Math 150B Lab Workbook - California State University

Math 150B Lab Workbook

6.1: Inverse Functions

1. For each of the following invertible functions f(x), find its inverse function f−1(x).

(a) f(x) =2x− 5

3x+ 4

(b) f(x) =√

5x− 1

2. For f(x) = x2 + 2x, where x is restricted to the interval [1,∞), find the inverse

function f−1(x). Also, what are the domain and range of f−1? Furthermore, what

is the largest interval containing [1,∞) on which f will be one-to-one and so have an

inverse function? (Hint: Drawing the graph y = x2 + 2x might be helpful.)

1

3. Let f be a differentiable one-to-one function on (−∞,∞) such that f(3) = 4, f(4) = 5,

f ′(3) = 6, f ′(4) = 7, and f ′(5) = 8. Find (f−1)′(4).

4. Let f(t) = t3 + 4t, for t ∈ (−∞,∞). Write down why f has an inverse function, and

find (f−1)′(5).

2

6.2* or 6.4: The Derivative of the Natural Logarithm

1. For each of the following functions, find f ′(x).

(a) f(x) = x2 lnx

(b) f(x) = ln(3x4 + x).

(c) f(x) =ln 2x

3x+ 4

3

2. Compute each of the following integrals.

(a)

∫5x3 − 5

x4 − 4x− 12dx

(b)

∫5x3 − 5√

x4 − 4x− 12dx

(c)

∫1

x(ln 2x)3dx

4

3. Find the slope of the line tangent to the curve

x2y + lnx+ ln(2y − 9) = 5

at the point (1,5).

4. Find all intervals of increase and all intervals of decrease for f(x) = 3x− 2 lnx.

5

5. Find the area of the region between the curves y = 2/x and y = 3− x .

6. Use logarithmic differentiation to differentiate f(x) =x4(2x− 1)5 sinx√

x2 + 9.

6

6.3* or 6.2: The Natural Exponential Function


(a) f(x) = (x3 + 5)ex

(b) f(x) = e1+3 tanx

7


(a)

∫2x2ex

3+6 dx

(b)

∫e2x√

3e2x + 5 dx

8

3. Use differentials or linear approximation to approximate e−0.02.

4. Let R be the region under the curve y = ex and the above the x-axis, 0 ≤ x ≤ 1. Find

the volume of the solid generated by revolving R about the line y = −2.

9

6.4* or 6.4: General Exponentials and Logarithms


(a) f(x) = x10x

(b) f(x) = log5(4x2 + 1)

(c) f(x) = x2x+1

10

2. Let R be the region under the curve y = 2(x2) and the above the x-axis, 0 ≤ x ≤ 1.

Find the volume of the solid generated by revolving R about the line y-axis.

6.5: Exponential Growth and Decay

1. The number of bacteria in a certain culture grows from 3000 to 9000 in eight hours.

Assuming the growth is exponential, find the amount of bacteria one day after the

start.

11

2. After one year, a sample of tritium-3 decays by 5.5% of its original amount. Find the

half-life of tritium-3. Write down your answer, including units, in a complete sentence

that does not include the term “half-life.”

3. A cup of coffee is made with boiling water. If the surrounding room temperature is

20◦C and the coffee cools to 80◦C in 3 minutes, find how long it will take to cool down

to 60◦C. Write down your answer in a complete sentence with units included.

12

6.6: Inverse Trigonometric Functions

1. Simplify the expression.

(a) tan(cos−1 x)

(b) sin(2 cos−1 x)

13


(a) f(x) = ln(1 + arcsinx)

(b) f(x) = x tan−1(x2)

14


(a)

∫1√

1− 3x2dx

(b)

∫1

3 + 6x2dx

15

4. Find all maxima and minima of f(x) = x− arctan 2x.

5. Find the area of the region that is below the curve y =4

1 + x2and above the line

y = 1.

