ࡱ>  b}n{ bjbjzz ;hHP $$%%%%%%8%(T%>:*\8B(`B`BC. 4 $x* 9%#N##$$`BCW˷˷˷#$>`Br%C˷#˷˷b".qCrɞ} \0> Jߤ JqJ%q@"b˷z@@@ٮ@@@>####J@@@@@@@@@ :  Chapter 02.01 Primer on Differentiation After reading this chapter, you should be able to: understand the basics of differentiation, relate the slopes of the secant line and tangent line to the derivative of a function, find derivatives of polynomial, trigonometric and transcendental functions, use rules of differentiation to differentiate functions, find maxima and minima of a function, and 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.  SHAPE \* MERGEFORMAT  Let  EMBED Equation.3  and  EMBED Equation.3  be two points on the curve as shown in Figure 1. The secant line is the straight line drawn through  EMBED Equation.3  and  EMBED Equation.3 .  SHAPE \* MERGEFORMAT  The slope of the secant line (Figure 2) is then given as  EMBED Equation.3   EMBED Equation.3  As  EMBED Equation.3  moves closer and closer to  EMBED Equation.3 , the limiting portion is called the tangent line. The slope of the tangent line  EMBED Equation.3  then is the limiting value of  EMBED Equation.3  as  EMBED Equation.3 .  EMBED Equation.3  Example 1 Find the slope of the secant line of the curve  EMBED Equation.3  between points (3,36) and (5,100). Figure 3 Calculation of the secant line for the function  EMBED Equation.3 . Solution The slope of the secant line between (3,36) and (5,100) is  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 2 Find the slope of the tangent line of the curve  EMBED Equation.3  at point (3,36). Solution The slope of the tangent line at (3,36) is  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   Figure 4 Calculation of the tangent line in the function  EMBED Equation.3 . The slope of the tangent line is       Derivative of a Function Recall from calculus, the derivative of a function  EMBED Equation.3  at  EMBED Equation.3  is defined as  EMBED Equation.3  Example 3 Find  EMBED Equation.3  if  EMBED Equation.3 . Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 4 Find  EMBED Equation.3  if  EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  from knowing that  EMBED Equation.3  Second Definition of Derivatives There is another form of the definition of the derivative of a function. The derivative of the function  EMBED Equation.3  at  EMBED Equation.3  is defined as  EMBED Equation.3  As  EMBED Equation.3 , the definition is nothing but the slope of the tangent line at  EMBED Equation.3 .  SHAPE \* MERGEFORMAT  Example 5 Find  EMBED Equation.3  if  EMBED Equation.3  by using the form  EMBED Equation.3  of the definition of a derivative. Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Finding equations of a tangent line One of the numerical methods used to solve a nonlinear equation is called the Newton-Raphson method. This method is based on the knowledge of finding the tangent line to a curve at a point. Let us look at an example to illustrate finding the equation of the tangent line to a curve. Example 6 Find the equation of the line tangent to the function  EMBED Equation.3  at  EMBED Equation.3 . Solution The line tangent is a straight line of the form  EMBED Equation.3  To find the equation of the tangent line, let us first find the slope  EMBED Equation.3  of the straight line.   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  To find the value of the  EMBED Equation.3 -intercept  EMBED Equation.3  of the straight line, we first find the value of the function at  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  The tangent line passes through the point  EMBED Equation.3 , so  EMBED Equation.3   EMBED Equation.3   Figure 6 Graph of function f(x) and the tangent line at x = 0.05. Hence,  EMBED Equation.3   EMBED Equation.3  is the equation of the tangent line. Other Notations of Derivatives Derivates can be denoted in several ways. For the first derivative, the notations are  EMBED Equation.3  For the second derivative, the notations are  EMBED Equation.3  For the  EMBED Equation.3  derivative, the notations are  EMBED Equation.3  Theorems of Differentiation Several theorems of differentiation are given to show how one can find the derivative of different functions. Theorem 1 The derivative of a constant is zero. If  EMBED Equation.3 , where  EMBED Equation.3  is a constant,  EMBED Equation.3 . Example 7 Find the derivative of  EMBED Equation.3 . Solution  EMBED Equation.3   EMBED Equation.3  Theorem 2 The derivative of  EMBED Equation.3 , where  EMBED Equation.3  is  EMBED Equation.3 . Example 8 Find the derivative of  EMBED Equation.3 . Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 9 Find the derivative of  EMBED Equation.3 . Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Theorem 3 The derivative of  EMBED Equation.3 , where  EMBED Equation.3  is a constant is  EMBED Equation.3 . Example 10 Find the derivative of EMBED Equation.3 . Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Theorem 4 The derivative of  EMBED Equation.3  is  EMBED Equation.3 . Example 11 Find the derivative of  EMBED Equation.3 . Solution       Theorem 5 The derivative of  EMBED Equation.3  is  EMBED Equation.3 . (Product Rule) Example 12 Find the derivative of  EMBED Equation.3  Solution Using the product rule as given by Theorem 5 where,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Taking the derivative of  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Taking the derivative of  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Using the formula for the product rule  EMBED Equation.3   EMBED Equation.3  Theorem 6 The derivative of  EMBED Equation.3  is  EMBED Equation.3  (Quotient Rule) Example 13 Find the derivative of  EMBED Equation.3 . Solution Use the quotient rule of Theorem 6, if  EMBED Equation.3  then  EMBED Equation.3  From  EMBED Equation.3  we have  EMBED Equation.3   EMBED Equation.3  Taking the derivative of  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Taking the derivative of  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Using the formula for the quotient rule,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Table of Derivatives ,  EMBED Equation.3  EMBED Equation.3  Chain Rule of Differentiation Sometimes functions that need to be differentiated do not fall in the form of simple functions or the forms described previously. Such functions can be differentiated using the chain rule if they are of the form  EMBED Equation.3 . The chain rule states  EMBED Equation.3  For example, to find  EMBED Equation.3  of  EMBED Equation.3 , one could use the chain rule.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Implicit Differentiation Sometimes, the function to be differentiated is not given explicitly as an expression of the independent variable. In such cases, how do we find the derivatives? We will discuss this via examples. Example 14 Find  EMBED Equation.3  if  EMBED Equation.3  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 15 If  EMBED Equation.3 , find the value of  EMBED Equation.3 . Solution        Higher order derivatives So far, we have limited our discussion to calculating first derivative,  EMBED Equation.3  of a function  EMBED Equation.3 . What if we are asked to calculate higher order derivatives of  EMBED Equation.3 . A simple example of this is finding acceleration of a body from a function that gives the location of the body as a function of time. The derivative of the location with respect to time is the velocity of the body, followed by the derivative of velocity with respect to time being the acceleration. Hence, the second derivative of the location function gives the acceleration function of the body. Example 16 Given EMBED Equation.3 , find the second derivative,  EMBED Equation.3  and the third derivative, EMBED Equation.3 . Solution Given  EMBED Equation.3  we have  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 17 If  EMBED Equation.3 , find the value of  EMBED Equation.3 . Solution From Example 15 we obtain  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  After substitution of  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3  Finding maximum and minimum of a function The knowledge of first derivative and second derivative of a function is used to find the minimum and maximum of a function. First, let us define what the maximum and minimum of a function are. Let  EMBED Equation.3  be a function in domain  EMBED Equation.3 , then  EMBED Equation.3  is the maximum of the function if  EMBED Equation.3  for all values of  EMBED Equation.3  in the domain  EMBED Equation.3 .  EMBED Equation.3  is the minimum of the function if  EMBED Equation.3  for all values of  EMBED Equation.3  in the domain  EMBED Equation.3 . The minimum and maximum of a function are also the critical values of a function. An extreme value can occur in the interval  EMBED Equation.3  at end points  EMBED Equation.3 . a point in  EMBED Equation.3  where  EMBED Equation.3 . a point in  EMBED Equation.3  where  EMBED Equation.3  does not exist. These critical points can be the local maximas and minimas of the function (See Figure 8). Example 18 Find the minimum and maximum value of  EMBED Equation.3  in the interval  EMBED Equation.3 .  SHAPE \* MERGEFORMAT  SHAPE \* MERGEFORMAT  Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  at  EMBED Equation.3 .  