integration tofind areas and volumes, volumes …math121/assignments/assignments...unit #13 -...

Unit #13 - Integration to Find Areas and Volumes, Volumes of Revolution

Some problems and solutions selected or adapted from Hughes-Hallett Calculus.

Areas

In Questions #1-8, find the area of one strip or slice,then use that to build a definite integral representingthe total area of the region. Where possible with thetechniques from the class, evaluate the integral.

1.

2.

3.

4.

5.

6.

7.

8.

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Volumes of Geometric Shapes

In Questions #9-14, write a Riemann sum and thena definite integral representing the volume of the re-gion, using the slice shown. Evaluate the integral ex-actly. (Regions are parts of cones, cylinders, spheres,and pyramids.)

9.

10.

11.

12. Note: for this example, the integration techniques fromclass will not be enough to allow you to evaluate theintegral. Simply set up the integral for the volume.

13.

14.

15. Find, by slicing, the volume of a cone whose height is3 cm and whose base radius is 1 cm.

16. Find the volume of a sphere of radius R by slicing.

17. Find, by slicing, a formula for the volume of a cone ofheight h and base radius r.

18. The figure below shows a solid with both rectangularand triangular cross sections.

(a) Slice the solid parallel to the triangular faces.Sketch one slice and calculate its volume in termsof x, the distance of the slice from one end. Thenwrite and evaluate an integral giving the volume ofthe solid.

(b) Repeat part (a) for horizontal slices. Instead of x,use h, the distance of a slice from the top.

2

Volumes of Revolution

In Questions #19-23, the region is rotated around thex-axis. Find the volume.

19. Bounded by y = x2, y = 0, x = 0, x = 1.

20. Bounded by y = (x+ 1)2, y = 0, x = 1, x = 2.

21. Bounded by y = 4− x2, y = 0, x = −2, x = 0.

22. Bounded by y =√

x+ 1, y = 0, x = −1, x = 1.

23. Bounded by y = ex, y = 0, x = −1, x = 1.

24. Find the volume obtained when the region bounded byy = x3, x = 1, y = −1 is rotated around the axisy = −1.

25. Find the volume obtained when the region bounded byy =

√

x, x = 1, y = 0 is rotated around the axis x = 1.

26. Find the volume obtained when the region bounded byy = x2, y = 1, and the y-axis is rotated around they-axis.

27. Find the volume obtained when the region bounded byy = x2, y = 1, and the y-axis is rotated around thex-axis.

28. Find the volume obtained when the region bounded byy = ex, the x-axis, and the lines x = 0 and x = 1 isrotated around the x-axis.

29. Find the volume obtained when the region bounded byy = ex, the x-axis, and the lines x = 0 and x = 1 isrotated around the line y = −3.

30. Rotating the ellipse x2/a2+y2/b2 = 1 about the x-axisgenerates an ellipsoid. Compute its volume.

31. (a) A pie dish is 9 inches across the top, 7 inches acrossthe bottom, and 3 inches deep, as shown in the fig-ure below. Compute the volume of this dish.

(b) Make a rough estimate of the volume in cubicinches of a single cut-up apple, and estimate thenumber of apples that is needed to make an applepie that fills this dish.

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Unit #14 - Center of Mass, Improper Integrals


Computing Totals With Slices

1. Find the mass of a rod of length 10 cm, with linealdensity δ(x) = e−x g/cm, where x is the distance incm from the left end of the rod.

2. A rod is 2 meters long. At a distance x meters from itsleft end, the density of the rod is given by δ(x) = 2+6xg/cm.

(a) Write a Riemann sum approximating the totalmass of the rod. (Do not evaluate this sum.)

(b) Find the exact mass by converting the sum into anintegral and evaluating it.

3. Find the mass of the block 0 ≤ x ≤ 10, 0 ≤ y ≤ 3, 0 ≤z ≤ 1, whose density δ, is given by

δ = 2− z mass units/unit volume, for 0 ≤ z ≤ 1.

4. Circle City, a typical metropolis, is densely populatednear its center, and its population gradually thins outtoward the city limits. In fact, its population densityis 10, 000(3−r) people/ square mile at distance r milesfrom the center.

(a) Assuming that the population density at the citylimits is zero, find the radius of the city.

(b) What is the total population of the city?

5. The density of oil in a circular oil slick on the surfaceof the ocean at a distance r meters from the center ofthe slick is given by δ(r) = 50/(1 + r) kg/ m2.

(a) If the slick extends from r = 0 to r = 10,000 m,find a Riemann sum approximating the total massof oil in the slick.

(b) Find the exact value of the mass of oil in the slickby turning your sum into an integral and evaluat-ing it.

(c) Within what distance r is half the oil of the slickcontained?

6. The soot produced by a garbage incinerator spreads outin a circular pattern. The depth, H(r), in millimeters,of the soot deposited each month at a distance r kilome-ters from the incinerator is given by H(r) = 0.115e−2r.

(a) Write a definite integral giving the total volume ofsoot deposited within 5 kilometers of the incinera-tor each month.

(b) Evaluate the integral you found in part (a), givingyour answer in cubic meters.

Center of Mass

7. A point mass of 2 grams located 3 centimeters to theleft of the origin and a point mass of 5 grams located4 centimeters to the right of the origin are connectedby a thin, light rod. Find the center of mass of thesystem.

8. Find the center of mass of a system containing threepoint masses of 5 g, 3 g, and 1 g located respectivelyat x = −10, x = 1, and x = 2.

9. A rod with density δ(x) = 2 + sin(x) lies on the x-axisbetween x = 0 and x = π. Find the center of mass ofthe rod.

10. A rod of length 1 meter has density δ(x) = 1 + kx2

grams/meter, where k is a positive constant. The rodis lying on the positive x-axis with one end at the ori-gin.

(a) Find the center of mass as a function of k.

(b) Show that the center of mass of the rod satisfies0.5 < x < 0.75.

11. A rod of length 2 meters and density δ(x) = 3 − e−x

kilograms per meter is placed on the x-axis with itsends at x = ±1.

(a) Will the center of mass of the rod be on the left orright of the origin? Explain.

(b) Find the coordinate of the center of mass.

12. A metal plate, with constant density 2 g/cm2, has ashape bounded by the curve y = x2 and the x-axis,with 0 ≤ x ≤ 1 and x, y in cm.

(a) Find the total mass of the plate.

(b) Sketch the plate, and decide, on the basis of theshape, whether x is less than or greater than 1/2.

(c) Find x.

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13. A metal plate, with constant density 5 g/cm2, has ashape bounded by the curve y =

√x and the x-axis,

with 0 ≤ x ≤ 1 and x, y in cm.

(a) Find the total mass of the plate.

(b) Find x and y.

14. An isosceles triangle with uniform density, altitude a,and base b is placed in the xy-plane as in the diagrambelow.

a

b/2

−b/2

Show that the center of mass is at x = a/3, y = 0.Hence show that the center of mass is independent ofthe triangle’s base.

Improper Integrals

15.

∫ ∞

1

1

5x+ 2dx

16.

∫ ∞

1

1

(x+ 2)2dx

17.

∫ ∞

0

xe−x2

dx

18.

∫ 0

−∞

ex

1 + exdx

19.

∫ 4

0

1√16− x2

dx

20.

∫ π/2

π/4

sinx√cosx

dx

21.

∫ ∞

1

1

x2 + 1dx

22.

∫ ∞

2

1

x lnxdx

23.

∫ 1

0

lnx

xdx

24.

∫ π

0

1√xe−

√x dx

25.

∫ ∞

4

1

x2 − 1dx

26. Given that

∫ ∞

−∞

e−x2

dx =√π, calculate the exact

value of∫ ∞

−∞

e−(x−a)2/b dx

27. The rate, r, at which people get sick during an epidemicof the flu can be approximated by r = 1000te−.5t,where r is measured in people/day and t is measuredin days since the start of the epidemic.

(a) Sketch a graph of r as a function of t.

(b) When are people getting sick fastest?

(c) How many people get sick altogether?

28. Suppose a function h is defined by

h(x) =

1

x

√x−

1

16if 0 < x ≤ 4

1

x2if x > 4.

Consider the following integrals:

(i)

∫ 2

0

h(x)dx (iii)

∫ ∞

4

h(x)dx

(ii)

∫ 4

2

h(x)dx (iv)

∫ ∞

0

h(x)dx

For each integral, determine the following:

• Is the integral improper?