16

6.8: Indeterminate Forms and L’Hospital’s Rule

1. Find the limit (finite, ∞, or −∞) or state “Does not exist.”

(a) limx→0

5x

5x− 2 sinx

(b) limx→1

1 + cos πx

x2 − 2x+ 1

(c) limt→0+

√t ln t

17

(d) limx→0+

[1

2 sinx− 1

sin 2x

]

(e) limx→∞

3x

4x− 2 lnx

(f) limx→∞

(1 +

2

x

)3x

18

2. Find the value of the constant c that makes f continuous at x = 2 if

f(x) =

ln(x− 1)

x2 − x− 2, x < 2

x2 + cx, x ≥ 2

19

7.1: Integration by Parts


(a)

∫x2/3 ln 2x dx

(b)

∫t2 cos 3t dt

20

(c)

∫x

(x+ 5)3/2dx

2. Let R be the region that is below the curve y = e3x and above the line y = −1 for

0 ≤ x ≤ 2. Find the volume of the solid generated by revolving R about the y-axis.

21

7.2: Trigonometric Integrals

1. Compute each of the following integrals. (Caution: Techniques from earlier sections

might be needed.)

(a)

∫cos5 2x dx

(b)

∫tan1/3t sec4t dt

22

(c)

∫x sin2x dx

(d)

∫sin 5x cos 3x dx

23

2. Let R be the region that is below the curve y = cosx and above the x-axis for 0 ≤ x ≤π/2. Find the volume of the solid generated by revolving R about the line y = 2.

7.3: Trigonometric Substitution


might be needed.)

(a)

∫x2

(9− x2)3/2dx

24

(b)

∫x3

(4 + x2)1/2dx

(c)

∫1

x(x2 − 16)1/2dx

25

7.4: Rational Functions


might be needed.)

(a)

∫9x2 + 3x+ 4

x3 + xdx

(b)

∫5x2 + 25x+ 36

x(x+ 3)2dx

26

2. Find the average value of the function f(t) =1

t2/3 − 1over the interval [8, 27].

27

7.5: Strategy for Integration

1. Write down which techniques of integration seem appropriate for evaluating the fol-

lowing integrals. Do not integrate.

(a)

∫x3

(4− x2)5/2dx

(b)

∫x3

(4− x2)2dx

(c)

∫x

(4− x2)5/2dx

(d)

∫tan4 3x sec4 3x dx

(e)

∫x sec2 3x dx

(f)

∫x(3x− 4)1/2 dx

(g)

∫x2

e5xdx

(h)

∫5

(2x− x2)1/2dx

(i)

∫1

sec4 xdx

28

7.7: Approximate Integration

1. Approximate the definite integral

∫ 6

−2

2x

x2 + 4dx

(a) using the Trapezoidal Rule with n = 4

(b) using the Midpoint Rule with n = 4

(c) using Simpson’s Rule with n = 4

2. Approximate the definite integral

∫ 3

0

cos(πx2) dx using Simpson’s Rule with n = 6.

29

3. The field pictured to the right is divided by

ropes into strips that are 6 feet wide. The

lengths of the ropes are as shown. Use the

Trapezoidal Rule to approximate the area

of the field.

4. Let f be a function that satisfies the table of values:

x -2 -1 0 1 2 3 4

f (x ) 10 12 16 8 4 2 1

(a) Approximate

∫ 4

−2f(x) dx using the trapezoidal rule with n = 3 subintervals.

(b) Approximate

∫ 4

−2f(x) dx using the Simpson’s rule with n = 6 subintervals.

30

7.8: Improper Integrals

1. For each of the following improper integrals, show whether the integral converges or

diverges. If an integral converges, compute its value.

(a)

∫ ∞1

x3

x4 + 5dx

(b)

∫ 1

−∞

x

(x2 + 3)3/2dx

31

(c)

∫ 4

1

3√2x− 2

dx

(d)

∫ 6

2

5x

(9− x2)5/3dx

32

2. For each of the following improper integrals, use the comparison theorem to determine

whether the integral converges or diverges. Write down an explanation of the reasoning

you used.

(a)

∫ ∞1

1

ex +√xdx

(b)

∫ ∞2

1√x− 1

dx

33

8.1: Arc Length

1. Find the length of the curve y = 3 + 8x3/2, 0 ≤ x ≤ 4.

2. Find the length of the curve x =1

4y4 +

1

8y−2, 1 ≤ y ≤ 3.

3. Express the length of the curve y = x3, 0 ≤ x ≤ 2, as an integral and then estimate

the length by applying Simpson’s Rule with n = 6 subintervals to the integral.

