The Math and Science Bible: Test Preparation Kit Version 2oddjob.utias.utoronto.ca/john/tutor/bible.pdfThe Math and Science Bible: Test Preparation Kit Version 2.2 John Gatsis University

The Math and Science Bible:Test Preparation Kit

Version 2.2

John GatsisUniversity of Toronto

[email protected]

January 9, 2007

John Gatsis, [email protected] and Science Tutor

Note to student:

I am a Ph.D. candidate at the University of Toronto Institute for AerospaceStudies and a long-time math and science tutor for high school and univer-sity students. These questions are designed to help you practice for yourhigh school math, physics, and chemistry exams as well as your universitycalculus and linear algebra exams. I have designed most of these questionsand have based some on actual high school and university exam questions.If you are proficient at completing these questions, then you should be ingood shape when facing most evaluations. If you have any suggestions foradditional questions or wish to report any corrections, please email me.

Good Luck!

John Gatsis, BASc, MASc

To download a newer version of this document, please visit:

http://oddjob.utias.utoronto.ca/john

To contact the author, please write to:

[email protected]

To buy software for learning pre-calculus topics visit:

www.connectwithmath.com

1

Contents

1 Pre-Calculus: Selected Topics 51.1 Basic Math Operations . . . . . . . . . . . . . . . . . . . . . . 51.2 Solving Equations With One Variable . . . . . . . . . . . . . . 51.3 The x− y Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Graphing Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . 51.6 Factoring Special Forms and Grouping . . . . . . . . . . . . . 51.7 Completing the Square . . . . . . . . . . . . . . . . . . . . . . 61.8 Equations of Parabolas and Circles . . . . . . . . . . . . . . . 6

2 Calculus: More Algebra 72.1 Polynomial Long Division . . . . . . . . . . . . . . . . . . . . 72.2 Factor and Remainder Theorems . . . . . . . . . . . . . . . . 72.3 Polynomial Equations and Inequalities . . . . . . . . . . . . . 8

3 Calculus: Limits 93.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Calculus: Differentiation 154.1 First Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.6 Multiple Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Higher-Order Derivatives . . . . . . . . . . . . . . . . . . . . . 17

2


4.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . 18

5 Calculus: Applications 195.1 Tangents and Normals . . . . . . . . . . . . . . . . . . . . . . 195.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Calculus: Transcendental Functions and Differentiation 216.1 Exponentials and Logarithms . . . . . . . . . . . . . . . . . . 216.2 Exponential and Logarithmic Equations . . . . . . . . . . . . 216.3 Differentiation of Exponential and

Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 236.4 Logarithmic Differentiation . . . . . . . . . . . . . . . . . . . . 246.5 Differentiation of Trigonometric Functions . . . . . . . . . . . 256.6 Differentiation of Inverse Trigonometric Functions . . . . . . . 266.7 Differentiation of Hyperbolic

Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 27

7 Calculus: Integration 297.1 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.5 Integration by Substitution . . . . . . . . . . . . . . . . . . . . 337.6 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 357.7 Trigonometric Powers . . . . . . . . . . . . . . . . . . . . . . . 367.8 Integrals with Trigonometric Substitutions . . . . . . . . . . . 387.9 Integrals with Partial Fractions . . . . . . . . . . . . . . . . . 397.10 Integrals with Rationalizing Substitutions . . . . . . . . . . . 40

8 Linear Algebra: Matrices 418.1 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . 418.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 428.3 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . 428.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 43

3


9 Linear Algebra: Geometric Applications 449.1 Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . 449.2 Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . 449.3 Equations of Planes . . . . . . . . . . . . . . . . . . . . . . . . 449.4 Intersection of Lines . . . . . . . . . . . . . . . . . . . . . . . 449.5 Intersection of Planes . . . . . . . . . . . . . . . . . . . . . . . 449.6 Intersection of Lines and Planes . . . . . . . . . . . . . . . . . 44

10 Linear Algebra: Vector Spaces 4510.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.2 Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3 Spanning Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4 Dimension and Rank . . . . . . . . . . . . . . . . . . . . . . . 4510.5 Null and Image Spaces . . . . . . . . . . . . . . . . . . . . . . 45

