Math 233 Calculus 3 - Fall 2017

§12.6 and §12.2 Recitation Problems

1. Match the graphs to the equations. For one of the equations,justify your choice of graph by describing the horizontal and /or vertical cross-sections of the graph and how they correspondto the equation. Do not explain equation 5 / graph G since wedid that one in class. TOO LONG, cut it down and do a coupleof them as examples!

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2. A boat is pulled onto shore using two ropes, as shown in thediagram. If a force of 255 N is needed, find the magnitude ofthe force in each rope.

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Extra Problem (if extra time – does not need to be handed in):

3. A motorboat traveling 4 m/s N45E encounters a current trav-eling 7 m/s N. (N45◦E means 45 degrees east of the northerlydirection.)

(a) What is the resultant velocity of the motorboat?

(b) If the river runs North-South and the width of the river is 80meters, then how much time does it take the boat to travelshore to shore?

(c) What distance downstream does the boat reach the oppositeshore?

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§12.3 and §12.4 Recitation Problems

1. Find two unit vectors that are at 45◦ angles to the vector< 3, 5 >.

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Extra Problems (if extra time – does not need to be handed in):

2. Use vectors to show that if two lines in the plane y = m1x + b1

and y = m2x + b2 are perpendicular to each other, then theirslopes are opposite reciprocals.

3. Consider the vector c = a × (a × b). Is ~c ⊥ ~a? Is ~c ⊥ ~b? Eitherprove or give a counterexample.

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4. Suppose that ~j × ~a = ~i. Give two possible solutions for ~a, anddiscuss other possible solutions.

5. Show that ~a is perpendicular to (~a −~b) × (~a +~b).

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6. Given ~a =< 1, 2, 3 > and~b =< 1,−1,−1 > , sketch or describe thecollection of all vectors c satisfying ~a × b = ~a × ~c.

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7. (a) Find all vectors ~v such that

< 1, 2, 1 > × ~v =< 3, 1,−5 >

(b) Explain why there is no vector ~v such that

< 1, 2, 1 > × ~v =< 3, 1, 5 >

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§13.1 Recitation Problems

Find a vector function (or parametric equations) that represents thecurve of intersection of the two surfaces. (Not to hand in.)

1. The parabolic cylinder y = x2 and the top half of the ellipsoidx2 + 4y2 + 4z2 = 16.

2. The paraboloid y = 4x2 + z2 and the plane x = 2z + 3

3. The hyperboloid z = x2− y2 and the cylinder x2 + y2 = 1

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Extra Problem

4. z = x2 + y2 and 2x − 4y − z − 1 = 0

Hint: complete the square

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CATEGORY A CATEGORY B CATEGORY C CATEGORY D

The line between ~r =< 2, 4, 1 > +t < 1, 2, 0 > x = 2t, y = 4t, z = 1x − 1

2=

y − 24

, z = 1

(0, 0, 1) and (1, 2, 1)

The line between ~r =< 1,−1, 2 > +t < 1, 2,−1 > x = 2 + t, y = 1 + 2t, z = 1 − tx − 1

2=

y + 14

=z − 2−2

(0,−3, 3) and (3, 3, 0)

The line between ~r =< 1, 2, 3 > +t < 0. − 1, 1 > x = 1, y = −t, z = 5 + t x = 1,y − 1−2

=z − 4

2(1, 3, 2) and (1,−1, 6)

The line between ~r =< 9, 6, 7 > +t < −3,−2,−1 > x = 6 − 6t, y = 4 − 4t, z = 6 − 2tx − 3

3=

y − 22

=z − 5

1(0, 0, 4) and (12, 8, 8)

The line between ~r =< 0, 0, 0 > +t < −1, 1,−3 > x = −1 + t, y = 1 − t, z = −3 + 3tx + 2−2

=y − 2

2=

z + 6−6

(−3, 3,−9) and (3,−3, 9)

The line between ~r =< 3, 3, 3 > +t < 7, 1, 2 > x = 10 + 7t, y = 4 + t, z = 5 + 2t,x + 4

14=

y − 22

=z − 1

4(−4, 2, 1) and (−11, 1,−1)The plane that contains x − 2y + z = 0 −3(x − 3) + 6(y − 4) − 3(z − 5) The plane that contains lines x = t + 2, y = 2, z = tthe points (5, 4, 3),(1, 2, 3), and (2, 2, 2)

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The line between (0, 0, 1) and (1, 2, 1)

~r =< 2, 4, 1 > +t < 1, 2, 0 >

x = 2t, y = 4t, z = 1

x − 12

=y − 2

4, z = 1

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The line between (0,−3, 3) and (3, 3, 0)