34

8.2: Area of a Surface of Revolution

1. Find the area of the surface obtained by revolving about the y-axis the part of the

curve y = 4− x2 that lies in the first quadrant.

2. Find the area of the surface obtained by revolving about the x-axis the curve y =√

7 + 6x, 0 ≤ x ≤ 8.

35

3. (Gabriel’s horn) Consider the surface obtained by revolving about the x-axis the curve

y = 1/x, x ≥ 1.

(a) Express the volume (of revolution) of the solid inside this surface as an improper

integral and evaluate this integral.

(b) Express the area of the surface as an improper integral and use the comparison test

to show that the integral diverges. (Hint: For x ≥ 1,

√1 +

1

x4≥√

1 + 0 = 1.)

(c) The above says that the volume of the horn is finite, but its surface area is infinite.

In practical terms this seems to indicate that the horn can be filled with a finite

amount of paint, but there would not be enough to paint its inside surface. Explain

what the flaw in this apparent contradiction is. Write down your explanation in

complete sentences.

36

8.3: Applications to Physics and Engineering

1. A rectangular swimming pool is 100 feet long and 30 feet wide. Its bottom is an

inclined plane with the water depth at the deep end being 12 feet while 100 feet away

at the shallow end the water depth is 3 feet. Find the hydrostatic force on the bottom

of the pool. Include units in your answer.

2. The sides of a 10 meter long trough are as shown in the

diagram on the right. (All dimensions are in meters.)

It is filled to a depth of 2 meters with water. Find the

hydrostatic force on one side of the trough. Include

units in your answer. What piece of information given

above is not used in computing your answer?

,,,

,,,

3

3

6

37

3. Find the centroid (x, y) of the region R that lies between the curves y = 2 − x2 and

y = −1.

4. Find the centroid (x, y) of the triangular region R with corners (0, 0), (1, 0), (0, 2).

38

8.5: Probability

1. A marksman shoots at a target. The distance X in centimeters from where his shot

lands to the center of the bullseye has the probability density function p(x) = .004x2e−.2x.

(a) Explain what the integral

∫ 2

0

.004x2e−.2x dx represents.

(b) What is the most likely distance from the center of the bullseye that the shot will

land?

39

2. The weights of a city’s garbage trucks vary between three tons (for an empty truck)

and seven tons (for a fully loaded one), according to the probability density function

p(x) = x/12− 1/6 for 3 ≤ x ≤ 7; p(x) = 0 otherwise, where x is measured in tons.

(a) Find the probability that a randomly chosen truck weighs more than five tons.

(b) What fraction of trucks weigh more than five tons?

(c) Find the mean weight of all of the garbage trucks.

(d) Set up an equation that, if solved, would determine the median weight of all of

the garbage trucks. Do not solve the equation.

40

3. The distribution of the amount water used per day at a small company has the density

function p(x) = 364x2(4− x) for 0 ≤ x ≤ 4, where x is measured in cubic meters.

(a) Find the proportion of days in which less than 3 cubic meters of water is used.

(b) Compute the value of an integral that verifies that p(x) is a legitimate probability

density function.

(c) Find the mean amount of water that the company uses per day.

41

4. The speeds of vehicles on a certain freeway at a certain time of day vary in a way

that is closely modeled by the probability density function f(x) = 0.04xe−0.2x, where

x represents the amount over 50 mph that a car is traveling. For example, x = 15

represents a car going 50 + 15 = 65 mph.

(a) Find the probability that a randomly selected car is speeding. The speed limit

on this freeway is 65 mph.

(b) Find the mean speed of all cars on this freeway at this time of day.

42

5. The function p(x) =1

π(1 + x2)for −∞ < x <∞ is known as the Cauchy density.

(a) If the random variable X has this probability density, find P (−√

3 < X <√

3).

(b) Find the median of the Cauchy density. (Hint: Note that p(x) is an even function.)

(c) Surprisingly, not all densities have a mean. Show that the mean of the Cauchy

density does not exist.

43

6. Heights of young adult Americans are approximately normally distributed. The mean

and standard deviation for each gender are given below, for heights measured in inches:

Mean Ht. (in.) S. D. of Ht. (in.)

