# 2 - 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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-- CHAPTER 2 Derivatives 2.1 The Derivative of a Function This chapter begins with the definition of the derivative. Two examples were in Chapter 1. When the distance is t2, the velocity is 2t. When f(t) = sin t we found v(t) = cos t. The velocity is now called the derivative off (t). As we move to a more formal definition and new examples, we use new symbols f' and dfldt for the derivative. 2A At time t, the derivative f '(t)or df /dt or v(t) is f '(t) = lim f Ct -t At) -f (0 (1) At+O The ratio on the right is the average velocity over a short time At. The derivative, on the left side, is its limit as the step At (delta t) approaches zero. Go slowly and look at each piece. The distance at time t + f (t + At). The distance at time t is f(t). Subtraction gives the change in distance, between those times. We often write A f for this difference: A f = + - f (t). The average velocity is the ratio AflAt-change in distance divided by change in time. The limit of the average velocity is the derivative, if this limit exists: df - lim -. Af dt At-0 This is the neat notation that Leibniz invented: Af/At approaches df /dt. Behind the innocent word "limit" is a process that this course will help you understand. Note that Af is not A times f! It is the change in f. Similarly is not A times t. It is the time step, positive or negative and eventually small. To have a one-letter symbol we replace At by h. The right sides of (1) and (2) contain average speeds. On the graph of f(t), the distance up is divided by the distance across. That gives the average slope Af /At. The left sides of (1) are instantaneous speeds dfldt. They give the slope at the instant This is the derivative (when At Af shrink to zero). Look again
---- 2.1 The Derivative of a Function at the calculation for f(t) = t 2: -- - f(t+At)-f(t) - t2+2tAt+(At)'-t2 Af - = 2t + At. At Important point: Those steps are taken before At goes to zero. If we set = 0 too soon, we learn nothing. The ratio Af/At becomes 010 (which is meaningless). The numbers Af and At must approach zero together, not separately. Here their ratio is 2t + At, the average speed. To repeat: Success came by writing out (t + At)2 and subtracting t2 and dividing by At. Then and only then can we approach At = 0. The limit is the derivative 2t.

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## This note was uploaded on 06/13/2009 for the course TAM 455 taught by Professor Petrina during the Fall '08 term at Cornell.

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

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