Preface

Introduction to the Kentucky Edition:

This text is a minor adaptation of Professor Neal Koblitz’s notes from his University of Washington Math 124 course. Professor Koblitz generously permitted their use in this extension to Kentucky of his idea of a complete calculus text for which students pay only the reproduction costs. Errors which have been introduced into the text during the adaptation process are the responsibility of the Affordable Text Project of the University of Kentucky, edited by Paul Eakin, Carl Eberhart, and Avinash Sathaye.

This project has been supported by the National Science Foundation ( NSF) Calculus and Advanced Technical Education ( ATE) programs.

Prerequisites: Topics You Should Have Studied Before Taking Ma 113. Basic High School Algebra, Plane Geometry and Trigonometry are prerequisites to MA113. If you have taken a pre-calculus course (perhaps called advanced math or functions in high school or Ma 110 Analytic Geometry and Trigonometry here at UK), you are prepared to take this course.

The following are a few of the ideas from basic algebra and geometry which you will encounter in MA113. The list is far from complete. You should have previously studied and understood all of them. Although you may have forgotten the details of most you should be confident that with a very brief review you could re-establish your ability to solve problems with these ideas.

If you have not previously encountered a significant number of the following topics, then you may want to speak to your instructor or teaching assistant. They can discuss your background and preparation with you and help you make sure that you are in the right course.

• Cartesian coordinates, coordinates of points in the plane

• slope of a line,slope of a line perpendicular to a given line

• equation of a line through two points,equation of line given point and slope

• area of circle, rectangle, triangle; surface area of sphere, cylinder, rectangular box • volume of sphere, rectangular box, cylinder, cone

• circumference of circle, perimeter of rectangle, triangle

• quadratic formula,multiplication and ”long division” of polynomials

• addition and multiplication of rational functions

• factorization of x^{2} − y^{2}, x^{3} − y^{3}, x^{3} + y^{3}

• equation of a circle with given center and radius

• points of intersection of a circle and a line

• factor x^{2} − y^{2} and x^{3} − y^{3}, expand (a + b)^{2} and (a + b)^{3}

• absolute value, right triangle, similar triangles

• Pythagorean Theorem, law of sines, law of cosines, radian measure

You should have worked with and previously explored the graphs of functions like the following. You should be able to graph these on a graphics calculator or computer.

• x,x^{2},x^{3},x^{2} − 3x + 2,e^{x}, e^{−x}

• x^{3} − x, 1/x, 1/x^{2}, x + 1, x^{2} + 1, x^{3} + 1

• 1/(x + 1), 1/(x^{2} + 1),x − 1, (x − 1)^{2} + 2, (x^{3} − 1) − 3, 1/(x − 1) + 1

• 1/(x^{2} − 1) + 1, |x|,|x − 2|, log(x), ln(x),e^{−x^{2}}

• sin(x), cos(x), tan(x)

Advice for Students

The overwhelming emphasis in this course is on applications — story problems. To master story problems one needs a tremendous amount of practice — doing homework, worksheet, and practice exam problems. For this reason, this course will require much more of your time than would a high school or community college calculus course that does not have such a focus on word problems.

Here are some pointers for doing story problems:

1. Carefully read each problem twice before writing anything.

2. Assign letters to quantities that are described only in words; draw a diagram if appropriate.

3. Decide which letters are constants and which are variables. A letter stands for a constant if its value remains the same throughout the problem.

4. Using math notation, write down what you know and then write down what you want to find.

5. Decide what category of problem it is (this might be obvious if the problem comes at the end of a particular chapter, but will not necessarily be so obvious if it comes on an exam covering several chapters).

6. Double check each step as you go along; don’t wait until the end to check your work.

7. Use common sense; if an answer is out of the range of practical possibilities, then check your work to see where you went wrong.

Suggestions for Using These Notes

1. Read the example problems carefully, filling in any steps that are left out (ask someone if you can’t follow the solution to a worked example).

2. Later use the worked examples to study by covering the solutions, and seeing if you can solve the problems on your own.

3. Give yourself the practice midterms and finals in an exam situation (without access to material other than a single sheet of notes, and with the same time limit as on the exam).

4. The answers to homework and practice exam problems that are in the back of the notes should be used only as a final check on your work, not as a crutch. Keep in mind that sometimes an answer could be expressed in various ways that are algebraically equivalent, so don’t assume that your answer is wrong just because it doesn’t have exactly the same form as the answer in the back.

Formulas Used in This Course

Algebra Completing the square: X^{2} + bX + c = (X + \frac{b}{2})^{2} − \frac{b^{2}}{4} + c . Quadratic formula: roots of aX^{2} + bX + c are \frac{−b ± \sqrt{b^{2} − 4ac}}{2a} .