Females 64.1 2.7

Males 69.3 2.9

Source: National Health and Nutrition Examination Survey (NHANES III)

(a) Sketch the normal density that models the distribution of heights for your gender.

Label the axis with the mean µ and values at µ±σ. (The latter two values occur

at the inflection points of the normal curve.)

(b) Find the proportion of each gender that is more than six feet tall according to

the normal distribution model.

44

9.3: Separable Equations

1. Solve the differential equationdy

dt= ky, for any non-zero constant k.

2. Find the solution of the differential equation (3x2 + sin x)y2 + y′ = 0 that satisfies the

initial condition y(0) = 1.

45

3. Find the solution of the differential equation −5 + (6xy2 + 6y2)y′ = 0 that satisfies the

initial condition y(0) = 2.

4. Consider the family of curves y = k/x2. Find the orthogonal trajectories of this

family by first finding a separable differential equation having the curves as solutions,

then writing down a second differential equation whose solutions are perpendicular the

original curves, and then solving this equation. Also, plot some of the original curves

along with some of the orthogonal trajectories on the same graph.

46

10.1: Curves Defined by Parametric Equations

1. Consider the curve given by the parametric equations x = 3t + 5, y = −2t + 1 for

−∞ < t <∞.

(a) Sketch the curve and indicate by an arrowhead the direction in which the curve

is traced out as t increases.

(b) Eliminate the parameter to find a Cartesian equation of the curve.

2. Consider the curve given by the parametric equations x = 2 + cos t, y = 3 + sin t for

0 ≤ t ≤ 2π.

(a) Sketch the curve and indicate by an arrowhead the direction in which the curve

is traced out as t increases.


47

3. Sketch the curve given by the parametric equations x = t cos 2t, y = t sin 2t for 0 ≤t ≤ 2π, and indicate by an arrowhead the direction in which the curve is traced out as

t increases.

4. Consider the circle of radius 3 centered at the origin.

(a) Find f, g, a and b such that the circle (traced out in the counterclockwise direction)

can be expressed parametrically as x = f(t), y = g(t) for a ≤ t ≤ b.

(b) Repeat part (a) with a different choice of f, g, a and b.

(c) Repeat part (a) with the circle being traced out in the clockwise direction.

48

5. Consider the graphs given to the right. Below, plot

the curve given parametrically by x = g(t), y = f(t)

for 0 ≤ t ≤ 4.

1 2x

1

2

y

1 2 3 4t

1

2

x

x = gHtL

1 2 3 4t

1

y

y = fHtL

10.2: Calculus with Parametric Equations

1. Assuming that the parametric equations x = t3 + 5t + 1, y = 6t + et, give y as a

function of x, finddy

dxand

d2y

dx2when x = 1.

49

2. Consider the curve given parametrically by x = −√t+ 1, y =

√3t. Find the equation

of the line tangent to this curve at the point (−2, 3).

3. Consider the curve given by the parametric equations x = 8t3/2, y = 2t for 1 ≤ t ≤ 9.

(a) Find the length of the curve.


(c) Use the Cartesian equation to again find the length of the curve.

50

4. Consider the curve given parametrically by x = t3 + 1, y = t2 + 2 for 0 ≤ t ≤ 1.

Express as an integral (In each case, do not evaluate the integral.)

(a) the length of the curve.

(b) the area of the surface generated by revolving the curve about the line x = −3.

5. Find the area of the surface generated by revolving about the y-axis the curve given

by x = 3 cos t, y = 3 sin t, for 0 ≤ t ≤ π/6.

51

10.3: Polar Coordinates

1. Convert the polar equation r = 5 sin θ into Cartesian coordinates and identify the

corresponding curve.

2. Convert the polar equation r cos(θ − π/6) = 4 into Cartesian coordinates and identify

the corresponding curve.

52

3. Sketch the curve given by the polar equation r = 2 + 3 cos θ.

4. Find the slope of the line tangent to the polar curve r = cos1

6θ at the point with

θ = 2π.

53

5. Find the points on the curve r = 3− 2 sin θ at which the tangent line is horizontal.

10.4: Areas and Lengths in Polar Coordinates

1. Find the area of the region enclosed by one loop of the curve r = 3 sin 2θ.

54

2. Find the area of the region that is inside the circle r = 2 cos θ and outside the circle

r = 1.

