CHAPTER 1 Mathematical Preliminaries and Error Analysis Figure 1.2 The tangent line has slope '(ro xo, f(ro)) Theorem 1.6 If the function f is differentiable at xo then f is continuous at xo. theorem attributed to michel The next theorems are of fundamental importance in deriving methods for error esti (1652-1719)appeared in mation. The proofs of these theorems and the other unreferenced results in this section can 1691 in a little-known treatise be found in any standard calculus text entitled Metode pour resound The set of all functions that have n continuous derivatives on X is denoted Cn(X), and es egalites. Rolle originally the set of functions that have derivatives of all orders on X is denoted C (X). Polynomia criticized the calculus that was developed by Isaac Newton and ational,trigonometric, exponential, and logarithmic functions are in C(X), where X Gottfried Leibniz. but later consists of all numbers for which the functions are defined. when X is an interval of the becam real line, we will again omit the parentheses in this notation. Theorem 1.7(Rolle's Theorem) upposef E Cla, b] and f is differentiable on(a, b). If f(a)=f(b), then a number c in (a, b)exists with f(c)=0.(See Figure 1.3.) fa)=f(b) Theorem 1.8(Mean Value Theorem) If f E Cla, b] and f is differentiable on(a, b), then a number c in(a, b) exists with(See f'(c) f(b-f(a) Copyright 2010 Cengage Learning. All Rights May no be copied, scanned, or duplicated, in whole or in part Due to maternally aftec the overall leaning expenence. Cengage Learning
4 CHAPTER 1 Mathematical Preliminaries and Error Analysis Figure 1.2 x y y f(x) (x0, f(x0)) f(x0) x0 The tangent line has slope f(x0) Theorem 1.6 If the function f is differentiable at x0, then f is continuous at x0. The next theorems are of fundamental importance in deriving methods for error estimation. The proofs of these theorems and the other unreferenced results in this section can be found in any standard calculus text. The theorem attributed to Michel Rolle (1652–1719) appeared in 1691 in a little-known treatise entitled Méthode pour résoundre les égalites. Rolle originally criticized the calculus that was developed by Isaac Newton and Gottfried Leibniz, but later became one of its proponents. The set of all functions that have n continuous derivatives on X is denoted Cn(X), and the set of functions that have derivatives of all orders on X is denoted C∞(X). Polynomial, rational, trigonometric, exponential, and logarithmic functions are in C∞(X), where X consists of all numbers for which the functions are defined. When X is an interval of the real line, we will again omit the parentheses in this notation. Theorem 1.7 (Rolle’s Theorem) Suppose f ∈ C[a, b] and f is differentiable on (a, b). If f (a) = f (b), then a number c in (a, b) exists with f (c) = 0. (See Figure 1.3.) Figure 1.3 x f(c) 0 a c b f(a) f(b) y y f(x) Theorem 1.8 (Mean Value Theorem) If f ∈ C[a, b] and f is differentiable on (a, b), then a number c in (a, b) exists with (See Figure 1.4.) f (c) = f (b) − f (a) b − a . Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
of calc 1.4 f(b)-f(a) Theorem 1.9(Extreme Value Theorem) Iff∈C[a,b].then llx∈[a,b In addition, if f is differentiable on(a, b), then the numbers ci and c2 occur either at the endpoints of [a, b] or where fis zero. (See Figure 1.5.) Figure 1.5 C2 Research work on the design of algorithms mathematics began in the 1960s. The first system to be operational, As mentioned in the preface, we will use the computer algebra system Maple wheneve in the 1970s, was a LISP-based appropriate. Computer algebra systems are particularly useful for symbolic differentiation system called MACSYMA and plotting graphs. Both techniques are illustrated in Example 1 Example 1 Use Maple to find the absolute minimum and absolute maximum values of f(x)=5cos 2x- 2rsin 2xf(x) on the intervals (a)[1, 2, and (b)[0.5, 11 Solution There is a choice of Text input or Math input under the Maple C 2D Math option. The Text input is used to document worksheets by standard text information in the document. The Math input option is used to execute Maple commands. Maple input Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. no be copied, scanned, or duplicated, in whole or in part Due to