• If improper, does it diverge?

• If it is convergent, or a proper integral, what nu-merical value does the integral converge to?

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Unit #15 - Differential Equations


Basic Differential Equations

1. Show that y = x+ sin(x) − π satisfies the initial valueproblem

dy

dx= 1 + cosx

2. Find the general solution of the differential equationdy

dx= x3 + 5

3. Find the general solution of the differential equation

dq

dz= 2 + sin z, given that q = 5 when z = 0.

4. A tomato is thrown upward from a bridge 25 m abovethe ground at 40 m/sec.

(a) Give formulas for the acceleration, velocity, andheight of the tomato at time t. (Assume that theacceleration due to gravity is g = 9.8 m/s2.)

(b) How high does the tomato go, and when does itreach its highest point?

(c) How long is it in the air?

5. Ice is forming on a pond at a rate given bydy

dt= k

√t

where y is the thickness of the ice in inches at time tmeasured in hours since the ice started forming, and kis a positive constant. Find y as a function of t.

6. If a car goes from 0 to 80 km/h in six seconds withconstant acceleration, what is that acceleration?

7. A car going 80 ft/s ( about 90 km/h) brakes to a stopin five seconds. Assume the deceleration is constant.

(a) Graph the velocity against time, t, for 0 ≤ t ≤ 5seconds.

(b) Represent, as an area on the graph, the total dis-tance traveled from the time the brakes are ap-plied until the car comes to a stop.

(c) Find this area and hence the distance traveled.

(d) Now find the total distance traveled using antid-ifferentiation.

8. A 727 jet needs to attain a speed of 200 mph to take off.If it can accelerate from 0 to 200 mph in 30 seconds,how long must the runway be? ( Assume constant ac-celeration.)

9. Pick out which functions are solutions to which differ-ential equations. (Note: Functions may be solutions tomore than one equation or to none; an equation mayhave more than one solution.)

(a)dy

dx= −2y (I) y = 2 sinx

(b)dy

dx= 2y (II) y = sin 2x

(c)d2y

dx2= 4y (III) y = e2x

(d)d2y

dx2= −4y (IV) y = e−2x

Modelling With Differential Equations

10. Match the graphs in the figure below with the followingdescriptions.

(a) The temperature of a glass of ice water left on thekitchen table.

(b) The amount of money in an interest- bearing bankaccount into which $50 is deposited.

(c) The speed of a constantly decelerating car.

(d) The temperature of a piece of steel heated in a fur-nace and left outside to cool.

11. Match the graphs in the figure below with the followingdescriptions.

(a) The population of a new species introduced onto atropical island

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(b) The temperature of a metal ingot placed in a fur-nace and then removed

(c) The speed of a car traveling at uniform speed andthen braking uniformly

(d) The mass of carbon-14 in a historical specimen

(e) The concentration of tree pollen in the air over thecourse of a year.

12. Show that y = A + Cekt is a solution to the equationdy

dt= k(y −A).

13. Show that y = sin 2t satisfies the differential equation

d2y

dt2+ 4y = 0

14. Find the value(s) of ω for which y = cosωt satisfies

d2y

dt2+ 9y = 0

15. Estimate the missing values in the table below if you

know thatdy

dt= 0.5y. Assume the rate of growth given

bydy

dtis approximately constant over each unit time

interval and that the initial value of y is 8.

t y0 81234

16. (a) For what values of C and n (if any) is y = Cxn asolution to the differential equation:

xdy

dx− 3y = 0?

(b) If the solution satisfies y = 40 when x = 2, whatmore (if anything) can you say about C and n?

Slope Fields

17. The slope field for the equation y′ = x(y − 1) is shownin in the figure below.

(a) Sketch the solutions passing through the points(i) (0, 1) (ii) (0, -1) (iii) (0, 0)

(b) From your sketch, write down the equation of thesolution with y(0) = 1 .

(c) Check your solution to part (b) by substituting itinto the differential equation.

18. The slope field for the equation y′ = x+ y is shown inFigure 11.17.

Figure 11.17: y′ = x+ y

(a) Sketch the solutions that pass through the points(i) (0, 0) (ii) (-3, 1) (iii) (-1, 0)

(b) From your sketch, write the equation of the solu-tion passing through (-1, 0).

(c) Check your solution to part (b) by substituting itinto the differential equation.

19. One of the slope fields on the diagram below has theequation y′ = (x + y)/(x− y). Which one?

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20. The slope field for the equation dP/dt = 0.1P (10−P ),for P ≥ 0, is in the figure below.

(a) Plot the solutions through the following points:(i) (0, 0)(ii) (1, 4)(iii) (4, 1)(iv) (-5, 1)(v) (-2, 12)(vi) (-2, 10)

(b) For which positive values of P are the solutions in-creasing? Decreasing? What is the limiting valueof P as t gets large?

21. Match the slope fields shown below with their differen-tial equations:

(a) y′ = −y

(b) y′ = y

(c) y′ = x

(d) y′ = 1/y

(e) y′ = y2

You will have to infer the vertical & horizontal scalingon the graphs.

22. Match the slope fields shown below with their differen-tial equations:

(a) y′ = 1 + y2

(b) y′ = x

(c) y′ = sinx

(d) y′ = y

(e) y′ = x− y

(f) y′ = 4− y

Each slope field is graphed for−5 ≤ x ≤ 5, −5 ≤ y ≤ 5.3

Euler’s Method

23. Consider the differential equation y′ = x+ y. Use Eu-ler’s method with ∆x = 0.1 to estimate y when x = 0.4for the solution curves satisfying

(a) y(0) = 1

(b) y(−1) = 0

24. Consider the differential equation y′ = (sinx)(sin y).

(a) Calculate approximate y-values using Euler’smethod with three steps and ∆x = 0.1, startingat each of the following points:(i) (0, 2) (ii)(0, π).

(b) Use the slope field below to explain your solutionto part (a)(ii).

25. Consider the differential equationdy

dx= f(x) with

initial value y(0) = 0. Explain why using Euler’smethod to approximate the solution curve gives thesame results as using left Riemann sums to approxi-

mate

∫x

0

f(t)dt.

26. Consider the solution of the differential equation y′ = ypassing through y(0) = 1.

(a) Sketch the slope field for this differential equation,and sketch the solution passing through the point(0, 1).

(b) Use Euler’s method with step size ∆x = 0.1 toestimate the solution at x = 0.1, 0.2, . . . , 1.

(c) Plot the estimated solution on the slope field; com-pare the solution and the slope field.

(d) Check that y = ex is the solution of y′ = y withy(0) = 1.

27. (a) Use Euler’s method to approximate the value of yat x = 1 on the solution curve to the differentialequation

dy

dx= x3 − y3

that passes through (0, 0). Use ∆x = 1/5 (i.e., 5steps).

(b) Using the slope field shown below, sketch the solu-tion that passes through (0, 0). Show the approxi-mation you made in part (a).

(c) Using the slope field, say whether your answer topart (a) is an overestimate or an underestimate.

Slope field fordy

dx= x3 − y3.

28. Consider the differential equationdy

dx= 2x, with initial

condition y(0) = 1.

(a) Use Euler’s method with two steps to estimate ywhen x = 1 . Then use four steps.

(b) What is the formula for the exact value of y?

(c) Does the error in Euler’s approximation behave aspredicted in the box?

Separable Differential Equations

29. Determine which of the following differential equationsis separable. Do not solve the equations.

(a) y′ = y

(b) y′ = x+ y

(c) y′ = xy

(d) y′ = sin(x+ y)

4

(e) y′ − xy = 0

(f) y′ = y/x

(g) y′ = ln(xy)

(h) y′ = (sinx)(cos y)

(i) y′ = (sinx)(cos xy)

(j) y′ = x/y

(k) y′ = 2x

(l) y′ = (x+ y)/(x+ 2y)

For Questions 30-37, find the particular solution to thedifferential equation.

30.dP

dt= −2P, P (0) = 1

31.dL

dp=

L

2, L(0) = 100

32.dy

dx+

y

3= 0, y(0) = 10

33.dm

dt= 3m, m = 5 when t = 1.

34.1

z

dz

dt= 5, z(1) = 5.