3. Find the length of the polar curve r = 2θ2, 0 ≤ θ ≤ 1.

55

11.1: Sequences

1. For each of the following, determine whether the sequence {an} converges or diverges.

If it converges, find its limit.

(a) an =1

n+ sin

π

2n

(b) an =n2 + 8n− 1

6n2 + 3n+ 7

(c) an =ln 5n

n

56

2. Let {bn} be a sequence such that limn→∞

bn =∞. Find limn→∞

bn2bn + 3

.

3. Let an =n3

2nfor n = 1, 2, . . . .

(a) Show that the sequence {an} is eventually monotonic (increasing or decreasing).

(b) Use the result of part (a) to prove that {an} converges.

57

4. Let {an} be the sequence defined by a1 = 2 and an+1 =2 + 3an

5for n = 1, 2, . . . .

(a) Show that the sequence {an} is bounded below by 1.

(b) Show that the sequence {an} is monotonic.

(c) What do the results of part (a) and part (b) say about {an} converging or diverg-

ing? If {an} converges, find its limit L.

58

11.2: Series - Introduction

1. For each of the following, determine whether the series converges or diverges. If it

converges, find its sum.

(a)∞∑n=1

2

5n

(b)∞∑k=1

(−3)k

(c)∞∑j=1

(1

j + 1− 1

j + 2

)

(d)1

4− 1

12+

1

36− 1

108+− · · ·

(e)∞∑n=1

n

n+ 4

59

2. A bug is crawling up a wall. In the first time period, it crawls up one inch, in the

second time period it crawls up 1/3 of an inch, in the third time period it crawls up

1/9 of an inch, and so on. In each time period it moves one third of the distance it

moved in the previous time period. If this is assumed to continue forever, what is the

total distance it moves, in the limit?

3. In one version of Zeno’s paradox, Achilles is racing after a tortoise. Achilles can run ten

times as fast as the tortoise, but the tortoise has a 100-yard head-start. Zeno claims

that that Achilles will never catch up to the tortoise because by the time Achilles runs

100 yards, the tortoise will have moved ahead by 10 yards, by the time Achilles runs

the 10 more yards, the tortoise will have moved ahead by 1 yard, and so on. Explain

why Achilles will actually catch up to the tortoise, and compute how many yards he

will have to run to do it.

4. Explain why if a series∞∑n=1

an converges, then the series∞∑n=1

1

1 + anmust diverge.

60

11.3: The Integral Test and Estimates of Sums

1. For each of the following, use the integral test to determine whether the series converges

or diverges.

(a)∞∑n=1

n

n2 + 10

(b)∞∑n=1

n

(n2 + 10)3/2

(c)1

2(ln 2)2+

1

3(ln 3)2+

1

4(ln 4)2+

1

5(ln 5)2+ · · ·

61

2. Explain why∞∑n=3

1

n2<

1

2. Hint: Consider the diagram.

1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

3. If S =∞∑n=3

1

n2and S100 =

100∑n=3

1

n2, explain why |S−S100| < 0.01. Hint: Draw a diagram.

62

11.4: The Comparison Tests

1. For each of the following, test whether the series converges or diverges. (Caution: Tests

from earlier sections might be needed.)

(a)∑n

| cosn|n2

(b)∑n

1

n3/2 + 5

(c)1

1 · 31+

1

2 · 32+

1

3 · 33+

1

4 · 34+ · · ·

63

(d)∑n

n3/2

n2 + 5

(e)∑k

k3/2

k3 − sin k

(f)∑n

(−1)nn2

n2 + 5

64

2. Suppose that an > 0 for all n and∑n

an converges. Prove that∑n

ln(1 + 3(an)2)

converges.

11.5: Alternating Series

1. For each of the following, test whether the series converges or diverges. (Caution: Tests

from earlier sections might be needed.)

(a)∞∑n=1

(−1)n+1

n2 + n

(b)∞∑n=1

(−1)n+1 n2 + 1

n2 + n

65

2. Show that the series∞∑k=1

(−1)k+1 1

3k + 2converges and find out how large n must be to

be sure that Sn =n∑

k=1

(−1)k+1 1

3k + 2is within 0.001 of the sum of the series.

