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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: Project

Forces on a Piston

Date:

Team Members:

The problem: In an internal combustion engine, each piston is housed in a cylinder and is attached to the crankshaft by a connecting rod as indicated in the diagram below. The piston moves back and forth in the cylinder, and, in response the crankshaft rotates. When the crankshaft is turning, even at a constant angular velocity the (linear) velocity and speed of the piston are constantly changing. Investigate the effect of the length of the connecting rod on the velocity and acceleration of the piston.

1. Suppose the crank is 1.5 inches in radius and the connecting rod is 6 inches long and the crank is turning at a constant 1 revolution per second. Assume the piston is moving on the x-axis and express the position of the point at which the rod connects to the piston (the wrist pin) as a function of t (in seconds).

Hint: Let r denote the crank radius, l the length of the connecting rod, θ(t) the crank angle at time t and assume θ(0) = 0 and that the rotation is counterclockwise. We can place the center of the crank at the origin. Then if x(t) is the position of the wrist pin at time t, the law of cosines says that

l(t)^{2} = r^{2} + x(t)^{2} − 1rx(t) cos(θ(t)).

Solve for x(t) and choose one of the solutions (e.g. the one for which x(t) = r + l when t = 0).

2. Calculate expressions for the position, velocity, and acceleration of the wrist pin as a function of time and plot all three on the same graph for t from 0 to 2 seconds. At what times are the velocity and acceleration maximal and minimal? What are the maximum velocity and acceleration?

3. Repeat the above calculations for various lengths of the connecting rod from 2 to 8 and observe the effect of these changes on the velocity and acceleration of the wrist pin. Note particularly the change in shape of the acceleration curve and formulate a conjecture as to length of the connecting rod for which marks the transition from one shape to the other. Develop additional evidence for your conjecture through additional experiments.