35.dy

dt= 0.5(y − 200), y = 50 when t = 0.

36.dm

dt= 0.1m+ 200, m(0) = 1000.

37.dw

dθ= θw2 sin(θ2), w(0) = 1.

For Questions 38-41, find the general solution to thedifferential equations. Assume a, b, and k are nonzeroconstants.

38.dR

dt= kR

39.dP

dt− aP = b

40.dy

dt= ky2(1 + t2)

41.dx

dt=

x lnx

t

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Unit #16 - Differential Equations


Growth and Decay

1. Each curve in in the figure below represents the balancein a bank account into which a single deposit was madeat time zero. Assuming continuously compounded in-terest, find:

(a) The curve representing the largest initial deposit.

(b) The curve representing the largest interest rate.

(c) Two curves representing the same initial deposit.

(d) Two curves representing the same interest rate.

2. The graphs in the figure below represent the tempera-ture, H (◦C), of four eggs as a function of time, t, inminutes. Match three of the graphs with the descrip-tions (a)-(c). Write a similar description for the fourthgraph, including an interpretation of any intercepts andasymptotes.

(a) An egg is taken out of the refrigerator (just above0◦C) and put into boiling water.

(b) Twenty minutes after the egg in part (a) is takenout of the fridge and put into boiling water, thesame thing is done with another egg.

(c) An egg is taken out of the refrigerator at the sametime as the egg in part (a) and left to sit on thekitchen table.

3. One model used in medicine is that the rate of growthof a tumor is proportional to the size of the tumor.

(a) Write a differential equation satisfied by S, the sizeof the tumor, in mm, as a function of time, t.

(b) Find the general solution to the differential equa-tion.

(c) If the tumor is 5 mm across at time t = 0, whatdoes that tell you about the solution?

(d) If, in addition, the tumor is 8 mm across at timet = 3, what does that tell you about the solution?

4. Hydrocodone bitartrate is used as a cough suppressant.After the drug is fully absorbed, the quantity of drug inthe body decreases at a rate proportional to the amountleft in the body. The half-life of hydrocodone bitartratein the body is 3.8 hours, and the usual oral dose is 10mg.

(a) Write a differential equation for the quantity, Q, ofhydrocodone bitartrate in the body at time t, inhours, since the drug was fully absorbed.

(b) Solve the differential equation given in part (a)

(c) Use the half-life to find the constant of proportion-ality, k.

(d) How much of the 10 mg dose is still in the bodyafter 12 hours?

5. The amount of land in use for growing crops increasesas the world’s population increases. Suppose A(t) rep-resents the total number of hectares of land in use inyear t. (A hectare is about 2 1

2acres.)

(a) Explain why it is plausible that A(t) satisfies theequation A′(t) = kA(t). What assumptions areyou making about the world’s population and itsrelation to the amount of land used?

(b) In 1950 about 1 · 109 hectares of land were in use;in 1980 the figure was 2 · 109. If the total amountof land available for growing crops is thought to be3.2 · 109 hectares, when does this model predict itis exhausted? (Let t = 0 in 1950.)

6. The radioactive isotope carbon-14 is present in smallquantities in all life forms, and it is constantly replen-ished until the organism dies, after which it decays tostable carbon-12 at a rate proportional to the amountof carbon-14 present, with a half-life of 5730 years.Suppose C(t) is the amount of carbon-14 present attime t.

(a) Find the value of the constant k in the differentialequation C′ = −kC .

1

(b) In 1988 three teams of scientists found that theShroud of Turin, which was reputed to be the burialcloth of Jesus, contained 91% of the amount ofcarbon-14 contained in freshly made cloth of thesame material.4 How old is the Shroud of Turin,according to these data?

7. Before Galileo discovered that the speed of a fallingbody with no air resistance is proportional to the timesince it was dropped, he mistakenly conjectured that

the speed was proportional to the distance it had fallen.

(a) Assume the mistaken conjecture to be true andwrite an equation relating the distance fallen, D(t),at time t, and its derivative.

(b) Using your answer to part (a) and the correct ini-tial conditions, show that D would have to be equalto 0 for all t, and therefore the conjecture must bewrong.

Other Applications of Differential Equations

8. A yam is put in a 200◦C oven and heats up accordingto the differential equation

dH

dt= −k(H − 200)

(a) If the yam is at 20◦C when it is put in the oven,solve the differential equation.

(b) Find k using the fact that after 30 minutes thetemperature of the yam is 120◦C.

9. A detective finds a murder victim at 9 am. The tem-perature of the body is measured at 90.3◦F. One hourlater, the temperature of the body is 89.0◦F. The tem-perature of the room has been maintained at a con-stant 68◦F. (For reference, normal body temperatureis 98.6◦F.)

(a) Assuming the temperature, T , of the body obeysNewton’s Law of Cooling, write a differential equa-tion for T .

(b) Solve the differential equation to estimate the timethe murder occurred.

10. At 1:00 pm one winter afternoon, there is a power fail-ure at your house in Wisconsin, and your heat does notwork without electricity. When the power goes out, itis 68◦F in your house. At 10:00 pm, it is 57◦F in thehouse, and you notice that it is 10◦F outside.

(a) Assuming that the temperature, T , in your homeobeys Newton’s Law of Cooling, write the differen-tial equation satisfied by T .

(b) Solve the differential equation to estimate the tem-perature in the house when you get up at 7:00 amthe next morning. Should you worry about yourwater pipes freezing?

(c) What assumption did you make in part (a) aboutthe temperature outside? Given this (probably in-correct) assumption, would you revise your esti-mate up or down? Why?

11. At time t = 0, a bottle of juice at 90◦F is stood in amountain stream whose temperature is 50◦F. After 5minutes, its temperature is 80◦F. Let H(t) denote thetemperature of the juice at time t, in minutes.

(a) Write a differential equation for H(t) using New-ton’s Law of Cooling.

(b) Solve the differential equation.

(c) When will the temperature of the juice havedropped to 60◦F?

12. Water leaks out of a barrel at a rate proportional tothe square root of the depth of the water at that time.If the water level starts at 36 inches and drops to 35inches in 1 hour, how long will it take for all of thewater to leak out of the barrel?

13. According to a simple physiological model, an athleticadult needs 20 calories per day per pound of bodyweight to maintain his weight. If he consumes moreor fewer calories than those required to maintain hisweight, his weight changes at a rate proportional tothe difference between the number of calories consumedand the number needed to maintain his current weight;the constant of proportionality is 1/3500 pounds percalorie. Suppose that a particular person has a con-stant caloric intake of I calories per day. Let W (t) bethe person’s weight in pounds at time t (measured indays).

(a) What differential equation has solution W (t)?

(b) Solve this differential equation.

(c) Graph W (t) if the person starts out weighing 160pounds and consumes 3000 calories a day.

14. Water leaks from a vertical cylindrical tank througha small hole in its base at a rate proportional to thesquare root of the volume of water remaining. If thetank initially contains 200 liters and 20 liters leak outduring the first day, when will the tank be half empty?How much water will there be after 4 days?

2

15. As you know, when a course ends, students start to for-get the material they have learned. One model (calledthe Ebbinghaus model) assumes that the rate at whicha student forgets material is proportional to the differ-ence between the material currently remembered andsome positive constant, a.

(a) Let y = f(t) be the fraction of the original mate-rial remembered t weeks after the course has ended.Set up a differential equation for y. Your equationwill contain two constants; the constant a is lessthan y for all t.

(b) Solve the differential equation.

(c) Describe the practical meaning (in terms of theamount remembered) of the constants in the so-lution y = f(t).

16. When people smoke, carbon monoxide is released intothe air. In a room of volume 60 m3, air containing5% carbon monoxide is introduced at a rate of 0.002m3/min. (This means that 5% of the volume of theincoming air is carbon monoxide.) The carbon monox-ide mixes immediately with the rest of the air and themixture leaves the room at the same rate as it enters.

(a) Write a differential equation for c(t), the concen-tration of carbon monoxide at time t, in minutes.

(b) Solve the differential equation, assuming there isno carbon monoxide in the room initially.

(c) What happens to the value of c(t) in the long run?

17. (Continuation of previous problem.) Medical textswarn that exposure to air containing 0.02% carbonmonoxide for some time can lead to a coma. How longdoes it take for the concentration of carbon monoxidein the room in Problem 16 to reach this level?