11 Statistics: General 4611.1 Single-Variable Population and Sample Data . . . . . . . . . . 4611.2 Two-Variable Population and Sample Data . . . . . . . . . . . 4611.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4 Probability Distributions . . . . . . . . . . . . . . . . . . . . . 46

12 Chemistry: Introduction 4712.1 Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.3 Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . 4712.4 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . 4712.5 Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.6 Acids and Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.7 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.8 Oxidation and Reduction . . . . . . . . . . . . . . . . . . . . . 4812.9 Organic Chemistry Nomenclature . . . . . . . . . . . . . . . . 4812.10Organic Chemistry Applications . . . . . . . . . . . . . . . . . 48

13 Solutions to Selected Problems 49

4

Chapter 1

Pre-Calculus: Selected Topics

1.1 Basic Math Operations

1.2 Solving Equations With One Variable

1.3 The x− y Plane

1.4 Graphing Lines

1.5 Solving Linear Equations

1.6 Factoring Special Forms and Grouping

1. Factor:16x4 − 625

2. Factor:27x3 + 64

3. Factor:6x6 + 9x5 − 4x− 6

4. Factor:x3 − 125

5. Factor:8x3 + 12x2 + 2x + 3

5


6. Factor:512− 125x12

1.7 Completing the Square

1.8 Equations of Parabolas and Circles

6

Chapter 2

Calculus: More Algebra

2.1 Polynomial Long Division

1. Evaluate using long division:

x3 + x2 − 5x + 3x + 3


x3 + x2 − 5x + 3x− 2


x4 − 2x2 + x + 1x2 − 2x + 1

2.2 Factor and Remainder Theorems

1. Is 2x−1 a factor of 6x3+x2−12x+5? Show using (i) the factor/remaindertheorem and (ii) long division.

2. Is 2x + 3 a factor of 6x4 + 23x3 + 7x2 − 27x− 9?

3. The graph of f(x) = 3x4 + 14x3 + px2 + qx + 24 has x-intercepts x = −4and x = 2. Determine the function.

7


4. Write the equation for the quartic function with roots 2 + 3i and 1 − 2iand whose graph passes through (−2, 325).

5. One root of 3x3 − 15x2 + kx− 4 = 0 is 2. Find k and the other roots.

6. Two factors of the polynomial 4x4 + 24x3 + kx2 + 4x− 15 are x + 1 andx + 3. What is the value of k?

7. Factor: 6x4 − 19x3 − 2x2 + 44x− 24

2.3 Polynomial Equations and Inequalities

1. Solve 2x4 − x3 = 8x2 − x− 6 for x.

2. Solve 2x4 + 5x3 + 7x2 + 7x + 3 < 0 for x.

3. The functions f(x) = −7x2 + 15x and g(x) = 9− x3 intersect. For whatvalues of x is f ≥ g? If these functions did not intersect, would this questionbe more difficult? Explain.

8

Chapter 3

Calculus: Limits

3.1 Limits

1. Evaluate:

limx→1

(1

1 + x

)2. Evaluate:

limx→1

x2 − 1x− 1

3. Evaluate:

limx→1

x3 − 1x− 1

4. Evaluate:

limx→3

x− 36− 2x

5. Evaluate:

limx→1

√x2 + 1−

√2

x− 16. Evaluate:

limx→2

x2 − 3x + 2x− 2

7. Evaluate:lim

x→−1

√x

8. Evaluate:

limx→−2

|x|x

9


9. Evaluate:

limx→9

x− 3√x− 3

10. Evaluate:

limx→3+

x + 3

x2 − 7x + 1211. Evaluate:

limx→0

x(1 +

1

x2

)12. Evaluate:

limx→0

sin 3x

2x

13. Evaluate:

limx→0

4x

tan 7x

14. Evaluate:

limx→1

x5 − 1x4 − 1

15. Evaluate:

limx→0

x2 + 1

x− 116. Evaluate:

limx→5

2− x2

4x

17. Evaluate:

limx→1

x2 − 1x3 − 1

18. Evaluate:

limx→0

x2

x2 + 1

19. Evaluate:

limx→∞

x2 + 2x + 1

x + 1

20. Evaluate:

limx→−∞

x + 1

x2 + 2x + 1

21. Evaluate:

limx→∞

sin x

x

10


22. Evaluate:

limx→∞

(sin x

x

)π23. Evaluate:

limx→0

sin x2

x2

24. Evaluate:

limx→π

4

1− cos xx

25. Evaluate:limx→0

x csc x

26. Evaluate:

limx→0

1− cos 4x9x2

27. Evaluate:

limx→0

sin x

x2

28. Evaluate:

limx→π

4

sin x

x

29. Evaluate:

limx→0

2x2 + x

sin x30. Evaluate:

limx→π

6

sin(x + π

3

)− 1

x− π6

31. Evaluate:

limx→π

4

sin(x + π

4

)− 1

x− π4

32. Evaluate:limx→0

|x|

33. Evaluate:

limx→0

3x2 + 7x3

x2 + 5x4

34. Evaluate:

limx→0

2 sin3 x

(5x)3

11


3.2 Asymptotes

1. Identify any vertical, horizontal and oblique asymptotes for:

f(x) =x2 + 2x + 1

x + 1


f(x) =2x + 1

3x− 1


f(x) =1

x + 2


f(x) = x2 − 3x− 4


f(x) =x3 + 3x + 5

x2 + 1

3.3 Continuity

For each of the following functions, determine whether or not the function iscontinuous at the indicated point. If not, describe the discontinuity.

1. Point: x = 3f(x) = x2 − 5x + 1

2. Point: x = −1

f(x) =

√x2 + 1

x− 13. Point: x = 5

f(x) = |x2 − 25|4. Point: x = 2

f(x) =

{x2 + 4, if x < 2x3, if x ≥ 2

12


5. Point: x = 2

f(x) =

x2 + 5, if x < 210, if x = 21 + x3, if x > 2

6. Point: x = −1f(x) =

{1

1+x, if x 6= −1

0, if x = −1For each of the following functions, sketch the graph and describe any dis-continuities.

8.

f(x) =x2 − 4x− 2

9.f(x) = |x− 3|

10.f(x) = |x2 − 9|

11.

f(x) =x− 4x−16

12.

f(x) =

{−x2, if x < 01−

√x, if x ≥ 0

13.

f(x) =

x− 1, if x < 10, if x = 1x2, if x > 1

14.

f(x) =

−1, if x < −1x3, if − 1 ≤ x ≤ 11, if x > 1

15.

f(x) =

1, if x ≤ −2x2, if − 2 < x < 4√x, if x ≥ 4

13


16.

f(x) =

2x + 9, if x < −2x2 + 1, if − 2 < x ≤ 13x− 1, if 1 < x < 3x + 6, if x > 3

17. Determine A for the following function to be continuous at x = 1

f(x) =

{x2, if x < 1Ax− 3, if x ≥ 1

18. Determine A and B for the following function to be continuous at x = 1,but discontinuous at x = 2

f(x) =

Ax−B, if x ≤ 13x, if 1 < x < 2Bx2 − A, if x ≥ 2

14

Chapter 4

Calculus: Differentiation

4.1 First Principles

1. Differentiate using first principles:

f(x) = 6x2 + 2x− 1

2. Differentiate using first principles:

f(x) =1

x− 33. Differentiate using first principles:

