0.181891363533595

4. (1 pt) setDemo/demo pr4.pgEnter here the expression 1

a + 1b .

Enter here the expression 1a+b .

Correct Answers:

• 1/a+1/b• 1/(a+b)

5. (1 pt) setDemo/demo pr5.pgEnter here the expression

a+12+b

Enter here the expressiona+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)

6. (1 pt) setDemo/demo pr6.pgEnter 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

Enter here the expression

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)


Enter here the expression

−b+√

b2−4ac2a

Note: this is an expression that gives the solution of a quadraticequation by the quadratic formula.

Correct Answers:• (-b+sqrt(b**2-4*a*c))/(2a)

1


2

Hsiang-Ping HuangMath 2210-90, Fall 2008WeBWorK Assignment 11 due 12/11/2008 at 10:59pmMSTSurface Integrals, Divergence and Stokes’ TheoremsThis assignment will cover the material from Chapters 14.5 –14.7.

1. (1 pt) set11/p12-1.pgLet F = 3xI+6yJ+5zK. Compute the divergence and the curl.

A. div F =B. curl F = I+ J+ KCorrect Answers:• 14• 0• 0• 0

2. (1 pt) set11/p12-2.pgLet F = (4yz)I+(7xz)J+(2xy)K. Compute the following:

A. div F =B. curl F = I+ J+ KC. div curl F =Note: Your answers should be expressions of x, y and/or z;

e.g. ”3xy” or ”z” or ”5”Correct Answers:• 0• (2 - 7)*x• (4 - 2)*y• (7 - 4)*z• 0

3. (1 pt) set11/p12-3.pgA fluid has density 3 and velocity field v =−yI+ xJ+1zK.Find the rate of flow outward through the sphere x2 +y2 +z2 = 4


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

Use Stokes’ theorem to evaluateZ Z

ScurlF · dS where

F(x,y,z) =−8yzI+8xzJ+17(x2 +y2)zK and S is the part of theparaboloid z = x2 + y2 that lies inside the cylinder x2 + y2 = 1,oriented upward.


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

Use Stokes’ Theorem to evaluateZ

CF · dr where F(x,y,z) =

xI + yJ + 9(x2 + y2)K and C is the boundary of the part of theparaboloid where z = 36−x2−y2 which lies above the xy-planeand C is oriented counterclockwise when viewed from above.

Correct Answers:

• 0

6. (1 pt) set11/p12-6.pgSuppose F = F(x,y,z) is a gradient field with F = ∇ 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 dS where S is the helicoid: r(u,v) =

ucos(v)I+usin(v)J+ vK, with 0≤ u≤ 3,0≤ v≤ 4π

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

Correct Answers:

• 150.79644737231

8. (1 pt) set11/p12-8.pgFind the surface area of the part of the sphere x2 + y2 + z2 = 1that lies above the cone z =

√x2 + y2

Correct Answers:

• 1.84030236902122

9. (1 pt) set11/p12-10.pgLet S be the part of the plane 3x + 2y + z = 2 which lies in thefirst octant, oriented upward. Find the flux of the vector fieldF = 2I+3J+2K across the surface S.

Correct Answers:

• 4.66666666666667

10. (1 pt) set11/p6-1.pgFind an equation of the tangent plane to the parametric surfacex = 4r cosθ, y = −4r sinθ, z = r at the point

(4√

2,−4√

2,2)

when r = 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 + 0.176776695296637 * (x- ((4)*sqrt(2))) + -0.176776695296637 * (y- ((-4)*sqrt(2)))


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Hsiang-Ping Huang Math 2210-90, Fall 2008 5. WeBWorK Assignment … · 2009-01-07 · Hsiang-Ping Huang Math 2210-90, Fall 2008 WeBWorK Assignment 1 due 09/04/2008 at 10:59pm MDT