18. An aquarium pool has volume 2× 106 liters. The poolinitially contains pure fresh water. At t = 0 minutes,water containing 10 grams/liter of salt is poured intothe pool at a rate of 60 liters/minute. The salt waterinstantly mixes with the fresh water, and the excessmixture is drained out of the pool at the same rate (60liters/minute).

(a) Write a differential equation for S(t), the mass ofsalt in the pool at time t.

(b) Solve the differential equation to find S(t).

(c) What happens to S(t) as t → ∞?

The Logistic Model

19. Table 1 below gives the percentage, P , of householdswith a DVD, as a function of year.

Table 1: Percentage of households with a VCR

Year 1998 1999 2000 2001 2002P (%) 1 5 13 21 35Year 2003 2004 2005 2006P (%) 50 70 75 81

(a) Explain why a logistic model is a reasonable one touse for this data.

(b) Use the data to estimate the point of inflection ofP . What limiting value L does this point of inflec-tion predict? Does this limiting value appear to beaccurate given the percentages for 1990 and 1991?

(c) The best logistic equation for this data turns outto be the following. What limiting value does thismodel predict?

P =86.395

1 + 316.75e−0.699t

20. The growth of an animal population is governed by theequation

1000

P

dP

dt= 100− P

where P (t) is the number of individuals in the colonyat time t. The initial population is known to be 200individuals. Sketch a graph of P (t). Will there everbe more than 200 individuals in the colony? Will thereever be fewer than 100 individuals? Explain.

21. A model for the population, P , of carp in a landlockedlake at time t is given by the differential equation

dP

dt= 0.25P (1− 0.0004P ).

(a) What is the long-term equilibrium population ofcarp in the lake?

(b) Under a plan to join the lake to a nearby river,the fish will be able to leave the lake. A net lossof 10% of the carp each year is predicted, but thepatterns of birth and death are not expected tochange. Revise the differential equation to takethis into account. Use the revised differential equa-tion to predict the future development of the carppopulation.

22. The population of a species of elk on Reading Islandin Canada has been monitored for some years. Whenthe population was 600, the relative birth rate was 35%and the relative death rate was 15%. As the popula-tion grew to 800, the corresponding figures were 30%

3

and 20%. The island is isolated so there is no huntingor migration.

(a) Write a differential equation to model the popula-tion as a function of time. Assume that relativegrowth rate is a linear function of population.

(b) Find the equilibrium size of the population. Todaythere are 900 elk on Reading Island. How do youexpect the population to change in the future?

(c) Oil has been discovered on a neighboring islandand the oil companies want to move 450 elk of thesame species to Reading Island. What effect wouldthis move have on the elk population on ReadingIsland in the future?

(d) Assuming the elk are moved to Reading Island,sketch the population on Reading Island as a func-tion of time. Start before the elk are transferredand continue for some time into the future. Com-ment on the significance of your results.

23. Consider the equation

dP

dt= 0.02P 2

− 0.08P.

(a) Sketch the slope field for this differential equationfor 0 ≤ t ≤ 50, 0 ≤ P ≤ 8.

(b) Use your slope field to sketch the general shape ofthe solutions to the differential equation satisfyingthe following initial conditions:(i) P (0) = 1, (ii) P (0) = 3 (iii) P (0) = 4(iv) P (0) = 5

(c) Are there any equilibrium values of the population?If so, are they stable?

24. Consider the equation

dP

dt= P 2

− 6P

(a) Sketch a graph ofdP

dtagainst P for positive P .

(b) Use the graph you drew in part (a) to sketchthe approximate shape of the solution curve withP (0) = 5. To do this, consider the following ques-

tion. For 0 < P < 6, isdP

dtpositive or nega-

tive? What does this tell you about the graph ofP against t? As you move along the solution curve

with P (0) = 5, how does the value ofdP

dtchange?

What does this tell you about the concavity of thegraph of P against t?

(c) Use the graph you drew in part (a) to sketch thesolution curve with P (0) = 8.

(d) Describe the qualitative differences in the behav-ior of populations with initial value less than 6 andinitial value more than 6. Why do you think P = 6is called the threshold population?

25. Consider a population satisfying

dP

dt= aP 2

− bP

(a) Sketch a graph ofdP

dtagainst P .

(b) Use this graph to sketch the shape of solutioncurves with various initial values. Use your graph

from part (a) to decide wheredP

dtis positive or

negative, and where it is increasing or decreas-ing. What does this tell you about the graph ofP against t?

(c) Why is P = b/a called the threshold population?What happens if P (0) = b/a? What happens inthe long-run if P (0) > b/a? What if P (0) < b/a?

4

Unit #17 - Functions of Two Variables


Functions of More Than One Variable

1. The balance, B, in dollars, in a bank account dependson the amount deposited, A dollars, the annual interestrate, r%, and the time, t, in months since the deposit,so B = f(A, r, t).

(a) Is f an increasing or decreasing function of A? Ofr? Of t?

(b) Interpret the statement f(1250, 1, 25) ≈ 1276.Give units.

2. The temperature adjusted for wind-chill is a tempera-ture which tells you how cold it feels, as a result of thecombination of wind and temperature. Some wind-chillvalues are shown below.

Temperature (oC)

Wind Speed (km/h)0 -5 -10 -15 -20 -25 -30 -35 -40 -45

10 -3 -9 -15 -21 -27 -33 -39 -45 -51 -5720 -5 -12 -18 -24 -30 -37 -43 -49 -56 -6230 -6 -13 -20 -26 -33 -39 -45 -52 -59 -6540 -7 -14 -21 -27 -34 -41 -48 -54 -61 -6850 -8 -15 -22 -29 -35 -42 -49 -56 -63 -6960 -9 -16 -23 -30 -36 -43 -50 -57 -64 -71

Source: Environment Canada

(a) If the temperature is -5oC and the wind speed is20 km/h, how cold does it feel?

(b) If the temperature is -10oC, what wind speedmakes it feel like -20oC?

(c) If the temperature is -15oC, what wind speedmakes it feel like -25oC?

(d) If the wind is blowing at 20 km/h, what temper-ature feels like -25oC?

3. A car rental company charges $40 a day and 15 centsa km for its cars.

(a) Write a formula for the cost, C, of renting a caras a function, f , of the number of days, d, and thenumber of km driven, m.

(b) If C = f(d,m), find f(5, 300) and interpret it.

4. The concentration, C, in mg per liter, of a drug in theblood as a function of x, the amount, in mg, of the druggiven and t, the time in hours since the injection. For0 ≤ x ≤ 4 and t ≥ 0, we have C = f(x, t) = te−t(5−x)

(a) Find f(3, 2). Give units and interpret in terms ofdrug concentration.

(b) Graph the following two single variable functionsand explain their significance in terms of drug con-centration.

(i) f(4, t)

(ii) f(x, 1)

Spatial Reasoning

5. A cube is located such that its top four corners havethe coordinates (-1, -2, 2), (-1, 3, 2), (4, -2, 2) and (4,3, 2). Give the coordinates of the center of the cube.

6. You are at the point (-1, -3, -3), standing upright andfacing the yz-plane. You walk 2 units forward, turnleft, and walk for another 2 units. What is your finalposition? From the point of view of an observer look-ing at the coordinate system in Figure 12.2, are you infront of or behind the yz-plane? To the left or to theright of the xz-plane? Above or below the xy-plane?

7. Sketch the graph of the equation x = −3 in 3-space.

8. Find a formula for the shortest distance between apoint (a, b, c) and the y-axis.

9. Find the equations of planes that just touch the sphere(x− 2)2 + (y − 3)2 + (z − 3)2 = 16 and are parallel to

(a) The xy-plane

(b) The yz-plane

(c) The xz-plane

10. Find an equation of the largest sphere contained in thecube determined by the planes x = 2, x = 6, y = 5,y = 9 and z = −1, z = 3.

1

Graphs of Functions of Two Variables

11. Without a calculator or computer, match the functionswith their graphs in the figure below.

(a) z = 2 + x2 + y2

(b) z = 2− x2 − y2

(c) z = 2(x2 + y2)

(d) z = 2 + 2x− y

(e) z = 2

In Problems 12-15, sketch a graph of the surface andbriefly describe it in words.