f(x) =√

x + 1

4. Calculate f ′(1) using first principles given:

f(x) =1√x

5. Calculate f ′(2) using first principles given:

f(x) =√

x + 7

4.2 Power Rule

1. Differentiate:f(x) = 5x2 − 7x− 3

15


2. Differentiate:

f(x) = −121x3 − 2x2 + 4x− 11x−2

3. Differentiate:f(x) = π +

√3

4. Differentiate:f(x) = xπ − x

√2

4.3 Product Rule

1. Differentiate:f(x) = (x2 + 1)(x3 − 2)

2. Differentiate:f(x) = (x2 − 2x + 3)(3x− 2)

3. Differentiate:f(x) = (x100 + 12)(x2 + x + 1)

4.4 Quotient Rule

1. Differentiate:

f(x) =x3 + 1

x + 10

2. Differentiate:

f(x) =3x− 1

x2

3. Differentiate:

f(x) =4x2 − 3x + 2

x + 5

4.5 Chain Rule

1. Differentiate:f(x) =

(x5 − 4x3 + 2x + 55

)52. Differentiate:

f(x) =(x3 + x2 + (x2 + 1)3 + 1

)216


3. Differentiate:

f(x) =((x + 1)π + (x2 − 1)2 + (x3 − 1)

√2)π2

4.6 Multiple Rules

1. Differentiate:

f(x) =

(x3 − 4x + 5

9x− 2

)32. Differentiate:

f(x) =(x + 1)(2x2 − 1)

(x− 3)2

3. Differentiate:

f(x) =√

5x + 3√

2x + 1− π4. Differentiate:

f(x) = (x2 + 1)(√

x + 1)(x3 + 1)

4.7 Higher-Order Derivatives

1. Find the second derivative of:

f(x) = x7 + 12x2 − 17x + 5


f(x) =√

1− x3 + x + 1x− 1


f(x) =x4 − 1x− 1

4. Find the third derivative of:

f(x) = x3 − 9x + 1x− x−3

5. Consider the function f(x) = xn where n is a positive integer. What isits nth derivative? What is its (n + 1)st derivative?

17


4.8 Implicit Differentiation

1. Find dydx

given:x3 + x2y + xy2 + y4 − 12 = 0

2. Find dydx

given:x2 + 1

y2 + 1= 1

3. Find dydx

given:3√

x + 1 = 3√

y − 1

4. Find d2y

dx2given:

x2y + xy2 + xy + y3 = 8

18

Chapter 5

Calculus: Applications

5.1 Tangents and Normals

. Find the equation of the tangent at x = to the function:

.

. Find the equation of the normal at x = to the function:

.

. Find the equation of the tangent at (., .) to the relation:

.

5.2 Optimization

. open top box.

. surface area of a can.

..

19


5.3 Related Rates

. sphere.

. cone.

. cylinder 1 step.

. cylinder 2 step.

. shadow.

. distance north and west.

. swimmer.

. pipeline.

5.4 Curve Sketching

. quartic.

. 2 va 1 ha zero.

. cubic / quadratic, slanted asymptote

.

20

Chapter 6

Calculus: TranscendentalFunctions and Differentiation

6.1 Exponentials and Logarithms

1. Simplify:log5 25

2

2. Simplify:log 1

28

3. Simplify:log6 2 + log6 3

4. Simplify:log3 27

5. Simplify:

log3 54− log3 6 + log3(

1

3

)6. Simplify:

3 log2√

8− 2 log2 4

6.2 Exponential and Logarithmic Equations

1. Solve for x:log x = 3 log 4

21


2. Solve for x:4x+2 = 86

3. Solve for x:3(8)2x−1 = 30

4. Solve for x:5(6)x = 182

5. Solve for x:7x = 35



8. Solve for x:2 log2 x− log2(x− 2) = 3

9. Solve for x:9x+2 = 33x−3

10. Solve for x:43x+1 = 77

11. Solve for x:log2(x + 6)− log2(x− 3) = 3

12. Solve for x:log7(x

2 − 2) = 1

13. Solve for x:x = log2 5

14. Solve for x:log2 x

2 = log2 9 + log2 16

22


6.3 Differentiation of Exponential and

Logarithmic Functions

1. Differentiate:f(x) = e−2x

2. Differentiate:f(x) = e

√x+1

3. Differentiate:f(x) = ln(x2 − 3x + 1)

4. Differentiate:f(x) = ex ln x

5. Differentiate:f(x) = x2 − 2x

6. Differentiate:f(x) = e4 ln x

7. Differentiate:f(x) = log2 x

8. Differentiate:f(x) = log

√x4 − 12x− 7

9. Differentiate:f(x) = ex + ln x + 4x + log2 x + x

5

10. Differentiate:f(x) = 2x+5

11. Differentiate:

f(x) =e2x − 1e2x + 1

12. Differentiate:f(x) = ln(2x− ln(x2 − 1))

13. Differentiate:f(x) = 24x3ln x

14. Differentiate:f(x) = (ex − e−x)3

23


15. Differentiate:f(x) = 22

x

16. Differentiate:

f(x) =ln x

ex − 4x2−317. Differentiate:

