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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 161

Practice Second Midterms 167

Practice Final Exams 171

Answers 179 1. Answers to Homework Problems 179 2. Answers to Practice Midterms 182 3. Answers to Practice Final Exams 184

Index 186

Laboratory Calculus I: Lab Project Bouncing Ball

Date:

Team Members:

Introduction: The famous magician Michael Moshen has an act in which he stands in a large triangular sculpture and bounces balls off the sides, keeping them in motion. This is a look at the elementary mathematics of what he is doing.

The Context to be Investigated: Two plane surfaces, one parallel to the ground, meet at an angle θ = 30 degrees. A ball is thrown against the upper plane at an angle β = 5 degrees with a velocity of 20 feet per second. It hits 10 feet from the corner and five feet above the ground. The ball bounces elastically, with no spin or air resistance but is acted upon by gravity.

Develop a mathematical model for the path the ball follows with and without the influence of gravity. First assume that there is no gravity and calculate the path of the ball as it bounces between the planes. Vary the initial velocity, angle of entry, and angle made by the two planes. Can the ball be made to bounce any number of times?

Repeat the investigation with the influence of gravity. Compare the paths of the ball with and without gravity for different initial velocities, angles of entry, angles between the planes, etc.

Finally, repeat the investigation for a ball that is not perfectly elastic. Suppose the ball always rebounds from a surface with the component of its velocity normal to the surface having only 75% of its magnitude prior to impact.