3. Suppose that∑n

an and∑n

bn are two convergent series

(a) Give an example that shows that∑n

anbn could be divergent.

(b) If an > 0 and bn > 0 for all n, prove that∑n

anbn must converge.

66

11.6: Absolute Convergence & the Ratio and Root Tests

1. For each of the following, test whether the series converges or diverges. When ap-

propriate, test whether convergence is absolute or conditional. (Caution: Tests from

earlier sections might be needed.)

(a)∑n

n5

3n

(b)1!

4 · 12+

2!

5 · 22+

3!

6 · 32+

4!

7 · 42+ · · ·

67

(c)∑n

(−1)n+1

1 + n3

(d)∑n

(−4)n

n!

(e)∑n

(1

2+

1

n

)n

68

11.7: Strategy for Testing Series

1. For each of the following, test whether the series converges or diverges. When appro-

priate, test whether convergence is absolute or conditional.

(a)∑n

1

3n +√n

(b)1

4 · 1!− 1

5 · 2!+

1

6 · 3!− 1

7 · 4!+− · · ·

69

(c)∑n

(−1)n+1

1 +√n

(d)∑n

n

2n+ 1

(e)∑n

n√2n3 + 1

70

(f)∑n

2(n!)

(2n)!

(g)∑n

(sinn)2

n3 + 5

71

11.8: Power Series

1. For each of the following power series, find the interval of convergence.

(a)∑n

(−2)n√n

(x− 4)n

(b)∑n

6n

n!(x− 1)n

72

(c)∑n

n4

2n(x+ 5)n

(d)1

1 · 5(x− 8) +

1

2 · 52(x− 8)2 +

1

3 · 53(x− 8)3 + · · ·

73

11.9: Representations of Functions as Power Series

1. Starting with the power series expansion for1

1− x, find power series expansions for

each of the following. Also find the interval on which each expansion holds.

(a)1

3− 5x

(b)1

(3 + 2x)2

(c) ln(1− 4x)

74

2. (a) Starting with the power series expansion for1

1− x, find a power series expansion

forx

1 + x6.

(b) Using the result of part (a), find a power series expansion for

∫x

1 + x6dx.

(c) Using the result of part(b), approximate

∫ 1/3

0

x

1 + x6dx, correct to within 0.0000001.

(d) Explain why the power series expansion cannot be used to approximate

∫ 2

0

x

1 + x6dx.

75

11.10: Taylor and Maclaurin Series

1. Let f(x) = (1− 5x)−1. Using the definition of Maclaurin series,

(a) find the first four nonzero terms of the Maclaurin series for f .

(b) find the (entire) Maclaurin series for f in sigma notation.

76

2. Using the definition of Taylor series, find the first six nonzero terms of the Taylor series

centered at a = π/6 for f(x) = cos x.

77

3. Starting with a standard Maclaurin series, find the Maclaurin series for each of the

following.

(a) f(x) = sin(2x3)

(b) f(x) = x2e3x

4. Use a standard Maclaurin series to find the sum of the series

π

3− π3

3!33+

π5

5!35− π7

7!37+− · · · .

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11.11: Applications of Taylor Polynomials

1. Let f(x) = (1 + x)1/2.

(a) Compute the Maclaurin polynomial of f of order 2.

(b) Use your polynomial from part (a) to approximate (0.8)1/2.

(c) Given the fact that |f ′′′(x)| ≤ 0.66 for all |x| ≤ 0.2, estimate the error in your

approximation.

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2. Derive the 3rd - degree Maclaurin polynomial for f(x) = sinx, and use it to approx-

imate sin(0.3) . Also, either using the remainder term for a Taylor series or using a

remainder estimate for a relevant convergence test, provide an upper bound on the

error in your approximation.

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3. Find the Taylor polynomial of degree 2 based at 4 of the function f(x) =√x, and use

it to approximate the value of√

3.

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4. Let f(x) = x10/3.

(a) Find the Taylor polynomial of f of degree 2 based at a = 1.

(b) Use the polynomial from part (a) to approximate (0.7)10/3.

(c) Estimate the error in your approximation.

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Documents

Math 150B Lab Workbook - California State University