f(x) = log7

(17− x

x + 1

)418. Consider the function f(x) = ekx where k is a real number. What is itsnth derivative? What is its (n + 1)st derivative?

6.4 Logarithmic Differentiation

1. Differentiate:

f(x) =(x + 1)2 (2x− 5)4 (3x + 1)3

(x2 − 2x− 2)5 (x3 + 1)3

2. Differentiate:

f(x) =(3x + x3)3 (ln x + ex)2

(x2 + 1)2 (log5 x + 5)3

3. Differentiate:f(x) = xx

4. Differentiate:f(x) = (ln x)ln x

5. Differentiate:f(x) = (ln x)x

2+1

6. Differentiate:f(x) = (x2 + 1)ln x

7. Differentiate:f(x) = xx

2

8. Differentiate:f(x) = (x + 1)x

24


6.5 Differentiation of Trigonometric Functions

1. Differentiate:f(x) = sin(x2 − 3)

2. Differentiate:f(x) = (cos x + sin x)3

3. Differentiate:

f(x) =cos x + 1

cos x− 14. Differentiate:

f(x) = tan(sin x + cos x)

5. Differentiate:f(x) = csc(x3 + 3x + 17)

6. Differentiate:f(x) = sec(2x)

7. Differentiate:f(x) = cot(ln x)

8. Differentiate:f(x) = sin(sin x)

9. Differentiate:f(x) = cos(x4 − 5x + 1)

10. Differentiate:f(x) = ln(tan(sin x3)))

11. Differentiate:f(x) = esin 2x

12. Differentiate:f(x) = sin(sin(x2 + 3x + 3))

13. Differentiate:f(x) = ln(cos e2x)

14. Differentiate:f(x) = sec(x2 + ex + 3x)

25


15. Differentiate:

f(x) =sin x + tan x

sec x + x2

16. Differentiate:f(x) = 8sin x

3

17. Differentiate:

f(x) =(sin2 x + 4x2)3 (csc x + sec x)4

(3x + ex)2 (log3 x + tan x)5

18. Differentiate:f(x) = (sin x)cos x

19. Differentiate:f(x) = (tan x)ln x


(√tan x

)x

6.6 Differentiation of Inverse Trigonometric

Functions

1. Differentiate:f(x) = sin−1 x

2. Differentiate:f(x) = cos−1 x

3. Differentiate:f(x) = tan−1 x

4. Differentiate:f(x) = csc−1 x

5. Differentiate:f(x) = sec−1 x

6. Differentiate:f(x) = cot−1 x

7. Differentiate:f(x) = sin−1(x3 − 2)

26


8. Differentiate:

f(x) =tan x

tan−1 x

9. Differentiate:f(x) = (x2 + sec−1 x)3 (ln x + ex)2

10. Differentiate:f(x) = tan−1

√x

11. Differentiate:f(x) = sin−1 x cos−1 x

12. Differentiate:f(x) = ln(tan−1 x)

13. Differentiate:f(x) = log3(sin x + sin

−1 x)

14. Differentiate:f(x) = ex sin−1 x

15. Differentiate:f(x) = esec

−1 x


(tan−1 x

)x

6.7 Differentiation of Hyperbolic

Trigonometric Functions

1. Differentiate:f(x) = sinh x

2. Differentiate:f(x) = cosh x

3. Differentiate:f(x) = sinh(ex + ln x)

4. Differentiate:f(x) = ln |1− cosh 2x|

27


5. Differentiate:f(x) = ex(cosh x + sinh x)

6. Differentiate:f(x) = e−x sinh 3x

7. Differentiate:

f(x) =sinh +1

sinh−18. Differentiate:

f(x) = tan−1(sinh x)