~r =< 1,−1, 2 > +t < 1, 2,−1 >

x = 2 + t, y = 1 + 2t, z = 1 − t

x − 12

=y + 1

4=

z − 2−2

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The line between (1, 3, 2) and (1,−1, 6)

~r =< 1, 2, 3 > +t < 0. − 1, 1 >

x = 1, y = −t, z = 5 + t

x = 1,y − 1−2

=z − 4

2

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The line between (0, 0, 4) and (12, 8, 8)

~r =< 9, 6, 7 > +t < −3,−2,−1 >

x = 6 − 6t, y = 4 − 4t, z = 6 − 2t

x − 33

=y − 2

2=

z − 51

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The line between (−3, 3,−9) and (3,−3, 9)

~r =< 0, 0, 0 > +t < −1, 1,−3 >

x = −1 + t, y = 1 − t, z = −3 + 3t

x + 2−2

=y − 2

2=

z + 6−6

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The line between (−4, 2, 1) and (−11, 1,−1)

~r =< 3, 3, 3 > +t < 7, 1, 2 >

x = 10 + 7t, y = 4 + t, z = 5 + 2t

x + 414

=y − 2

2=

z − 14

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The plane that contains the points (5, 4, 3),(1, 2, 3), and (2, 2, 2)

x − 2y + z = 0

−3(x − 3) + 6(y − 4) − 3(z − 5)

The plane that contains lines x = t +2, y = 2, z = t + 2

andx = 3t + 2, y = 2 + 2, z = t + 2

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The plane that contains the points (0, 3, 1),(1, 0, 3), and (2, 1, 0)

7x + 5y + 4z = 19

−7(x − 3) − 5(y + 2) − 4(z − 2) = 0

The plane that contains lines x = t, y =−3t + 3, z = 2t + 1

andx = t + 1, y = t, z = −3t + 3

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The plane that contains the points (0, 2, 2),(1, 2, 3), and (0, 0, 0)

−x − y + z = 0

2(x − 5) + 2(y − 4) − 2(z − 9) = 0

The plane that contains lines x = t, y =2t, z = 3t

andx = t + 1, y = 2, z = t + 3

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§14.2 Recitation Problems

Evaluate the limit or show that it does not exist.

1. lim(x,y)→(0,0)

x2 + 3xy2x2 + 3y2

2. lim(x,y)→(1,1)

x2 + 3xy2x2 + 3y2

3. lim(x,y)→(0,0)

xy√x2 + y2

Extra Problem from Lecture (not to hand in)

4. lim(x,y)→(0,0)

xy2

x2 + y4

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§14.3 Recitation Problems - When Clairout’s TheoremFails

Usually fxy = fyx. But in this activity, you will investigate an unusualfunction for which fxy does NOT equal fyx at (0, 0):

f (x, y) =

(xy)(x2− y2

x2 + y2

)if (x, y) , (0, 0)

0 if (x, y) = (0, 0)

1. Find∂ f∂x

(x, y) when (x, y) , (0, 0) and simplify your answer.

2. Find∂ f∂x

(0, y) when y , 0 by plugging in 0 for x above.

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3. Find∂ f∂x

(0, 0). Since f is piecewise defined at (0, 0), you will need

to use the limit definition of∂ f∂x

.

4. Conclude that∂ f∂x

(0, y) = −y for y , 0 and for y = 0.

5. Find∂ f∂y

(x, y) when (x, y) , (0, 0) and simplify. If you are tired of

doing algebra, you can argue based on part 1 and the symmetryof the function.

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6. Find∂ f∂y

(x, 0) when x , 0 by plugging in 0 for y above.

7. Find∂ f∂y

(0, 0). Since f is piecewise defined at (0, 0), you will need

to use the limit definition of∂ f∂y

.

8. Conclude that∂ f∂y

(x, 0) = x for x , 0 and for x = 0.

9. Find∂2 f∂y∂x

(0, y)

10. Find∂2 f∂x∂y

(x, 0)

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11. Observe that∂2 f∂y∂x

is not equal to∂2 f∂x∂y

at (0, 0). Why doesn’t

this contradict Clairout’s Theorem?

Hint: It turns out that when (x, y) , (0, 0),

∂2 f∂y∂x

=∂2 f∂x∂y

=x6 + 9x4y2

− 9x2y4− y6

(x2 + y2)3

.

Here is a picture of z =x6 + 9x4y2

− 9x2y4− y6

(x2 + y2)3

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As a reference, here is a picture of the original function

f (x, y) =

(xy)(x2− y2

x2 + y2

)if (x, y) , (0, 0)

0 if (x, y) = (0, 0)

Extra Problems - Not to be handed in

1. The following surfaces, labeled a, b, and c are graphs of a func-tion f and its partial derivatives fx and fy. Identify each surfaceand give reasons for your choices.