12. z = 3

13. z = x2 + y2 + 4

14. z = y2

15. x2 + y2 = 4

16. Without a calculator or computer, match the functionswith their graphs in the figure below.

(a) z =1

x2 + y2

(b) z = −e−x2−y

2

(c) z = x+ 2y + 3

(d) z = −y2

(e) z = x3 − sin y.

17. Without a computer or calculator, match the equations(a)-(i) with the graphs (I)-(IX).

2

(a) z = xye−(x2+y2)

(b) z = cos(√

x2 + y2)

(c) z = sin y

(d) z = −1

x2 + y2

(e) z = cos2 x cos2 y

(f) z =sin(x2 + y2)

x2 + y2

(g) z = cos(xy)

(h) z = |x||y|

(i) z = (2x2 + y2)e1−x2−y

2

18. For each of the following functions, decide whether itcould be a bowl, a plate, or neither. Consider a plateto be any fairly flat surface and a bowl to be anythingthat could hold water, assuming the positive z-axis isup.

(a) z = x2 + y2

(b) z = 1− x2 − y2

(c) x+ y + z = 1

(d) z = −√

5− x2 − y2

(e) z = 3

19. By setting one variable constant, find a plane that in-tersects the graph of z = 4x2 − y2 + 1 in a:

(a) Parabola opening upward

(b) Parabola opening downward

(c) Pair of intersecting straight lines

20. By setting one variable constant, find a plane that in-tersects the graph of z = (x2 + 1) sin(y) + xy2 in a:

(a) Parabola

(b) Straight line

(c) Sine curve

3

Unit #18 - Level Curves, Partial Derivatives


Contour Diagrams

1. Figure 1 shows the density of the fox population P (infoxes per square kilometer) for southern England.

Draw two different cross-sections along a north-southline and two different cross-sections along an east-westline of the population density P .

Figure 1

In Problems 2-6, sketch a contour diagram for the func-tion with at least four labeled contours. Describe inwords the contours and how they are spaced.

2. f(x, y) = x+ y

3. f(x, y) = 3x+ 3y

4. f(x, y) = x2 + y2

5. f(x, y) = −x2 − y2 + 1

6. f(x, y) = cos√

x2 + y2

7. Match the contour diagrams (a)-(d) with the surfaces(I)-(IV). Give reasons for your choice.

8. Match the pairs of functions (a)-(d) with the contourdiagrams (I)-(IV). In each case, show which contoursrepresent f and which represent g. (The x- and y-scalesare equal.)

(a) f(x, y) = x+ y, g(x, y) = x− y

(b) f(x, y) = 2x+ 3y, g(x, y) = 2x− 3y

(c) f(x, y) = x2 − y, g(x, y) = 2y + ln |x|

(d) f(x, y) = x2 − y2, g(x, y) = xy

1

9. Match the surfaces (a)-(e) with the contour diagrams(I)-(V) below.

10. Match Tables A-D with the contour diagrams (I)-(IV).

Table A

y|x -1 0 1-1 2 1 20 1 0 11 2 1 2

Table B

y|x -1 0 1-1 0 1 00 1 2 11 0 1 0

Table C

y|x 1 0 1-1 2 0 20 2 0 21 2 0 2

Table D

y|x -1 0 1-1 2 2 20 0 0 01 2 2 2

11. Match each Cobb-Douglas production function (a)-(c)with a graph (I)-(III) and a statement (D)-(G).

a. F (L,K) = L0.25K0.25

b. F (L,K) = L0.5K0.5

c. F (L,K) = L0.75K0.75

D. Tripling each input triples output.

E. Quadrupling each input doubles output.

G. Doubling each input almost triples output.

2

12. Below is the contour diagram of f(x, y).

Sketch the contour diagram of each of the followingfunctions.

(a) 3f(x, y)

(b) f(x, y)− 10

(c) f(x− 2, y − 2)

(d) f(−x, y)

Linear Functions

13. The charge, C, in dollars, to use an Internet service isa function of m, the number of months of use, and t,the total number of minutes on-line:

C = f(m, t) = 35 + 15m+ 0.05t

(a) Is f a linear function?

(b) Give units for the coefficients of m and t, and in-terpret them as charges.

(c) Interpret the intercept 35 as a charge.

(d) Find f(3, 800) and interpret your answer.

Which of the contour diagrams in Problems 14-17 couldrepresent linear functions?

14.

15.

16.

17.

3

18. Each column of the table below is linear with the sameslope, m = ∆z

∆x= 4/5. Each row is linear with the same

slope,n = ∆z∆y

= 3/2. We now investigate the slope ob-tained by moving through the table along lines that areneither rows nor columns.

y4 6 8 10 12

5 3 6 9 12 1510 7 10 13 16 1915 11 14 17 20 2320 15 18 21 24 2725 19 22 25 28 31

(a) Move down the diagonal of the table from the up-per left corner (z = 3) to the lower right corner(z = 31). What do you notice about the changesin z? Now move diagonally from z = 6 to z = 27.What do you notice about the changes in z now?

(b) Move in the table along a line right one step, uptwo steps from z = 19 to z = 9. Then move inthe same direction from z = 22 to z = 12. Whatdo you notice about the changes in z?

(c) Show that ∆z = m∆x+n∆y . Use this to explainwhat you observed in parts (a) and (b).

Which of the tables of values in Exercises 19 -22 couldrepresent linear functions?

19.

y0 1 2

0 0 1 41 1 0 12 4 1 0

20.

y0 1 2

0 10 13 161 6 9 122 2 5 8

21.

y0 1 2

0 0 5 101 2 7 122 4 9 14

22.

y0 1 2

0 5 7 91 6 9 122 7 11 15

23. Find the linear function whose graph is the planethrough the points (4, 0, 0), (0, 3, 0) and (0, 0, 2).

24. Find the equation of the linear function z = c+mx+nywhose graph intersects the xz=plane in the line z =3x+4 and intersects the yz-plane in the line z = y+4.

25. Find the equations for the linear function described bythe table below.

x|y 10 20 30 40100 3 6 9 12200 2 5 8 11300 1 4 7 10400 0 3 6 9

The Partial Derivative

26. A drug is injected into a patient’s blood vessel. Thefunction c = f(x, t) represents the concentration of thedrug at a distance x mm in the direction of the bloodflow measured from the point of injection and at timet seconds since the injection.

For the following partial derivatives,

• What are the units of the following partial deriva-

tives?

• What are their practical interpretations?

• What do you expect their signs to be?

(a)∂c

∂x

(b)∂c

∂t

4

27. The quantity, Q, of beef purchased at a store, in kilo-grams per week, is a function of the price of beef, b, andthe price of chicken, c, both in dollars per kilogram.

(a) Do you expect∂Q

∂bto be positive or negative?

Explain.

(b) Do you expect∂Q

∂cto be positive or negative?

Explain.

(c) Interpret the statement∂Q

∂b= −213 in terms of

quantity of beef purchased.

28. Below is a contour diagram for z = f(x, y). Is fx posi-tive or negative? Is fy positive or negative? Estimatef(2, 1), fx(2, 1), and fy(2, 1).

29. An experiment to measure the toxicity of formaldehydeyielded the data in the table below. The values showthe percent, P = f(t, c), of rats surviving an exposureto formaldehyde at a concentration of c (in parts permillion, ppm) after t months.

Estimate ft(18, 6) and fc(18, 6) . Interpret your an-swers in terms of formaldehyde toxicity.

Time t (months)14 16 18 20 22 24

0 100 100 100 99 97 95Conc. c 2 100 99 98 97 95 92(ppm) 6 96 95 93 90 86 80

15 96 93 82 70 58 36

30. The surface z = f(x, y) is shown in the graph below.The points A and B are in the xy-plane.

(a) What is the sign of

(i) fx(a)?

(ii) fy(A)?

(b) The point P in the xy-plane moves along astraight line from A to B. How does the signof fx(P ) change? How does the sign of fy(P )change?

NOTE: the axes are not positioned in the usual

location! Positive x values are back and left, and pos-itive y values are down and left. This affects your in-terpretation of the slope.

31. Consider the graph below:

(a) What is the sign of fx(0, 5)?

(b) What is the sign of fy(0, 5)?