9. Differentiate:f(x) = xcosh x

10. Differentiate:f(x) = (sinh x)x

28

Chapter 7

Calculus: Integration

7.1 Antiderivatives

1. Find the antiderivate of:

f(x) = sin x


f(x) = cos x


f(x) = sec2 x


f(x) = csc x cot x


f(x) = sec x tan x


f(x) = csc2 x

7. Find the antiderivate of:f(x) = 1

29


8. Find the antiderivate of:f(x) = x

9. Find the antiderivate of:f(x) = x2


f(x) =1

x


f(x) =1

1 + x2


f(x) =1√

1− x2


f(x) = sinh x


f(x) = cosh x

7.2 Definite Integrals

1. Evaluate: ∫ 10

(x5 + x4 + x3 + x2 + x + 1) dx

2. Evaluate: ∫ 21

2− xx2

dx

3. Evaluate: ∫ π0

1

2cos x dx

4. Evaluate: ∫ π/3π/6

sec x tan x dx

30



− csc2 x dx

6. Evaluate: ∫ 40

3√

x dx

7. Evaluate: ∫ 41

dx

x

8. Evaluate: ∫ 52|x− 3| dx

9. Evaluate: ∫ 02

dx

(x + 1)2

7.3 Indefinite Integrals

1. Evaluate: ∫(x3 − 3x− 2) dx

2. Evaluate: ∫(x2 + 1)(x5 − 1) dx

3. Evaluate: ∫ x2 − 9x + 12x2

dx

4. Evaluate: ∫ 4x

dx

5. Evaluate: ∫ −71 + x2

dx

6. Evaluate: ∫cos 2x dx

7. Evaluate: ∫sin 3x dx

31


8. Evaluate: ∫ sin xcos2 x

dx

9. Evaluate: ∫ dx√1 + x

10. Evaluate: ∫(x− a)(x− b) dx

11. Evaluate: ∫(3 +

√x)(3−

√x) dx

12. Find the function f(x) that satisfies: f ′(x) = 2x− 1, f(3) = 4.

13. Find the function f(x) that satisfies: f ′(x) = cos x, f(π) = 3.

14. Find the function f(x) that satisfies: f ′′(x) = sin x, f ′(0) = −2, f(0) = 1.

15. Find the function f(x) that satisfies: f ′′(x) = 2x − 3, f(2) = −1,f(0) = 3.

7.4 Areas

1. Find the area between the function f(x) = x√

x+1 and the x-axis on thedomain x ∈ [1, 9].

