mws_gen_dif_bck_primer - Chapter 02.01 Primer on...

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Chapter 02.01 Primer on Differentiation After reading this chapter, you should be able to: 1. understand the basics of differentiation, 2. relate the slopes of the secant line and tangent line to the derivative of a function, 3. find derivatives of polynomial, trigonometric and transcendental functions, 4. use rules of differentiation to differentiate functions, 5. find maxima and minima of a function, and 6. apply concepts of differentiation to real world problems. In this primer, we will review the concepts of differentiation you learned in calculus. Mostly those concepts are reviewed that are applicable in learning about numerical methods. These include the concepts of the secant line to learn about numerical differentiation of functions, the slope of a tangent line as a background to solving nonlinear equations using the Newton-Raphson method, finding maxima and minima of functions as a means of optimization, the use of the Taylor series to approximate functions, etc. Introduction The derivative of a function represents the rate of change of a variable with respect to another variable. For example, the velocity of a body is defined as the rate of change of the location of the body with respect to time. The location is the dependent variable while time is the independent variable. Now if we measure the rate of change of velocity with respect to time, we get the acceleration of the body. In this case, the velocity is the dependent variable while time is the independent variable. Whenever differentiation is introduced to a student, two concepts of the secant line and tangent line (Figure 1) are revisited. 02.01.1
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02.01.2 Chapter 02.01 Let and Q be two points on the curve as shown in Figure 1. The secant line is the straight line drawn through and . P P Q The slope of the secant line (Figure 2) is then given as a h a a f h a f m PQ + + = ) ( ) ( ) ( secant , x P Q a a+h ) ( x f Figure 2 Calculation of the secant line. Q P x f ( x ) secant line tangent line Figure 1 Function curve with tangent and secant lines.
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Primer on Differentiation 02.01.3 h a f h a f ) ( ) ( + = As Q moves closer and closer to P , the limiting portion is called the tangent line. The slope of the tangent line then is the limiting value of as . tangent , PQ m secant , PQ m 0 h h a f h a f m h PQ ) ( ) ( lim 0 tangent , + = Example 1 Find the slope of the secant line of the curve between points (3,36) and (5,100). 2 4 x y = 0 50 100 150 200 250 - 2 - 1012345678 x f(x) (5,100) (3,36) Figure 3 Calculation of the secant line for the function . 2 4 x y = Solution The slope of the secant line between (3,36) and (5,100) is 3 5 ) 3 ( ) 5 ( = f f m 3 5 36 100 = 32 = Example 2 Find the slope of the tangent line of the curve at point (3,36). 2 4 x y = Solution The slope of the tangent line at (3,36) is
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02.01.4 Chapter 02.01 h f h f m h ) 3 ( ) 3 ( lim 0 + = h h h 2 2 0 ) 3 ( 4 ) 3 ( 4 lim + = h h h h 36 ) 6 9 ( 4 lim 2 0 + + = h h h h 36 24 4 36 lim 2 0 + + = h h h h ) 24 4 ( lim 0 + = ) 24 4 ( lim 0 + = h h 24 = 0 10 20 30 40 50 60 70 -2 -1 0 1 2 3 4 5 x f(x) Figure 4 Calculation of the tangent line in the function .
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mws_gen_dif_bck_primer - Chapter 02.01 Primer on...

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