1.1 Review of Calculus 5 Figure 1.4 y a c b x Slope f(c) Parallel lines Slope b a f(b) f(a) y f(x) Theorem 1.9 (Extreme Value Theorem) If f ∈ C[a, b], then c1, c2 ∈ [a, b] exist with f (c1) ≤ f (x) ≤ f (c2), for all x ∈ [a, b]. In addition, if f is differentiable on (a, b), then the numbers c1 and c2 occur either at the endpoints of [a, b] or where f is zero. (See Figure 1.5.) Figure 1.5 y a c2 c1 b x y f(x) Research work on the design of algorithms and systems for performing symbolic mathematics began in the 1960s. The first system to be operational, in the 1970s, was a LISP-based system called MACSYMA. As mentioned in the preface, we will use the computer algebra system Maple whenever appropriate. Computer algebra systems are particularly useful for symbolic differentiation and plotting graphs. Both techniques are illustrated in Example 1. Example 1 Use Maple to find the absolute minimum and absolute maximum values of f (x) = 5 cos 2x − 2x sin 2xf (x) on the intervals (a) [1, 2], and (b) [0.5, 1] Solution There is a choice of Text input or Math input under the Maple C 2D Math option. The Text input is used to document worksheets by adding standard text information in the document. The Math input option is used to execute Maple commands. Maple input Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
CHAPTER 1- Mathematical Preliminaries and Error Analysis The Maple development project can either be typed or selected from the pallets at the left of the Maple screen. We will began at the University of show the input as typed because it is easier to accurately describe the commands For pallet Waterloo in late 1980. Its goal input instructions you should consult the Maple tutorials In our presentation, Maple input was to be accessible to researchers in mathematics ommands appear in italic type, and Maple responses appear in cyan type. To ensure that the variables we use have not been previously assigned, we first issue engineering, and science, but additionally to students for educational purposes. To be effective it needed to be portable. as well as space and time to clear the Maple memory. We first illustrate the graphing capabilities of Maple. To access efficient. Demonstrations of the he graphing package, enter the command system were presented in 1982, and the major paper setting out the design criteria for the to load the plots subpackage. Maple responds with a list of available commands in the MAPLE system was presented in package. This list can be suppressed by placing a colon after the with(plots)command 1983 ICGGG The following command defines f(r)=5cos 2x-2x sin 2x as a function of ∫:=x→5cos(2x)-2x:sin(2x) and Maple responds with x-5 cos(2x)-2x sin(2x) We can plot the graph of f on the interval [0.5, 2] with the command Figure 1.6 shows the screen that results from this command after doing a mouse click on the graph. This click tells Maple to enter its graph mode, which presents options for various views of the graph. We can determine the coordinates of a point of the graph by moving the mouse cursor to the point. The coordinates appear in the box above the left of the plot(f 0.5.. 2)command. This feature is useful for estimating the axis intercepts and extrema of The absolute maximum and minimum values of f(x)on the interval [a, b]can occur only at the endpoints, or at a critical point. (a) When the interval is [1, 2] we have f(1)=5cos2-2sin2=-3.899329036andf(2)=5c0s4-4sin4=-0.241008123. A critical point occurs when f'(r)=0. To use Maple to find this point, we first define a function fp to represent f with the command f:=x→df(f(x),x and Maple nds with To find the explicit representation of f(r) we enter the command and Maple gives the derivative as To determine the critical point we use the comman fsolve(p(x), x, 1.. 2) Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. May no be copied, scanned, or duplicated, in whole or in part Due to
