APPLICATIONS OF DIFFERENTIATION

Transcription

1 4 APPLICATIONS OF DIFFERENTIATION

2 APPLICATIONS OF DIFFERENTIATION 4.8 Newton s Method In this section, we will learn: How to solve high degree equations using Newton s method.

3 INTRODUCTION Suppose that a car dealer offers to sell you a car for $18,000 or for payments of $375 per month for five years. You would like to know what monthly interest rate the dealer is, in effect, charging you.

4 INTRODUCTION Equation 1 To find the answer, you have to solve the equation 48(1 + ) 60 - (1 + ) = 0 How would you solve such an equation?

5 HIGH-DEGREE POLYNOMIALS For a quadratic equation a 2 + b + c = 0, there is a well-known formula for the roots. For third- and fourth-degree equations, there are also formulas for the roots. However, they are etremely complicated.

6 HIGH-DEGREE POLYNOMIALS If f is a polynomial of degree 5 or higher, there is no such formula.

7 TRANSCENDENTAL EQUATIONS Likewise, there is no formula that will enable us to find the eact roots of a transcendental equation such as cos =.

8 APPROXIMATE SOLUTION We can find an approimate solution by plotting the left side of the equation. Using a graphing device, and after eperimenting with viewing rectangles, we produce the graph below.

9 ZOOMING IN We see that, in addition to the solution = 0, which doesn t interest us, there is a solution between and Zooming in shows that the root is approimately

10 ZOOMING IN If we need more accuracy, we could zoom in repeatedly. That becomes tiresome, though.

11 NUMERICAL ROOTFINDERS A faster alternative is to use a numerical rootfinder on a calculator or computer algebra system. If we do so, we find that the root, correct to nine decimal places, is

12 NUMERICAL ROOTFINDERS How do those numerical rootfinders work? They use a variety of methods. Most, though, make some use of Newton s method, also called the Newton-Raphson method.

13 NEWTON S METHOD We will eplain how the method works, for two reasons: To show what happens inside a calculator or computer As an application of the idea of linear approimation

14 NEWTON S METHOD The geometry behind Newton s method is shown here. The root that we are trying to find is labeled r.

15 NEWTON S METHOD We start with a first approimation 1, which is obtained by one of the following methods: Guessing A rough sketch of the graph of f A computergenerated graph of f

16 NEWTON S METHOD Consider the tangent line L to the curve y = f() at the point ( 1,f( 1 )) and look at the -intercept of L, labeled 2.

17 NEWTON S METHOD Here s the idea behind the method. The tangent line is close to the curve. So, its -intercept, 2, is close to the -intercept of the curve (namely, the root r that we are seeking). As the tangent is a line, we can easily find its -intercept.

18 NEWTON S METHOD To find a formula for 2 in terms of 1, we use the fact that the slope of L is f ( 1 ). So, its equation is: y - f( 1 ) = f ( 1 )( - 1 )

19 SECOND APPROXIMATION As the -intercept of L is 2, we set y = 0 and obtain: 0 - f( 1 ) = f ( 1 )( 2-1 ) If f ( 1 ) 0, we can solve this equation for 2 : 2 1 f( ) f 1 '( ) 1 We use 2 as a second approimation to r.

20 THIRD APPROXIMATION Net, we repeat this procedure with 1 replaced by 2, using the tangent line at ( 2,f( 2 )). This gives a third approimation: 3 2 f( ) f 2 '( ) 2

21 SUCCESSIVE APPROXIMATIONS If we keep repeating this process, we obtain a sequence of approimations 1, 2, 3, 4,...

22 SUBSEQUENT APPROXIMATION Equation/Formula 2 In general, if the nth approimation is n and f ( n ) 0, then the net approimation is given by: n 1 n f( ) f n '( ) n

23 CONVERGENCE If the numbers n become closer and closer to r as n becomes large, then we say that the sequence converges to r and we write: lim n n r

24 CONVERGENCE The sequence of successive approimations converges to the desired root for functions of the type illustrated in the previous figure. However, in certain circumstances, it may not converge.

25 NON-CONVERGENCE Consider the situation shown here. You can see that 2 is a worse approimation than 1. This is likely to be the case when f ( 1 ) is close to 0.