32. The figure below shows a contour diagram for themonthly payment P as a function of the interest rate, r

%, and the amount, L, of a 5-year loan. Estimate∂P

∂r

and∂P

∂Lat the following points. In each case, give the

units and the everyday meaning of your answer.

(a) r = 8, L = 4000

(b) r = 8, L = 6000

(c) r = 13, L = 7000

5

Computing Partial Derivatives

Find the partial derivatives in Problems 33–40. As-sume the variables are restricted to a domain on whichthe function is defined.

33. fx and fy if f(x, y) = 5x2y3 + 8xy2 − 3x2

34.∂

∂x(a√x)

35.∂

∂B

(

1

u0

B2

)

36. Fv if F =mv2

r

37.∂T

∂lif T = 2π

√

l

g

38. fa if f(a, b) = ea sin(a+ b)

39. gx if g(x, y) = ln(yexy)

40.∂Q

∂Kif Q = c(a1K

b1 + a2Lb2)γ

41. Money in a bank account earns interest at a continu-ous rate, r. The amount of money, $B, in the accountdepends on the amount deposited, $P , and the time, t,it has been in the bank according to the formula

B = Pert

Find∂B

∂tand

∂B

∂Pand interpret each in financial

terms.

42. The Dubois formula relates a person’s surface area, s,in m2, to weight, w, in kg, and height, h, in cm, by

s = f(w, h) = 0.01w0.25h0.75

Find f(65, 160), fw(65, 160), and fh(65, 160) . Inter-pret your answers in terms of surface area, height, andweight.

43. Is there a function f which has the following partialderivatives? If so what is it? Are there any others?

fx(x, y) = 4x3y2 − 3y4

fy(x, y) = 2x4y − 12xy3

6

Unit #19 - Functions of Many Variables, and Vectors in R2 and R

3


Functions of Three Variables

1. Write the level surface x2 + y+√z = 1 as the graph of

a function f(x, y).

In Problems 2-7, decide if the level surfaces given canbe expressed as the graph of a two-variable function,f(x, y).

2. z2 = x2 + y2

3. The bottom half of the ellipsoid x2 + y2 +z2

2= 1

4. 2x+ 3y − 5z − 10 = 0

5. x2 + y2 + z2 − 1 = 0

6. 4x− y − 2z = 6.

7. The top half of the sphere x2 + y2 + z2 − 10 = 0

8. Find a formula for a function f(x, y, z) whose level sur-face f = 4 is a sphere of radius 2, centered at the origin.

9. Find a formula for a function f(x, y, z) whose level sur-faces are spheres centered at the point (a, b, c).

Local Linearity

In Problems 10-13, find the equation of the tangentplane at the given point to each function given.

10. z =1

2(x2 + 4y2) at the point (2, 1, 4).

11. z = yex/y at the point (1, 1, e).

12. z = ey + x+ x2 + 6 at the point (1, 0, 9).

13. [Duplicate question: removed.]

For Problems 14-16, find the differentials of the givenfunctions.

14. f(x, y) = sin(xy)

15. h(x, t) = e−3t sin(x+ 5t)

16. f(x, y) = e−x cos(xy)

For Problems 17-18, find the differentials of the givenfunctions at the given point.

17. f(x, y) = xe−y at (1, 0)

18. g(x, t) = x2 sin(2t) at (2, π/4).

19. A student was asked to find the equation of the tan-gent plane to the surface z = x3 − y2 at the point(x, y) = (2, 3). The student’s answer was

z = 3x2(x − 2)− 2y(y − 3)− 1

(a) At a glance, how do you know this is wrong?

(b) What mistake did the student make?

(c) Answer the question correctly.

20. (a) Write a formula for the number π using only theperimeter L and the area A of a circle.

(b) Suppose that L and A are determined experimen-tally. Show that if the relative, or percent, errorsin the measured values of L and A are λ and µ, re-spectively, then the maximum relative, or percent,error in π is 2λ+ µ.

(Note that λ and µ would be positive, e.g. “5%error” is how relative errors are usually reported,even though the actual error could be in the pos-itive or negative direction relative to the originalmeasurement.)

21. One mole of ammonia gas is contained in a vesselwhich is capable of changing its volume (a compart-ment sealed by a piston, for example). The total en-ergy U (in joules) of the ammonia, is a function of thevolume V (in m3) of the container, and the tempera-ture T (in K) of the gas. The differential dU is givenby

dU = 840dV + 27.32dT

(a) How does the energy change if the volume is heldconstant and the temperature is increased slightly?

(b) How does the energy change if the temperature isheld constant and the volume is increased slightly?

(c) Find the approximate change in energy if the gasis compressed by 100 cm3 and heated by 2 K.

1

Vectors

For Problems 22-26, say whether the given quantitycould be represented only as a vector, or if a scalarcould be used to represent it.

22. The population of the US.

23. The distance from Toronto to Vancouver.

24. The magnetic field at a point on the earth’s surface.

25. The temperature at a point on the earth’s surface.

26. The populations of each of the 13 provinces and terri-tories.

For Problems 27-34, find the resulting vector, and writeit as both a combination of the unit vectors ~i, ~j, ~k, aswell as in component-only form, 〈a, b, c〉.

27. −4(~i− 2~j)− 0.5(~i− ~k)

28. 2(0.45~i− 0.9~j − 0.01~k)− 0.5(1.2~i− 0.1~k)

29. (3~i− 4~j + 2~k)− (6~i+ 8~j − ~k)

30. (4~i+ 2~j)− (3~i−~j)

31. (~i+ 2~j) + (−3)(2~i+~j)

32. −4(~i− 2~j)− 0.5(~i− ~k)

33. 2(0.45~i− 0.9~j − 0.01~k)− 0.5(1.2~i− 0.1~k)

34. (4~i− 3~j + 7~k)− 2(5~i+~j − 2~k)

For Problems 35-38, perform the indicated operationson the following vectors:

~a = 〈0, 2, 1〉, ~b = 〈3, 5, 4〉~c = 〈1, 6, 0〉, ~x = 〈2, 9, 0〉~y = 〈4,−7, 0〉, ~z = 〈1,−3,−1〉

35. Find ~a+ ~z.

36. Find 2~c+ ~x.

37. Find 2~a+ 7~b− 5~z.

38. Find ||~y − ~x||.39. Find a vector with length 2 that points in the same

direction as ~i−~j + 2~k.

For Problems 40-43, consider the following scenario. Acat on the ground at the point (1, 4, 0) watches a squir-rel at the top of a tree. The tree is one unit high withits base at (2, 4, 0). Find the displacement vectors forthe points described in Problems 40-43.

40. From the origin to the cat.

41. From the bottom of the tree to the squirrel.

42. From the bottom of the tree to the cat.

43. From the cat to the squirrel.

For Problems 44-48, find the length of the vectorsgiven.

44. ~v =~i−~j + 2~k

45. ~v =~i− 3~j − ~k

46. ~v = 〈1,−1, 3〉

47. ~v = 7.2~i− 1.5~j + 2.1~k

48. ~v = 〈1.2,−3.6, 4.1〉49. For each of the four statements below, answer the fol-

lowing questions: Does the statement make sense? Ifyes, is it true for all possible choices of ~a and ~b ? If no,why not?

(a) ~a+~b = ~b+ ~a

(b) ~a+ ||~b|| = ||~a+~b||(c) ||~b + ~a|| = ||~a+~b||(d) ||~a+~b|| = ||~a||+ ||~b||

50. For what values of t are the following pairs of vectorsparallel?

(a) 2~i+ (t2 + (2/3)t+ 1)~j + t~k, 6~i+ 8~j + 3~k

(b) t~i+~j + (t− 1)~k, 2~i− 4~j + ~k

(c) 2t~i+ t~j + t~k, 6~i+ 3~j + 3~k

51. Find all vectors ~v in 2 dimensions having ||~v|| = 5 andfor which the ~i-component of ~v is 3~i.

52. The diagram below shows a rectangular box contain-ing several vectors. Are the following statements trueor false? Explain.

(a) ~c = ~f

(b) ~a = ~d

(c) ~a = −~b(d) ~g = ~f + ~a

(e) ~e = ~a−~b

(f) ~d = ~g − ~c

2

Vector Applications

53. Shortly after takeoff, a plane is climbing northwestthrough still air at an airspeed of 200 km/hr, and risingat a rate of 300 m/min. Resolve its velocity vector intocomponents. The x-axis points east, the y-axis pointsnorth, and the z-axis points up.

