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Name _____________________________________

Math 103 February 10, 2012Kraines Test 1

Honor Code: After you finish this exam, sign the following statement:

I have neither given nor received aid on this test. ______________________Justify your answers for full credit. Simplification of answers is not required.

1.(15 points) Consider the pointsP(2,2,3), Q(1,1,1), and R(2,1,4).

a. Compute the cross product PQPR.

b. Find the area of the triangle with verticesP, Q, andR.

c. Find the equation of the plane that containsP, Q, andR.

d. Find the distance from the point (1, 0, 0) to this plane.

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2 (15 points)

Let L1 be the line given by the symmetric equations2

3x=

4

y=

3

1 z

a. Find a point Q and vectorv so that L2 has vector equation r = OQ + tv.

Consider the lineL2 through the point P(1, -1, 1) and parallel to the vector w = for some numberc.

b. Show thatL1 and L2 are not parallel for any numberc.

c. For which c do the lines L1 and L2 intersect?

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3.(15 points) Austin stands 24 feet (measured horizontally) from the 10 foot high basket and releases hisjump shot at a height of 8 feet. The velocity at release is vo= 24i + 18k.

a. Write down parametric equations for the path of the ball with the origin at Austins feet. (Assume no airresistance and that the acceleration of gravity is g= 32 ft/s2)

b. Find the curvature of the path at the moment of release.

c. How long did it take for the ball to reach the basket? (not how long did itseem to take)

b. Find an expression in terms of a definite integral for the distance that the basketball travels from release

until it reaches the basket. (not the expression of Roy Williams or Coach K)

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4. (15 points) Consider the function defined by 4),( 22 ++= yxyxf .

a. Sketch and label three level curves of the surface Sdescribed by z = f( x, y).

b. Identify and make a sketch of this surface in R3

c. Calculate )1,2( yf and give a geometrical interpretation of the result.

.

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5 (15 points) Consider the surface Sdefined by z = yxyxyyxf 2),( 222 ++= .

a. Find the vector equation of the line throughP(1, -1, 3) that is tangent to Sand parallel to thexzplane

(i.e. the tangent line of the path on the surface throughPand headed East).

b. Find the equation of the tangent plane to Sat the pointP.

c. Find and plot all critical points of this function.

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6 (15 points) Consider the curve given by r(t) = < t, t2, 2t>. Calculate the following:

a. Velocity v(t) = ______________

b. Speed v(t) = ______________

c. Unit tangent vector T(t)

d. Acceleration a(t) = ______________

e. Curvature at t= 1, (1) = ______________

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7. (10 points) Let22

22

),(yx

yxyxg

+

= for (x,y) (0,0) and let 0)0,0( =g .

a. For what points in the plane isgcontinuous? Justify your answer.

b. Use the definition of gx(0,0) to determine whethergx(0,0) is defined. Do NOT attempt to use thequotient rule.

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Name _____________________________________

Math 103 Test 2Kraines March 23, 2012

Honor Code: After you finish this exam, sign the following statement:

I have neither given nor received aid on this test. ______________________1 (10 points) Find and classify all critical points off(x,y) =x3 + y3+ 3xy +4.

Find the maximum and minimum of

3 (10 points) Assume that z=f(x, y) has continuous second partial derivatives and that x = 2 u v3 and

y = uv.

a. Findu

f

in terms of

x

f

,

y

f

, u and v.

b Find2

2

u

f

in terms of

2

2

x

f

,

yx

f

2,

2

2

y

f

,

x

f

,

y

f

, u and v.

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5 (10 points) Let T be the solid bounded by x2 + (y-1)2 = 1, z 0, and z y.

a. Set up an iterated integral of the form dxdyyxg ),( that gives the volume of T. Do not evaluate

b. Set up an iterated integral of the form drdrh ),( that gives the volume of T. Do not evaluate

6. (5 points) Consider the rectangleR with corners (1,1,0), (3,1,0), (1,5,0), (3,5,0). Let T be the solid

formed by revolvingR about thex axis.

Use Pappas Theorem to find the volume of T.

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6 The temperature of a square plate is given by T(x, y) = 22 yx for 0 x 4 and 0 y 4. An ant,

starting atP(1,2), crawls in the direction of greatest decrease in temperature.

a. What is the unit vector in the direction that the ant atPis headed?

b. If the ant were to crawl along the curvex

y2

= within this square, explain why she would be always

crawling in the direction of greatest heat increase or greatest heat decrease.

Sketch the path described by this curve.

c. Give the equation of the curve throughPalong which the ant should crawl to stay at a constant

temperature.

Sketch the path described by this curve.

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7. (15 points) Consider the iterated integral I =

4/

4/

)sec(2

0

3 )(cos

drdr

a. Sketch the region of integration.

b. Evaluate this integral.

c. Change to an iterated integral in rectangular coordinates. Do not evaluate

.

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8 (15 points) LetR be the lamina bounded the cardiod r= 2 + 2cos().Assume that the density of the lamina is given by = 3 +y.

a. Sketch the lamina.

b. Give an iterated integral for the mass ofR. Do not evaluate.

c. Give an iterated integral for polar moment of inertia ofR. Do not evaluate

Consider the integral M = +

+3

0

9

0 0

222 22x yx

dzdydxyx

a. Describe the region in space T over which this is evaluated.

b. IfM represents the mass of T, describe the composition of T.

b Change the integral +

+3

0

9

0 0

222 22x yx

dzdydxyx to cylindrical coordinates. Do not evaluate

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Name _____________________________________

Math 103.1 Test 3

Kraines April 25, 2012

Honor Code: After you finish this exam, sign the following statement:

I have neither given nor received aid on this test. ______________________

1. (10 points) a. State the flux/divergence form of Greens Theorem giving all the hypotheses.

b. Give a physical interpretation of this theorem.

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2 (15 points) Let G(x, y, z) = y2 i + (2xy + 3z)j + (3y+2z) k

a. Show that = 0.

b. Find a functiong(x, y, z) so that g = G.

c. Find the value of C

G Tds where C is the portion of the curve r(t) = < cos( t), t2 , t3+1>

for 10 t . Justify your answer.

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3 (20 points) Let and r= |r| and assume that G = r2r is a velocity field.

Let S be the sphere with radius a and center (0,0,0).

a Express the flux ofG across S with outward pointing normal as a surface integral.

b. What is the value of the integral above? Justify your answer.

c. Calculate

d. Use spherical coordinates to evaluate whereB is the solid ball with boundary

S.

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4 (15 points) Let S be the surfacedescribed in parametric polar coordinates by

>=< ),sin(),cos(),( rrrR with 31 r and 40 .

a Sketch Sanddescribe this surface in words.

b. Show that = >< r),cos(),sin( is a normal vector to the surface S at R( r, ).

c. Write down an iterated integral that represents the surface area of S. Do not evaluate

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5. (20 points) Consider the double integral K = R

dAyx )( 44

whereR is the region bounded by 1 xy 2 and 2 x2y2 4.

a Sketch the regionR.

b. Let u =xy and v = x2y2 . Calculate),(

),(

yx

vu

c. Write K as an integral of the form S

dudvvuf ),( .

d. Evaluate the integral in part c.

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6 . (20 points) Let Sbe the portion of the paraboloid and belowz= 4. Let Cbe the boundarycurve ofS.a Express the outward pointing (away from the zaxis) unit normal vectors n to S.

b. Give a parameterization of C so that C and S are coherently oriented.

c. Let F = ( . Compute the curl ofF.

c. Use Stokes Theorem to evaluate C

F Tds