EMBED Equation.3  exists everywhere in  EMBED Equation.3 . So the critical points are  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Hence, the minimum value of  EMBED Equation.3  occurs at  EMBED Equation.3 , and the maximum value occurs at  EMBED Equation.3 .  SHAPE \* MERGEFORMAT Figure 10 shows an example of a function that has no minimum or maximum value in the domain EMBED Equation.3 .  SHAPE \* MERGEFORMAT Figure 11 shows the maximum of the function occurring at a singular point. The function  EMBED Equation.3  has a sharp corner at  EMBED Equation.3 .  SHAPE \* MERGEFORMAT Example 19 Find the maximum and minimum of  EMBED Equation.3  in the interval  EMBED Equation.3 . Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  on  EMBED Equation.3 . So the critical points are  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  So the minimum value of  EMBED Equation.3  is at  EMBED Equation.3 , and the maximum value is at  EMBED Equation.3 . The point(s) where the second derivative of a function becomes zero is a way to know whether the critical point found in the first derivative test is a local minimum or maximum. Let  EMBED Equation.3  be a function in the interval  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3  is a local maximum of the function if  EMBED Equation.3 .  EMBED Equation.3  is a local minimum of the function if  EMBED Equation.3 . If  EMBED Equation.3 , then the second derivative does not offer any insight into the local maxima or minima. Example 20 Remember Example 18 where we found  EMBED Equation.3 at  EMBED Equation.3  for  EMBED Equation.3  in the interval  EMBED Equation.3 . Is  EMBED Equation.3  a local maxima or minima of the function? Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  at  EMBED Equation.3   EMBED Equation.3  So the  EMBED Equation.3  is the local minimum of the function. Applications of Derivatives Below are some examples to show real-life applications of differentiation. Example 21 A rain gutter cross-section is shown below.  SHAPE \* MERGEFORMAT  What angle of  EMBED Equation.3  would make the cross-sectional area of ABCD maximum? Note that common sense or intuition may lead us to believe that  EMBED Equation.3  would maximize the cross-sectional area of ABCD. Question your intuition. Solution  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  When is  EMBED Equation.3 ?  EMBED Equation.3   EMBED Equation.3  The angle at which the area is maximum is  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  For the interval of  EMBED Equation.3 , the area at the end points is  EMBED Equation.3  Example 22 A classic example of the application of differentiation is to find the dimensions of a circular cylinder for a specific volume but which uses the least amount of material. Do this classic problem for a volume of  EMBED Equation.3 . Solution The total surface area,  EMBED Equation.3  of the cylinder is  EMBED Equation.3  = top surface + side surface + bottom surface  EMBED Equation.3   EMBED Equation.3  The volume,  EMBED Equation.3  of the cylinder is  EMBED Equation.3  since  EMBED Equation.3 . We can write  EMBED Equation.3  This gives the surface area just in terms of  EMBED Equation.3  as  EMBED Equation.3   EMBED Equation.3   SHAPE \* MERGEFORMAT To find the minimum, take the first derivative of  EMBED Equation.3  with respect to  EMBED Equation.3  as  EMBED Equation.3   EMBED Equation.3  Solving for  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Since  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3  But does this value of  EMBED Equation.3  correspond to a minimum?  EMBED Equation.3   EMBED Equation.3  This value  EMBED Equation.3  for  EMBED Equation.3 . As per the second derivative test,  EMBED Equation.3  corresponds to a minimum. DIFFERENTIATIONTopicPrimer on DifferentiationSummaryThese are textbook notes of a primer on differentiationMajorGeneral EngineeringAuthorsAutar Kaw, Luke SnyderDateJuly 17, 2008 Web Site HYPERLINK "http://numericalmethods.eng.usf.edu" http://numericalmethods.eng.usf.edu     02.01. PAGE 24 Chapter 02.01 Primer on Differentiation 02.01. PAGE 23 02.01.1 Q P f(x) EMBED Equation.3 Q   EMBED Equation.3  secant line tangent line Figure 1 Function curve with tangent and secant lines. x P Q a a+h  EMBED Equation.3  Figure 2 Calculation of the secant line.  EMBED Equation.3  P Q  EMBED Equation.3  a  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x Figure 5 Graph showing the second definition of the derivative. x maximum minimum  EMBED Equation.3  x Figure 7 Graph illustrating the concepts of maximum and minimum. Domain = [c,d] c d f(x) % % % % % % % x Figure 8 The plot shows critical points of  EMBED Equation.3 in  EMBED Equation.3  . 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