54. An airplane is flying at an airspeed of 500 km/hr ina wind blowing at 60 km/hr toward the southeast. Inwhat direction should the plane head to end up goingdue east? What is the airplane’s speed relative to theground?

55. An airplane is flying at an airspeed of 600 km/hr ina cross-wind that is blowing from the northeast at aspeed of 50 km/hr. In what direction should the planehead to end up going due east?

56. A car drives counter-clockwise around the track in thefigure below, slowing down at the curves and speedingup along the straight portions. Sketch velocity vectorsat the points P , Q, and R.

57. A large ship is being towed by two tugs. The larger tugexerts a force which is 25% greater than the smaller tugand at an angle of 30 degrees north of east. Which di-rection must the smaller tug pull to ensure that theship travels due east?

58. A man wishes to row the shortest possible distancefrom north to south across a river which is flowing at4 km/hr from the east. He can row at 5 km/hr.

(a) In which direction should he steer?

(b) If there is also a wind that would blow the rowerat a rate of 10 km/hr towards the northwest, inwhich direction should he steer to try and go di-rectly across the river? What happens?

3

Unit #20 - Directional Derivatives and the Gradient


The Dot Product

For Problems 1-8, perform the following operations onthe given 3-dimensional vectors.

~a = 〈0, 2, 1〉 , ~b = 〈−3, 5, 4〉 , ~c = 〈1, 6, 0〉~y = 〈4,−7, 0〉 , ~z = 〈1,−3,−1〉

1. ~a · ~y

2. ~c · ~y

3. ~a ·~b

4. ~a · ~z

5. ~a · (~c+ ~y)

6. ~c · ~a+ ~a · ~y

7. (~a ·~b)~a

8. (~a · ~y)(~c · ~z)

9. Let ~v = 〈2, 3〉. Using only two-dimensional vectors,find a unit vector in the same direction as ~v, and thenfind another vector perpendicular to ~v.

10. Which pairs (if any) of vectors from the following list

(a) Are perpendicular?

(b) Are parallel?

(c) Have an angle less than π/2 between them?

(d) Have an angle of more than π/2 between them?

~a = 〈1,−3,−1〉 ,~b = 〈1, 1, 2〉 ,~c = 〈−2,−1, 1〉 , ~d = 〈−1,−1, 1〉

11. Which pairs of the vectors√3~i+~j, 3~i+

√3~j, ~i−

√3~j

are parallel and which are perpendicular?

12. Compute the angle between the vectors ~i + ~j + ~k and~i−~j − ~k.

13. (a) Give a 2-dimensional vector that is parallel to, butnot equal to, ~v = 〈4, 3〉.

(b) Give a vector that is perpendicular to ~v .

14. For what values of t are ~u = 〈t,−1, 1〉 and ~v = 〈t, t,−2〉perpendicular? Are there values of t for which ~u and ~vare parallel?

15. Removed - duplicate

Gradients and the Directional Derivative

In Problems 16-28 find the gradient of the given func-tion; if a point is also given, evaluate the gradient atthat specific point. Assume the variables are restrictedto a domain on which the function is defined.

16. f(x, y) =3

2x5 − 4

7y6

17. f(x, y, z) = 1/(x2 + y2 + z2)

18. f(x, y, z) = xey sin z

19. f(x, y) =√

x2 + y2

20. z = sin(x/y)

21. f(α, β) =2α+ 3β

2α− 3β

22. f(x, y, z) = xey + ln(xz)

23. f(x1, x2, x3) = x2

1x3

2x4

3

24. f(x, y, z) = xyz at (1, 2, 3)

25. f(x, y, z) = sin(xy) + sin(yz), at (1, π,−1)

26. f(x, y) = x2y + 7xy3, at (1, 2)

27. f(r, h) = 2πrh+ πr2, at (2, 3)

28. f(x, y) = 1/(x2 + y2), at (−1, 3)

In Problems 29-30, find the directional derivative

f~u(1, 2) for the function f with ~u =

⟨

3

5,−4

5

⟩

.

29. f(x, y) = 3x− 4y

30. f(x, y) = xy + y3

In Problems 31-36, use the contour diagram of f(x, y)shown below to decide if the specified directionalderivative is positive, negative, or approximately zero.

1

31. At the point (−2, 2), in direction ~i.

32. At the point (0,−2), in direction ~j.

33. At the point (−1, 1), in direction~i+~j

34. At the point (−1, 1), in direction −~i+~j.

35. At the point (0,−2), in direction ~i+ 2~j.

36. At the point (0, −2), in direction ~i− 2~j.

In Problems 37-38, check that the point (2, 3) lies onthe curve. Then, viewing the curve as a contour off(x, y), use ∇f(2, 3) to find a vector normal to thecurve at (2, 3) and an equation for the tangent line tothe curve at (2, 3).

37. x2 + y2 = 13

38. x2 − y = 1

39. The contour diagram below represents the level curvesf(x, y).

In each of the following parts, decide whether the givenquantity is positive, negative or zero. Explain your an-swer.

(a) The value of ∇f ·~i at P .

(b) The value of ∇f ·~j at P .

(c)∂f

∂xat Q.

(d)∂f

∂yat Q.

40. The temperature at any point in the plane is given bythe function

T (x, y) =100

x2 + y2 + 1

(a) What shape are the level curves of T ?

(b) Where on the plane is it hottest? What is thetemperature at that point?

(c) Find the direction of the greatest increase in tem-perature at the point (3, 2). What is the magni-tude of that greatest increase?

(d) Find the direction of the greatest decrease in tem-perature at the point (3, 2).

(e) Find a direction at the point (3, 2) in which thetemperature does not increase or decrease.

2

Unit #21 - The Chain Rule, Higher Partial Derivatives & Optimization


The Chain Rule

For Problems 1-2, finddz

dtusing the chain rule. As-

sume the variables are restricted to domains on whichthe functions are defined.

1. z = xy2, x = e−t, y = sin t

2. z = xey, x = 2t,y = 1− t2

For Problems 3-4, find∂z

∂uand

∂z

∂v. The variables are

restricted to domains on which the functions are de-fined.

3. z = xey, x = u2 + v2,y = u2 − v2

4. z = sin(x/y), x = lnu,y = v

5. Corn production, C, is a function of rainfall, R, andtemperature, T . Figures A and B show how rain-fall and temperature are predicted to vary with timebecause of global warming. Suppose we know that∆C ≈ 3.3.∆R− 5∆T . Use this to estimate the changein corn production between the year 2020 and the year

2021. Hence, estimatedC

dtwhen t = 2020.

Figure A - Rainfall as a function of time

Figure B - Temperature as a function of time

6. The voltage, V , (in volts) across a circuit is given byOhm’s law: V = IR, where I is the current (in amps)flowing through the circuit and R is the resistance (inohms). If we place two circuits, with resistance R1 andR2, in parallel, then their combined resistance, R, isgiven by

1

R=

1

R1

+1

R2

Suppose the current is 2 amps and increasing at 10−2

amp/sec and R1 is 3 ohms and increasing at 0.5ohm/sec, while R2 is 5 ohms and decreasing at 0.1ohm/sec. Calculate the rate at which the voltage ischanging.

Second-Order Partial Derivatives

In Problems 7-11, calculate all four second-order par-tial derivatives and check that fxy = fyx. Assume thevariables are restricted to a domain on which the func-tion is defined.

7. f(x, y) = 3x2y + 5xy3

8. f(x, y) = (x+ y)3

9. f(x, y) = e2xy

10. f(x, y) = sin(x2 + y2)

11. f(x, y) = 3 sin(2x) cos(5y)

In Problems 12-13, find the quadratic Taylor polyno-mials about (0, 0) for the given functions.

12. f(x, y) = (x− y + 1)2

13. f(x, y) = ex cos(y)

In Problems 14-23, use the level curves of the functionz = f(x, y) to decide the sign (positive, negative, orzero) of each of the following partial derivatives at thepoint P . Assume the x- and y-axes are in the usualpositions.

(a) fx(P )

1

(b) fy(P )

(c) fxx(P )

(d) fyy(P )

(e) fxy(P )

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

Local and Global Extrema

In Problems 24-26, do the functions shown have globalmaxima and minima on the region shown? (Assumethe domain is limited to the region shown on the con-tour diagram, and its boundary.)

