MATH 205 Lecture Notes - Lecture 19: Periodic Function, Asymptote, Even And Odd Functions

■Guidelines for Sketching a Curve The following checklist is intended as a guide to sketching a curve y f(x) by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. A. Domain It's often useful to start by determining the domain D of f, that is, the set of values of x for which f(x) is defined. B. Intercepts The y-intercept is f(0) and this tells us where the curve intersects the y-axis. To find the x-intercepts, we set y0 and solve for x. (You can omit this step if the equation is difficult to solve.) -x C. Symmetry (i) If f(-x) =f(x) for all x in D, that is, the equation of the curve is unchanged when x is replaced by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for x0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a)]. Here are some examples: y-x, y x", y-lal, and y cos x. metry (ii) If f(-x) =-f(x) for all t in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180° about the origin; see Figure 3(b).] Some simple examples of odd functions are y-x,y-x, y = x, and y-sin x. nmetry

pul aill al thes information together to sketch graphs that reveal the important You might ask: Why don't we just use a graphing calculator or computer to graph a It's true that modern technology is capable of producing very accurate graphs. But K 21+1+2 features of functions curve? Why do we need to use calculus? even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on tech- nology. (See "Graphing Calculators and Computers" at www.stewartcalcelus.com, espe 4 discover the most interesting aspects of graphs and in many cases to and minimum points and inflection points exocily instead of approximately cially Examples 1, 3.4, and 5. See also Section 4.6.) The use of calculus enables us to calculate maximum FIGURE 1 For instance, Figure I shows the gr aph of /(x)-記-21x, + 18x + 2.At first glance it seems reasonable: It has the same shape as cubic curves like y x and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x-0.75 and a minimum when x 1. Indeed if we zoom in to this portion of the graph, we see that behavior exhibited in Pigure 2 Without calculus, we could easily have overlooked it. 6a, FIGURE 2 In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the foilowing information. We don't assume that you have a graphing device, but if you do have one you should use it as a check on your work. ■Guidelines for Sketching a Curve every item is relevant to every function. (For instance, a to make a sketch that displays the most important aspects of the function. The following checklist is intended as a guide to sketching a curve y- f(x) by hand. Not given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need A. Domain It's often useful to start by determining the domain D of f, that is, the set B. Intercepts The y-intercept is f(o) and this tells us where the curve intersects the of values of x for which f(x) is defined y-axis. To find the x-intercepts, we set y-0and solve for x、(You can omit this step if the equation is difficult to solve.) ry (a) Even fusction: reflectional symmetry all x in D, that is, the equation of the curve is unchanged when x is replacea by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for 0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a). Here are some examples: y-ey-ry x). and for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180° about the origin; see Figure 3(b).] Some b) Oud function rotational symmetry FIGURE 3 simple examples of odd functions are y-xy-x'y - and y-sin

EXAMPLE 1 Use the guidelines to sketch the curve (A) The domain is {x1 x. 1-0) = {x | x-±1} = (-0, (B) The x- and y-intercepts are both 0 (C) Since r-x) =代x), the function is l-Select-- . The curve is symmetric about the y-axis o0) 4.x2 (D) lim 11m = 111m is a horizontal asymptote. Since the denominator is 0 when x = ±1, we Therefore the line y compute the following limits: 4.x2 4.22 1 4.r2 1 4.22 lin | Select--$ | lim--=1..-Select--$ Therefore the lines x =-1 and are vertical asymptotes. This information about limits and asymptotes enables us to draw the asymptotes in the top figure Video Example) Tutorial Online Textbook (E) f, (r) = 82(r2-1)-4r2 (2r) -82 Since f (x) >0 when x 0 (x1), fis increasing on (0, and ( ,) and decreasing on , and(