6 CHAPTER 1 Mathematical Preliminaries and Error Analysis can either be typed or selected from the pallets at the left of the Maple screen. We will show the input as typed because it is easier to accurately describe the commands. For pallet input instructions you should consult the Maple tutorials. In our presentation, Maple input commands appear in italic type, and Maple responses appear in cyan type. To ensure that the variables we use have not been previously assigned, we first issue the command. The Maple development project began at the University of Waterloo in late 1980. Its goal was to be accessible to researchers in mathematics, engineering, and science, but additionally to students for educational purposes. To be effective it needed to be portable, as well as space and time efficient. Demonstrations of the system were presented in 1982, and the major paper setting out the design criteria for the MAPLE system was presented in 1983 [CGGG]. restart to clear the Maple memory. We first illustrate the graphing capabilities of Maple. To access the graphing package, enter the command with(plots) to load the plots subpackage. Maple responds with a list of available commands in the package. This list can be suppressed by placing a colon after the with(plots) command. The following command defines f (x) = 5 cos 2x − 2x sin 2x as a function of x. f := x → 5 cos(2x) − 2x · sin(2x) and Maple responds with x → 5 cos(2x) − 2x sin(2x) We can plot the graph of f on the interval [0.5, 2] with the command plot(f , 0.5 . . 2) Figure 1.6 shows the screen that results from this command after doing a mouse click on the graph. This click tells Maple to enter its graph mode, which presents options for various views of the graph. We can determine the coordinates of a point of the graph by moving the mouse cursor to the point. The coordinates appear in the box above the left of the plot(f , 0.5 . . 2) command. This feature is useful for estimating the axis intercepts and extrema of functions. The absolute maximum and minimum values of f (x) on the interval [a, b] can occur only at the endpoints, or at a critical point. (a) When the interval is [1, 2] we have f (1) = 5 cos 2 − 2 sin 2 = −3.899329036 and f (2) = 5 cos 4 − 4 sin 4 = −0.241008123. A critical point occurs when f (x) = 0. To use Maple to find this point, we first define a function fp to represent f with the command fp := x → diff(f (x), x) and Maple responds with x → d dx f (x) To find the explicit representation of f (x) we enter the command fp(x) and Maple gives the derivative as −12 sin(2x) − 4x cos(2x) To determine the critical point we use the command fsolve(fp(x), x,1..2) Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
1.1 Review of calculus 7 Figure 1.6 +u限L田田 and Maple tells us that f(x)=fp(x)=0 for x in [1, 2] when x is 1.358229874 We evaluate f(r)at this point with the command The is interpreted as the last Maple response. The value of f at the critical point is 5675301338 As a consequence, the absolute maximum value of f(x)in[1, 2]is f(2)=-0.241008123 and the absolute minimum value is f(1.358229874)=-5675301338, accurate at least to the places listed (b) When the interval is [0.5, 1] we have the values at the endpoints given by f(0.5)=5cos1-lsin1=1.860040545andf(1)=5c0s2-2sin2=-3.899329036 However, when we attempt to determine the critical point in the interval [0.5, 1] with the ommand fsolve (p(r), x, 0.5. 1 Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. May no be copied, scanned, or duplicated, in whole or in part Due to
1.1 Review of Calculus 7 Figure 1.6 and Maple tells us that f (x) = fp(x) = 0 for x in [1, 2] when x is 1.358229874 We evaluate f (x) at this point with the command f (%) The % is interpreted as the last Maple response. The value of f at the critical point is −5.675301338 As a consequence, the absolute maximum value of f (x) in [1, 2] is f (2) = −0.241008123 and the absolute minimum value is f (1.358229874) = −5.675301338, accurate at least to the places listed. (b) When the interval is [0.5, 1] we have the values at the endpoints given by f (0.5) = 5 cos 1 − 1 sin 1 = 1.860040545 and f (1) = 5 cos 2 − 2 sin 2 = − 3.899329036. However, when we attempt to determine the critical point in the interval [0.5, 1] with the command fsolve(fp(x), x, 0.5 . . 1) Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
