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 to6 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