2. A contour map is given below. Use it to estimate fx(2, 1) andfy(2, 1).

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§14.5 Recitation Problems - Chain Rule

No problems need to be handed in this week.

1. Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and v are differentiable,u(1, 2) = 5, us(1, 2) = 4, ut(1, 2) = −3, v(1, 2) = 7, vs(1, 2) = 2,vt(1, 2) = 6, Gu(5, 7) = 9, Gv(5, 7) = −2. Find Rs(1, 2) and Rt(1, 2).

2. Suppose f is a differentiable function of x and y and g(u, v) =f (eu +sin v, eu +cos v). Use the table of values to calculate gu(0, 0)and gv(0, 0).

f (x, y) g(u, v) fx fy

(0, 0) 3 6 4 8(1, 2) 6 3 2 5

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Extra Problems

3. The pressure of 1 mole of an ideal gas is increasing at a rate of0.05 kPa/s and the temperature is increasing at a rate of 0.15 K/s.Use the equation PV = 8.31T to find the rate of change of thevolume when the pressure is 20 kPa and the temperature is 320K.

4. The speed of sound traveling through ocean water with salinity35 parts per thousand has been modeled by the equation

C = 1449.2 + 4.6T − 0.055T2 + 0.00029T3 + 0.016D

where C is the speed of sound (in meters per second), T is thetemperature (in degrees Celcius), and D is the depth below theocean surface (in meters). A scuba diver began a leisurely diveinto the ocean water; the diver’s depth and the surrounding wa-ter temperature over time are recorded in the following graphs.Estimate the rate of change (with respect to time) of the speed

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of sound through the ocean water experienced by the diver 20minutes into the dive. Give units.

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§14.7 Recitation Problems - Absolute Maximum andMinimum Values

Notes on how to find an absolute max or min value:

Ex: Maximize f (x, y) = 2x3 + y4 on the disk D = {(x, y)|x2 + y2≤ 1}.

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1. Find the absolute max and min value of f (x, y) = x2− 2xy + 2y

on the triangular region with vertices (0, 2), (3, 0), (3, 2).

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2. For each region of the plane, specify if the region is (i) closedor not closed, (ii) bounded or unbounded, and (iii) would acontinuous function f (x, y) be guaranteed to achieve an absolutemaximum and minimum value on this region?

A. {(x, y) | y ≥ 2}

B. {(x, y) | −2 ≤ x ≤ 2 and 1 ≤ y ≤ 4}

Optional Extra problems not to turn in

C. {(x, y) | x2 + y2 > 25}

D. {(x, y) | x ≥ 0 and y ≥ 0 and x + y < 1}

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S14.7 and S14.8 Recitation Problems

1. Three alleles (alternative versions of a gene) A, B, and O deter-mine the four blood types A (AA or AO), B (BB or BO), O (OO)and AB. The Hardy-Weinberg Law states that the proportion ofindividuals in a population who carry two different alleles is

P = 2pq + 2pr + 2rq

where p, q, and r represent the proportions of A, B, and O in thepopulation. Use the fact that p + q + r = 1 to show that P is atmost 2

3 .

2. Which point on the surface1x

+1y

+1z

= 1 is closest to the origin?

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Recitation: Chapter 14 Review - Concepts

1. Select the graph of the contour lines and the graph of the surfacethat best match the equations.

A. -4 -2 0 2 4

-4

-2

0

2

4

B. C. -4 -2 0 2 4

-4

-2

0

2

4

I. II. III.

1. f (x, y) = x2 + sin(y) 2. f (x, y) =√

4 − x2 − y2

3. f (x, y) = cos(x + y2) 4. f (x, y) = ln(x2 + y2 + 1)

5. f (x, y) = x2√

y2 − x2 6. f (x, y) = x2y

2. (§14.3: 73) Use the table of values of f (x, y) to estimate the valuesof fx(3, 2), fx(3, 2.2) and fxy(3, 2).

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3. (§14.3: 74) Level curves are shown for a function f . Determinewhether the following partial derivatives are positive or nega-tive at the point P.

(a) fx(b) fy

(c) fxx

4. (§14.4: 22) The wave heights h in the open sea depend on thespeed v of the wind and the length of time t that the wind hasbeen blowing at that speed. Values of the function h = f (v, t) arerecorded in feet in the following table. Use the table to find alinear approximation to the wave height function when v is near40 knots and t is near 20 hours. Then estimate the wave heightswhen the wind has been blowing for 24 hours at 43 knots.