CHAPTER 1- Mathematical Preliminaries and Error Analysis Maple gives the response f solve(-12 sin(2r)-4.x cos(2x),x,. 1) This indicates that Maple is unable to determine the solution. The reason is obvious once the graph in Figure 1.6 is considered. The function f is always decreasing on this interval so no solution exists. Be suspicious when Maple returns the same response it is given; it is as if it was questioning your request. In summary, on [0.5, 1] the absolute maximum value is f(0.5)=1.86004545 and the absolute minimum value is f(1)=-3899329036, accurate at least to the places The following theorem is not generally presented in a basic calculus course, but is derived by applying Rolle's Theorem successively to f, f,..., and, finally, to fn-l) This result is considered in Exercise 23 Theorem 1.10(Generalized Rolle's Theorem Suppose f E Cla, b] is n times differentiable on(a, b). If f(r)=0 at the n+ l distinct numbers a xo <x1 <.. xn< b, then a number c in(xo, xn), and hence in(a, b) exists with fm)(c)=0 We will also make frequent use of the Intermediate Value Theorem. Although its state ment seems reasonable, its proof is beyond the scope of the usual calculus course. It can however, be found in most analysis texts Theorem 1.11(Intermediate Value Theorem) If f E Cla, b] and K is any number between f(a) and f(b), then there exists a number c in(a, b) for which f(c)=K Value Theorem. In this example there are two other possibilitie teed by the Intermediate Figure 1.7 shows one choice for the number that is guara 1.7 (a, f(a)) f(b) (b, f(b)) Example 2 Show that x5-2x3+3x2-1=0 has a solution in the interval [0, 1 Solution Consider the function defined by f(x)=x5-2x3+3x2-1. The function f continuous on [0, 1. In addition Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. May no be copied, scanned, or duplicated, in whole or in part Due to
8 CHAPTER 1 Mathematical Preliminaries and Error Analysis Maple gives the response f solve(−12 sin(2x) − 4x cos(2x), x, .5 . . 1) This indicates that Maple is unable to determine the solution. The reason is obvious once the graph in Figure 1.6 is considered. The function f is always decreasing on this interval, so no solution exists. Be suspicious when Maple returns the same response it is given; it is as if it was questioning your request. In summary, on [0.5, 1] the absolute maximum value is f (0.5) = 1.86004545 and the absolute minimum value is f (1) = −3.899329036, accurate at least to the places listed. The following theorem is not generally presented in a basic calculus course, but is derived by applying Rolle’s Theorem successively to f , f , ... , and, finally, to f (n−1) . This result is considered in Exercise 23. Theorem 1.10 (Generalized Rolle’s Theorem) Suppose f ∈ C[a, b] is n times differentiable on (a, b). If f (x) = 0 at the n + 1 distinct numbers a ≤ x0 < x1 < ... < xn ≤ b, then a number c in (x0, xn), and hence in (a, b), exists with f (n) (c) = 0. We will also make frequent use of the Intermediate Value Theorem. Although its statement seems reasonable, its proof is beyond the scope of the usual calculus course. It can, however, be found in most analysis texts. Theorem 1.11 (Intermediate Value Theorem) If f ∈ C[a, b] and K is any number between f (a) and f (b), then there exists a number c in (a, b) for which f (c) = K. Figure 1.7 shows one choice for the number that is guaranteed by the Intermediate Value Theorem. In this example there are two other possibilities. Figure 1.7 x y f(a) f(b) y f(x) K (a, f(a)) (b, f(b)) a c b Example 2 Show that x5 − 2x3 + 3x2 − 1 = 0 has a solution in the interval [0, 1]. Solution Consider the function defined by f (x) = x5 − 2x3 + 3x2 − 1. The function f is continuous on [0, 1]. In addition, Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it