2. Find the area enclosed by the function y = x and y = x2.

3. Find the area enclosed by y = sin x and the x-axis for the domainx ∈ [0, π].

4. Sketch the region bounded by these curves and find its area: y = 8,y = x2 + 2x.

5. Sketch the region bounded by these curves and find its area: y = 8− x2,y = x2.

32


6. Sketch the region bounded by these curves and find its area: y = x + 1,y = cos x, x = π.

7. Sketch the region bounded by these curves and find its area: y = sin x,y = πx− x2.

7.5 Integration by Substitution

1. Evaluate: ∫x(1 + x2)3 dx

2. Evaluate: ∫ x dx(4x2 + 9)2

3. Evaluate: ∫5x(x2 + 1)−3 dx

4. Evaluate: ∫x√

x + 1 dx

5. Evaluate: ∫(sin3 x + 1) cos x dx

6. Evaluate: ∫cos(3x− 1) dx

7. Evaluate: ∫csc2 πx dx

8. Evaluate: ∫x sec2 x2 dx

9. Evaluate: ∫ sin√x√x

dx

10. Evaluate: ∫ dxsin2 x

33


11. Evaluate: ∫csc(1− 2x) cot(1− 2x) dx

12. Evaluate: ∫ sin x√1 + cos x

dx

13. Evaluate: ∫x2 cos(3x3 − 11) dx

14. Evaluate: ∫ xn−1√a + bxn

dx

15. Evaluate: ∫ ln xx

dx

16. Evaluate: ∫ ex1 + e2x

dx

17. Evaluate: ∫(x + 1) cosh(x2 + 2x + 1) dx

18. Evaluate: ∫ex cosh(2− ex) dx

19. Evaluate: ∫axex dx

20. Evaluate: ∫ 10

2x dx

1 + x2


x + 3√x + 1

dx


x2√x + 1

dx

23. Evaluate: ∫ π0

x cos x2 dx

34



csc x cot x dx


x√x2 + 16

dx

26. Evaluate: ∫ 1/31/4

sec2 πx dx

27. Evaluate: ∫ 0−1

x3(x2 + 1)6 dx


√ln x

xdx

29. Evaluate: ∫ π/40

1 + sin x

cos2 xdx

30. Evaluate: ∫ 2−1

x

x2 + 4dx


tan−1 x

1 + x2dx

7.6 Integration by Parts

1. Evaluate: ∫ln x dx

2. Evaluate: ∫xex dx

3. Evaluate: ∫x2ex dx

4. Evaluate: ∫ 20

x2x dx

35


5. Evaluate: ∫ x2√1− x

dx

6. Evaluate: ∫ dxx(ln x)3

7. Evaluate: ∫ √x ln x dx

8. Evaluate: ∫ 10

ln(1 + x2) dx

9. Evaluate: ∫x3 sin x2 dx

10. Evaluate: ∫ 1/40

sin−1 2x dx

11. Evaluate using the substitution u =√

x:∫cos

√x dx

12. Evaluate: ∫cos(ln x) dx

13. Evaluate: ∫ 2e1

x2(ln x)2 dx

7.7 Trigonometric Powers

1. Evaluate: ∫sin3 x dx

2. Evaluate: ∫sin2 cos3 dx


cos 2x sin 3x dx

36




6. Evaluate: ∫sin 5x sin 2x dx

7. Evaluate: ∫ 1/20

cos πx cos(

π

2x)