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5. (§14.5: 49) Show that any function of the form z = f (x + at) +g(a − xt) is a solution to the wave equation

∂2 f∂t2 = a2∂

2z∂x2

Hint: let u = x + at, v = x − at.

6. (§14.6: 38) Sketch the gradient vector ∇ f (4, 6) for the functionf whose level curves are shown. Explain how you chose thedirection and length of this vector.

7. (§14.7: 4) Use the level curves in the figure to predict the criticalpoints of f (x, y) and to determine whether f has a saddle pointor a local maximum or minimum at each critical point. Thenuse the Second Derivatives Test to confirm your prediction.

f (x, y) = 3x − x3− 2y2 + y4

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S15.3 and S15.6 Recitation Problems

1. Warm-up (not to hand in)

SET UP the bounds of integration to evaluate∫ ∫ ∫

E(x − y) dV

where E is enclosed by the surfaces z = x2− 1, z = 1 − x2, y = 0,

and y = 2.

TURN OVER

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2. Evaluate the integral∫ ∫ ∫

Ty2 dV where T is the solid tetra-

hedron with vertices (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2). Hint:you will need to find the equation of the plane that forms thetop surface of the tetrahedron.

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3. Evaluate the integral by changing to polar coordinates:

∫ 1/2

0

∫ √1−y2

√3y

xy2 dx dy

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Extra Problems (not to hand in)

4. Evaluate the integral∫ ∫ ∫

Ex dV where E is bounded by the

paraboloid x = 4y2 + 4z2 and the plane x = 4.

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S15.8 and 16.1 Recitation Problems

No problems need to be turned in.

1. Match the vector field with the plot.

A) B)

C) D)

1) ~F(x, y) = cos(x + y)~i + x~j 3) ~H(x, y) = y~i + (x − y)~j

2) ~G(x, y) =~i + x~j 4) ~J(x, y) =y√

x2 + y2~i +

x√x2 + y2

~j

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2. A solid E consists of the intersection of a ball of radius a cen-tered at the origin, and a cone above the xy-plane where thesides of the cone make an angle of π/6 with the z-axis. Set uprepeated integrals for

∫ ∫ ∫E z dV in all three coordinate systems

(Cartesian, cylindrical, and spherical) and evaluate it in one ofthe three coordinate systems.

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Extra Problems

3. Change to spherical coordinates and integrate:

∫ 2

−2

∫ √

4−x2

0

∫ √4−x2−y2

0y√

x2 + y2 + z2 dz dy dx

4. Find the average distance from a point in a ball of radius a to itscenter. Does the answer agree with your intuition?

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S16.3 Recitation Problems

At the end of class on Thursday, we stated that:if ~F(x, y) = P(x, y)~i + Q(x, y)~j is a vector field on an open, simply con-nected region D, and P and Q have continuous first partial deriva-tives, then ~F(x, y) is conservative if and only if Py = Qx.

1. Warm-up (not to hand in):

(a) Determine if ~F(x) = (yex + sin y)~i + (ex + x cos y + y)~j is con-servative.

(b) If it is conservative, find a potential function.

(c) Compute∫

C~F ◦ d~r where C is the circle x = cos t, y = sin t for

0 ≤ t ≤ 2π.

45

2. Explain why if ~F is conservative, then Py = Qx.Assume ~F =< P,Q > and P and Q have continuous first partialderivatives.

TURN OVER

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3. (a) Determine if ~F(x) = (ln y +yx

+ 1)~i + (ln x +xy

+ 2)~j is conser-

vative.

(b) If it is conservative, find a potential function.

(c) Compute∫

C~F◦d~r where C is the curve y = x2 from the point

(1, 1) to the point (3, 9).

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4. (a) Determine if ~F(x) = (xy + y2)~i + (x2 + 2xy)~j is conservative.

(b) If it is conservative, find a potential function.

(c) Compute∫

C~F ◦ d~r where C is the line from (0, 0) to (2, 3).

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S16.4 Recitation Problems

How can Green’s Theorem be used to calculate the centriod of alamina? (The centroid is the center of mass for a lamina of uniformdensity 1.)

1. Use Green’s theorem to find the centroid of the triangle withvertices (0, 0), (a, 0) and (a, b), where a > 0 and b > 0.

TURN OVER

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Extra Problem

2. Use Green’s theorem to evaluate∫

Cx2y2 dx + xy dy where C con-

sists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and theline segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0).

S16.6 Recitation Problems

Fill in the following chart for Line Integrals.with respect to arclength with respect to dx, dy, or dz with respect to d~r

Notation:∫

C f (x, y, z) ds”Unpacked fomula”

Changes signwhen parammetrized

backwards?Interpretation N/A Work

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