26 NON-CONVERGENCE It might even happen that an approimation falls outside the domain of f, such as 3. Then, Newton s method fails. In that case, a better initial approimation 1 should be chosen.

27 NON-CONVERGENCE See Eercises for specific eamples in which Newton s method works very slowly or does not work at all.

28 NEWTON S METHOD Eample 1 Starting with 1 = 2, find the third approimation 3 to the root of the equation = 0

29 NEWTON S METHOD Eample 1 We apply Newton s method with f() = and f () = Newton himself used this equation to illustrate his method. He chose 1 = 2 after some eperimentation because f(1) = -6, f(2) = -1 and f(3) = 16

30 NEWTON S METHOD Equation 2 becomes: Eample 1 n 1 n 3 n n n 3 2

31 NEWTON S METHOD Eample 1 With n = 1, we have: (2) 5 2 3(2) 2

32 NEWTON S METHOD Eample 1 With n = 2, we obtain: It turns out that this third approimation is accurate to four decimal places (2.1) 5 2 3(2.1) 2

33 NEWTON S METHOD The figure shows the geometry behind the first step in Newton s method in the eample. As f (2) = 10, the tangent line to y = at (2, -1) has equation y = So, its -intercept is 2 = 2.1

34 NEWTON S METHOD Suppose that we want to achieve a given accuracy say, to eight decimal places using Newton s method. How do we know when to stop?

35 NEWTON S METHOD The rule of thumb that is generally used is that we can stop when successive approimations n and n+1 agree to eight decimal places. A precise statement concerning accuracy in the method will be given in Eercise 37 in Section 11.11

36 ITERATIVE PROCESS Notice that the procedure in going from n to n + 1 is the same for all values of n. It is called an iterative process. This means that the method is particularly convenient for use with a programmable calculator or a computer.

37 NEWTON S METHOD Eample 2 Use Newton s method to find to eight decimal places. 6 2 correct First, we observe that finding is equivalent to finding the positive root of the equation 6 2 = So, we take f() = 6 2 Then, f () = 6 5

38 NEWTON S METHOD Eample 2 So, Formula 2 (Newton s method) becomes: 6 n n 1 n 6 5 n 2

39 NEWTON S METHOD Eample 2 Choosing 1 = 1 as the initial approimation, we obtain: As 5 and 6 agree to eight decimal places, we conclude that to eight decimal places

40 NEWTON S METHOD Eample 3 Find, correct to si decimal places, the root of the equation cos =. We rewrite the equation in standard form: cos = 0 Therefore, we let f() = cos Then, f () = sin 1

41 NEWTON S METHOD So, Formula 2 becomes: Eample 3 n 1 cos n n n sin 1 n n cos n n sin 1 n

42 NEWTON S METHOD Eample 3 To guess a suitable value for 1, we sketch the graphs of y = cos and y =. It appears they intersect at a point whose -coordinate is somewhat less than 1.

43 NEWTON S METHOD So, let s take 1 = 1 as a convenient first approimation. Then, remembering to put our calculator in radian mode, we get: As 4 and 5 agree to si decimal places (eight, in fact), we conclude that the root of the equation, correct to si decimal places, is Eample

44 NEWTON S METHOD Instead of using this rough sketch to get a starting approimation for the method in the eample, we could have used the more accurate graph that a calculator or computer provides.

45 NEWTON S METHOD This figure suggests that we use 1 = 0.75 as the initial approimation.

46 NEWTON S METHOD Then, Newton s method gives: So we obtain the same answer as before but with one fewer step.

47 NEWTON S METHOD VS. GRAPHING DEVICES You might wonder why we bother at all with Newton s method if a graphing device is available. Isn t it easier to zoom in repeatedly and find the roots as we did in Section 1.4?

48 NEWTON S METHOD VS. GRAPHING DEVICES If only one or two decimal places of accuracy are required, then indeed the method is inappropriate and a graphing device suffices. However, if si or eight decimal places are required, then repeated zooming becomes tiresome.

49 NEWTON S METHOD VS. GRAPHING DEVICES It is usually faster and more efficient to use a computer and the method in tandem. You start with the graphing device and finish with the method.