Exponents: a^{b} · a^{c} = a^{b+c}; \frac{a^{b}}{a^{c}} = a^{b−c}; (a^{b})^{c} = a^{bc}; a^{1/b} = \sqrt[b]{a} .

Geometry Circle: circumference = 2πr; area = πr^{2} . Sphere: vol = \frac{4}{3}πr^{3}; surface area = 4πr^{2} . Cylinder: vol = πr^{2}h; lateral area = 2πrh; total surface area = 2πrh + 2πr^{2}. Cone: vol = \frac{1}{3}πr^{2}h; lateral area = πr\sqrt{r^{2} + h^{2}}; total surface area = πr\sqrt{r^{2} + h^{2}} + πr^{2}. Analytic geometry Point-slope formula for straight line: y = y_{0} + m(x − x_{0}). Circle centered at (h, k): (x − h)^{2} + (y − k)^{2} = r^{2}. Ellipse with semimajor axis along x-axis and semiminor axis along y-axis: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1. Trigonometry sin = \frac{opposite}{hypotenuse}; cos = \frac{adjacent}{hypotenuse}; tan = \frac{opposite}{adjacent}; sec = \frac{1}{cos}; csc = \frac{1}{sin}; cot = \frac{1}{tan}; tan = \frac{sin}{cos}; cot = \frac{cos}{sin}; sin x = cos (\frac{π}{2} − x); cos x = sin (\frac{π}{2} − x); sin(x + π) = − sin x; cos(x + π) = − cos x. Law of cosines: a^{2} = b^{2} + c^{2} − 2bc cos A. Law of sines: \frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} . Sine of sum of angles: sin(x + y) = sin x cos y + cos x sin y. sin^{2} and cos^{2} formulas: sin^{2} + cos^{2} = 1; sin^{2} x = \frac{1 − cos(2x)}{2}; cos^{2} x = \frac{1 + cos(2x)}{2}.

iv PREFACE

Contents

Preface iii Introduction to the Kentucky Edition: iii Advice for Students iv

Chapter 1. Straight Lines, Polygonal Lines, Slopes, Distance, Circles 1 1. Distance Between Two Points; Circles 4

Chapter 2. Functions 9 1. Shifts and Expansions 13

Chapter 3. Instantaneous Rate Of Change: The Derivative 19

Chapter 4. The Derivative Function 25 1. Corners and Cusps 26 2. Graphical Differentiation 27

Chapter 5. The Tangent Line Approximation 31 1. The Power Rule 31 2. The Tangent Line Approximation 31

Chapter 6. Rules For Finding Derivatives 39 1. Linearity of the Derivative 39 2. The Product Rule 40 3. The Quotient Rule 41 4. The Second Derivative and Acceleration 43 5. The Chain Rule 45

Chapter 7. Implicit Functions and Implicit Differentiation 49 1. Implicit Functions 49 2. Implicit Differentiation 49

Chapter 8. Trig Functions and Sinusoidal Functions 55 1. Graphs of Trig Functions 56 2. Sinusoidal Functions 56

Chapter 9. Derivative of Trig Functions 63 1. The Derivative of sin x 63 2. Other Trig Derivatives 64

Chapter 10. Curves Given Parametrically 67 1. Trajectories 68 2. The Velocity Vector 70 Chapter 11. Combined Parametric Motions 79

Chapter 12. Related Rates 87

Chapter 13. Curve Sketching 95 1. Maxima/Minima 95 2. The Second Derivative Test 97 3. The First Derivative Test 97 4. Concavity and Inflection 97 5. Asymptotes and Other Things to Look For 98

Chapter 14. Maximum/Minimum Problems 105

Chapter 15. Newton’s Method 117

Chapter 16. Indefinite Integrals 123 1. From Acceleration to Velocity to Distance; Falling Bodies 124 2. antiderivative of sinx and cosx 126 3. Substitution 126

Chapter 17. Definite Integrals: Area 131 1. Riemann Sums 131 2. Computing the Area Under a Curve 132 3. Substitution 135 4. Some Properties of Definite Integrals 136 5. Application to Degree-Days 136 6. Velocity and Distance Traveled As Area 137

Chapter 18. The Fundamental Theorem 139 1. The Fundamental Theorem together with the Chain Rule 140

Lab Projects for Ma113 143

Practice First Midterms 151

Practice Second Midterms 155

Practice Final Exams 159

Answers 169 1. Answers to Homework Problems 169 2. Answers to Practice Midterms 172 3. Answers to Practice Final Exams 174

Index 186

168 CONTENTS