24.

2

25.

26.

In Problems 27-28, find the global maximum and min-imum of the function on −1 ≤ x ≤ 1, −1 ≤ y ≤ 1 , andsay whether it occurs on the boundary of the square.[Hint: Use graphs.]

27. z = −x2 − y2

28. z = x2 − y2

In Problems 29-33, do the given functions have globalmaxima and minima on their natural domains?

29. f(x, y) = x2 − 2y2

30. g(x, y) = x2y2

31. h(x, y) = x3 + y3

32. f(x, y) = −2x2 − 7y2

33. f(x, y) = x2/2 + 3y3 + 9y2 − 3x

3

Unit #22 - Unconstrained Optimization


Local Extrema

1. Which of the points A, B, C in the contour diagrambelow appear to be critical points? Classify those thatare critical points.

For Problems 2-5, find the critical points and classifythem as local maxima, local minima, saddle points, ornone of these.

2. f(x, y) = 400− 3x2 − 4x+ 2xy − 5y2 + 48y

3. f(x, y) = x3 − 3x+ y3 − 3y

4. f(x, y) = x3 + y3 − 6y2 − 3x+ 9

5. f(x, y) = 8xy −1

4(x+ y)4

6. Find A and B so that f(x, y) = x2 +Ax + y2 +B hasa local minimum value of 20 at (1, 0).

7. Letf(x, y) = kx2 + y2 − 4xy

Determine the values of k (if any) for which the criticalpoint at (0, 0) is:

(a) A saddle point

(b) A local maximum

(c) A local minimum

For Problems 8-10, use the contours of f in the figurebelow.

8. Decide whether you think each point is a local maxi-mum, local minimum, saddle point, or none of these.

(a) P

(b) Q

(c) R

(d) S

9. Sketch the direction of ||∇f || at several points aroundeach of P , Q, and R.

10. At the points where ||∇f || is largest, put arrows show-ing the direction of . f.

11. The behavior of a function can be complicated near acritical point where D = 0. Suppose that

f(x, y) = x3 − 3xy2

Show that there is one critical point at (0, 0) and thatD = 0 there. Show that the contour for f(x, y) = 0consists of three lines intersecting at the origin andthat these lines divide the plane into six regions aroundthe origin where f alternates from positive to negative.Sketch a contour diagram for f near (0, 0). The graphof this function is called a monkey saddle

Optimization

1

12. A missile has a guidance device which is sensitive toboth temperature, to C, and humidity, h. The range inkm over which the missile can be controlled is given by

Range = 27, 800− 5t2 − 6ht− 3h2 + 400t+ 300h

What are the optimal atmospheric conditions for con-trolling the missile?

13. An open rectangular box has volume 32 cm3. Whatare the lengths of the edges giving the minimum sur-face area? (You may assume that the answer you findis a local minimum for the surface area.)

14. A closed rectangular box with faces parallel to the co-ordinate planes has one bottom corner at the originand the opposite top corner in the first octant on theplane

3x+ 2y + z = 1

What is the maximum volume of such a box? (Verifythat your answer is a local maximum for the volume.)

15. Find the point on the plane

3x+ 2y + z = 1

that is closest to the origin by minimizing the squareof the distance. Confirm that your answer is a localminimum of the distance.

16. Two products are manufactured in quantities q1 and q2and sold at prices of p1 and p2, respectively. The costof producing them is given by

C = 2q21 + 2q22 + 10

(a) Find the maximum profit that can be made, as-suming the prices are fixed. (Verify that the criticalpoint you choose is in fact a local maximum.)

(b) Find the rate of change of that maximum profitas p1 increases. I.e. for every dollar you increasethe price p1, what will the effect be on your profit,assuming you are at the maximum for the currentprices?

2

Unit #2 - Limits, Continuity, and the Derivative


Lagrange Multipliers

In Problems 1−4, use Lagrange multipliers to find themaximum and minimum values of f subject to thegiven constraint, if such values exist. Make an argu-ment supporting the classification of your minima andmaxima.

1. f(x, y) = x+ y, x2 + y2 = 1

2. f(x, y) = xy, 4x2 + y2 = 8

3. f(x, y) = x2 + y, x2 − y2 = 1

4. f(x, y) = x2 + 2y2, x2 + y2 ≤ 4

20. (a) Draw contours of f(x, y) = 2x + y for z =−7,−5,−3,−1, 1, 3, 5, 7.

(b) On the same axes, graph the constraint x2 + y2 =5.

(c) Use the graph to approximate the points at whichf has a maximum or a minimum value subject tothe constraint x2 + y2 = 5.

(d) Use Lagrange multipliers to find the maximumand minimum values of f(x, y) = 2x + y subjectto x2 + y2 = 5.

5. A company manufactures x units of one item and y

units of another. The total cost in dollars, C, of pro-ducing these two items is approximated by the function

C = 5x2 + 2xy + 3y2 + 800

(a) If the production quota for the total number ofitems (both types combined) is 39, find the mini-mum production cost.

(b) Estimate the additional production cost or savingsif the production quota is raised to 40 or loweredto 38.

6. A firm manufactures a commodity at two different fac-tories. The total cost of manufacturing depends on thequantities, q1 and q2, supplied by each factory, and isexpressed by the joint cost function,

C = f(q1, q2) = 2q21+ q1q2 + q2

2+ 500

The company’s objective is to produce 200 units, whileminimizing production costs. How many units shouldbe supplied by each factory?

7. Each person tries to balance his or her time betweenleisure and work. The trade-off is that as you workless your income falls. Therefore each person has in-difference curves which connect the number of hours

of leisure, l, and income, s. If, for example, you areindifferent between 0 hours of leisure and an income of$1125 a week on the one hand, and 10 hours of leisureand an income of $750 a week on the other hand, thenthe points l = 0, s = 1125, and l = 10, s = 750 both lieon the same indifference curve. The table below givesinformation on three indifference curves, I, II, and III.

Weekly income Weekly leisure hoursI II III I II III

1125 1250 1375 0 20 40750 875 1000 10 30 50500 625 750 20 40 60375 500 625 30 50 70250 375 500 50 70 90

(a) Graph the three indifference curves.

(b) You have 100 hours a week available for work andleisure combined, and you earn $10/ hour. Writean equation in terms of l and s which representsthis constraint.

(c) On the same axes, graph this constraint.

(d) Estimate from the graph what combination ofleisure hours and income you would choose un-der these circumstances. Give the correspondingnumber of hours per week you would work.

8. The director of a neighborhood health clinic has anannual budget of $600,000. He wants to allocate hisbudget so as to maximize the number of patient vis-its, V , which is given as a function of the number ofdoctors, D, and the number of nurses, N , by

V = 1000D0.6N0.3

A doctor’s salary is $40,000; nurses get $10,000.

(a) Set up the director’s constrained optimizationproblem.

(b) Describe, in words, the conditions which must be

satisfied by∂V

∂Dand

∂V

∂Nfor V to have an opti-

mum value.

(c) Solve the problem formulated in part (a)

(d) Find the value of the Lagrange multiplier and in-terpret its meaning in this problem.

(e) At the optimum point, what is the marginal costof a patient visit (that is, the cost of an additionalvisit)? Will that marginal cost rise or fall with thenumber of visits? Why?

1

9. A mountain climber at the summit of a mountain wantsto descend to a lower altitude as fast as possible. Thealtitude of the mountain is given approximately by

h(x, y) = 3000−1

10, 000(5x2 + 4xy + 2y2) meters

where x, y are horizontal coordinates on the earth (inmeters), with the mountain summit located above theorigin. In thirty minutes, the climber can reach anypoint (x, y) on a circle of radius 1000 m. In which di-rection should she travel in order to descend as far as

possible?

10. For each value of λ the function h(x, y) = x2 + y2 −

λ(2x+ 4y − 15) has a minimum value m(λ).

(a) Find m(λ).

(b) For which value of λ is m(λ) the largest and whatis that maximum value?

(c) Find the minimum value of f(x, y) = x2 + y2

subject to the constraint 2x + 4y = 15 using themethod of Lagrange multipliers and evaluate λ.

(d) Compare your answers to parts (b) and (c).

2

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