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Chapter 3 - 5E-03(pp 126-135 1:49 PM Page 126 CHAPTER 3 By...

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Derivatives C H A P T E R 3 By measuring slopes at points on the sine curve, we get strong visual evidence that the derivative of the sine function is the cosine function. 5E-03(pp 126-135) 1/17/06 1:49 PM Page 126
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In this chapter we begin our study of differential calculus, which is concerned with how one quantity changes in rela- tion to another quantity. The central concept of differential calculus is the derivative , which is an outgrowth of the velocities and slopes of tangents that we considered in Chapter 2. After learning how to calculate derivatives, we use them to solve prob- lems involving rates of change and the approximation of functions. |||| 3.1 Derivatives In Section 2.6 we defined the slope of the tangent to a curve with equation at the point where to be We also saw that the velocity of an object with position function at time is In fact, limits of the form arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. Definition The derivative of a function at a number , denoted by , is if this limit exists. If we write , then and approaches if and only if approaches . Therefore, an equivalent way of stating the definition of the derivative, as we saw in finding tangent lines, is f a lim x l a f x f a x a 3 a x 0 h h x a x a h f a lim h l 0 f a h f a h f a a f 2 lim h l 0 f a h f a h v a lim h l 0 f a h f a h t a s f t m lim h l 0 f a h f a h 1 x a y f x 127 |||| is read “ prime of .” a f f a 5E-03(pp 126-135) 1/17/06 1:49 PM Page 127
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128 ❙❙❙❙ CHAPTER 3 DERIVATIVES EXAMPLE 1 Find the derivative of the function at the number . SOLUTION From Definition 2 we have Interpretation of the Derivative as the Slope of a Tangent In Section 2.6 we defined the tangent line to the curve at the point to be the line that passes through and has slope given by Equation 1. Since, by Defini- tion 2, this is the same as the derivative , we can now say the following. The tangent line to at is the line through whose slope is equal to , the derivative of at . Thus, the geometric interpretation of a derivative [as defined by either (2) or (3)] is as shown in Figure 1. If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve at the point : EXAMPLE 2 Find an equation of the tangent line to the parabola at the point . SOLUTION From Example 1 we know that the derivative of at the number is . Therefore, the slope of the tangent line at is 3, 6 f a 2 a 8 a f x x 2 8 x 9 3, 6 y x 2 8 x 9 y f a f a x a a , f a y f x FIGURE 1 Geometric interpretation of the derivative 0 x y y=ƒ ƒ-f(a) x-a x a P (b) fª(a)= lim = slope of tangent at P x = a ƒ-f(a) x-a 0 x y y=ƒ f(a+h)-f(a) h a+h a P h = 0 (a) fª(a)= lim = slope of tangent at P f(a+h)-f(a) h = slope of curve at P = slope of curve at P a f f a a , f a a , f a y f x f a m P P a , f a y f x 2 a 8 lim h l 0 2 ah h 2 8 h h lim h l 0 2 a h 8 lim h l 0 a 2 2 ah h 2 8 a 8 h 9 a 2 8 a 9 h lim h l 0 a h 2 8 a h 9 a 2 8 a 9 h f a
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