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# 3 - Contents CHAPTER 1 Introduction to Calculus Velocity...

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CHAPTER 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 CHAPTER 2 CHAPTER 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Contents Introduction to Calculus Velocity and Distance Calculus Without Limits The Velocity at an Instant Circular Motion A Review of Trigonometry A Thousand Points of Light Computing in Calculus Derivatives The Derivative of a Function Powers and Polynomials The Slope and the Tangent Line Derivative of the Sine and Cosine The Product and Quotient and Power Rules Limits Continuous Functions Applications of the Derivative Linear Approximation Maximum and Minimum Problems Second Derivatives: Minimum vs. Maximum Graphs Ellipses, Parabolas, and Hyperbolas Iterations x, + , = F(x,) Newton's Method and Chaos The Mean Value Theorem and l'H8pital's Rule

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C H A P T E R Applications of the Derivative Chapter 2 concentrated on computing derivatives. This chapter concentrates on using them. Our computations produced dyldx for functions built from xn and sin x and cos x. Knowing the slope, and if necessary also the second derivative, we can answer the questions about y = f(x) that this subject was created for: 1. How does y change when x changes? 2. What is the maximum value of y? Or the minimum? 3. How can you tell a maximum from a minimum, using derivatives? The information in dyldx is entirely local. It tells what is happening close to the point and nowhere else. In Chapter 2, Ax and Ay went to zero. Now we want to get them back. The local information explains the larger picture, because Ay is approximately dyldx times Ax. The problem is to connect the finite to the infinitesimal-the average slope to the instantaneous slope. Those slopes are close, and occasionally they are equal. Points of equality are assured by the Mean Value Theorem-which is the local-global connection at the center of differential calculus. But we cannot predict where dyldx equals AylAx. Therefore we now find other ways to recover a function from its derivatives-or to estimate distance from velocity and acceleration. It may seem surprising that we learn about y from dyldx. All our work has been going the other way! We struggled with y to squeeze out dyldx. Now we use dyldx to study y. That's life. Perhaps it really is life, to understand one generation from later generations. 3.1 Linear Approximation The book started with a straight line f = vt. The distance is linear when the velocity is constant. As soon as v begins to change, f = v t falls apart. Which velocity do we choose, when v(t) is not constant? The solution is to take very short time intervals, 91
3 Applications of the Derivative in which v is nearly constant: f = vt is completely false Af = vAt is nearly true df = vdt is exactly true. For a brief moment the functionf(t) is linear-and stays near its tangent line. In Section 2.3 we found the tangent line to y = f(x). At x = a, the slope of the curve and the slope of the line are f'(a). For points on the line, start at y = f(a). Add the slope times the "increment" x - a: Y = f(a) + f '(a)(x - a).

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3 - Contents CHAPTER 1 Introduction to Calculus Velocity...

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