dx

8. Evaluate: ∫ dx(x2 + 1)3

9. Evaluate: ∫(sin 3x− sin x)2 dx

10. Evaluate: ∫tan3 x dx

11. Evaluate: ∫csc2 2x dx

12. Evaluate: ∫tan2 x sec2 x dx

13. Evaluate: ∫cot2 x csc xdx

14. Evaluate: ∫ex tan2(ex) sec2(ex) dx

15. Evaluate: ∫cot4 x csc4 x dx


tan2 2x dx


cot2 x dx

37



tan3 x sec2 x dx

7.8 Integrals with Trigonometric Substitutions

1. Evaluate: ∫ dx(x2 + 1)3

2. Evaluate: ∫ dx((x + 1)2 + 1)2

3. Evaluate: ∫ dx((2x + 1)2 + 9)2

dx

4. Evaluate: ∫ dx(x2 + 2)

32

5. Evaluate: ∫ 20

x3√16− x2

dx

6. Evaluate: ∫ dx√x2 − 1

7. Evaluate: ∫ex√

e2x − 1 dx

8. Evaluate: ∫ 64

dx

x√

x2 − 49. Evaluate: ∫ dx

(x2 − 4x + 4) 3210. Evaluate: ∫ x

(x2 + 2x + 5)2dx

11. Evaluate: ∫ √6x− x2 − 8 dx

38


12. Evaluate: ∫x sin−1 x dx

7.9 Integrals with Partial Fractions

1. Evaluate: ∫ 7(x− 2)(x + 5)

dx

2. Evaluate: ∫ x(x + 1)(x + 2)(x + 3)

dx

3. Evaluate: ∫ x(x + 1)2

dx

4. Evaluate: ∫ 2x2 + 3x2(x− 1)

dx

5. Evaluate: ∫ xx3 − 1

dx

6. Evaluate: ∫ x3 + x2 + x + 3(x2 + 1)(x2 + 3)

dx

7. Evaluate: ∫ x + 1x3 + x2 − 6x

dx

8. Evaluate: ∫ 42

x4 − x3 − x− 1x3 − x2

dx

9. Evaluate: ∫ 20

x3

(x2 + 2)2dx

39


7.10 Integrals with Rationalizing Substitutions

1. Evaluate: ∫ dx1−

√x

2. Evaluate: ∫ √x1 + x

dx

3. Evaluate: ∫(x− 1)

√x + 2 dx

4. Evaluate: ∫ √x√x− 1

dx

5. Evaluate: ∫x(1 + x)

13 dx

6. Evaluate: ∫ 40

x32

x + 1dx

7. Evaluate: ∫ 10

√x

1 +√

xdx

8. Consider the substitution u = tan x2. Through a series of steps you can

show that: sin x = 2u1+u2

, cos x = 1−u2

1+u2, and dx = 2du

1+u2. Now evaluate:

∫ dxsin x + tan x

9. Evaluate: ∫ 1− cos x1 + sin x

dx


dx

sin x− cos x− 1

40

Chapter 8

Linear Algebra: Matrices

8.1 Basic Operations

1. Evaluate:

[1 22 3

]+

[-3 4-1 -5

]−[

5 -32 -2

]−[

2 -5-8 6

]

2. Evaluate:

[1 22 3

] [-3 4-1 -5

]

3. Evaluate:

[1 22 3

]3

4. Evaluate:

[-1 -53 7

] [4 2

-3 5

]

5. Evaluate:

1 9 -1-2 8 54 -2 -7

+ 5 6 64 -3 3

-4 8 -3

6. Evaluate:

-2 3-1 87 -5

− 6 65 -2

-2 3

41


7. Evaluate:

3 -1 -1-2 -2 51 2 -7

5 3 12 3 -7

-4 -1 3

[ ]

8.2 Linear Systems

1. Solve the linear system:

x + y = 2

x− y = 3

2. Solve the linear system:

[1 11 -1

∣∣∣∣∣ 23]

3. Compare your results from questions 1 and 2.

8.3 Elementary Operations

8.4 Determinants

1. Evaluate:

∣∣∣∣∣ 1 22 3∣∣∣∣∣

2. Evaluate:

∣∣∣∣∣∣∣0 -3 3

-2 0 -10 -6 -2

∣∣∣∣∣∣∣

42


3. Evaluate:

∣∣∣∣∣∣∣3 -1 -1

-2 -2 51 2 -7

∣∣∣∣∣∣∣

8.5 Eigenvalues and Eigenvectors

43

Chapter 9

Linear Algebra: GeometricApplications

9.1 Vector Operations

9.2 Equations of Lines

9.3 Equations of Planes

9.4 Intersection of Lines

9.5 Intersection of Planes

9.6 Intersection of Lines and Planes

44

Chapter 10

Linear Algebra: Vector Spaces

10.1 Vector Spaces

10.2 Subspace

10.3 Spanning Sets

10.4 Dimension and Rank

10.5 Null and Image Spaces

45

Chapter 11

Statistics: General

11.1 Single-Variable Population and Sample

Data

11.2 Two-Variable Population and Sample Data

11.3 Probability

11.4 Probability Distributions

46

Chapter 12

Chemistry: Introduction

12.1 Periodic Table

12.2 Nomenclature

12.3 Molecular Structure

12.4 Chemical Reactions

CH4 + 3O2 → 2CO2 + 2H2O

47


12.5 Stoichiometry

12.6 Acids and Bases

12.7 Heat

12.8 Oxidation and Reduction

12.9 Organic Chemistry Nomenclature

12.10 Organic Chemistry Applications

48

Chapter 13

Solutions to Selected Problems

Chapter 1

1.1.1 test

1.1.2 test

1.1.3 test

1.2.1 test

49

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The Math and Science Bible: Test Preparation Kit Version 2oddjob.utias.utoronto.ca/john/tutor/bible.pdfThe Math and Science Bible: Test Preparation Kit Version 2.2 John Gatsis University