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MATH116 Lecture Notes - Lecture 4: Tx Network, Recurrence Relation

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9 Dec 2015

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University of Waterloo

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Wednesday, october 28 lecture 18 : newton"s iterative procedure. (refers to section 4. 1 in your text) Students who have mastered the content of this lecture know about: newton"s iterative formula for approximating roots of simple functions. 18. 1 newton"s iterative procedure for finding roots let f : be a continuous function. We sketch the curve of y = f (x). At first glance we can approximate a root x* of y = f (x) by roughly examining where the curve of y = f (x) intersects the x-axis. Say our first approximation of the zero x* is denoted by x1 and we attempt to improve on this approximation. If x1 is reasonably near the root the sequence {xn : n = 1, 2, } of numbers will converge to the root. This method of finding roots is called newton"s iterative procedure.

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Related Questions

Please look at all the numbers 1-9

Sketch using these guidelines

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)-è¨-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. â 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 xã(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 180Â° 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

Complete the first problem using the information above as well as the product rule.

Math 151 - Sections 34, 35, 36 - Workshop 1:2 This workshop has to do with the indefinite integral Jf(x) da. We have looked at the formula f f(x) dx = g(x) + constant, where g'(x) = f(x). Basically, finding the indefinite integral J f(x) dz is the problem of finding g(x) such that g,(z) = f(z). Any time you find such a g(x), it is easy to check whether you made a mistake in finding g(x). Just differentiate g(r) and see whether you get f(x) as the answer. For many f(x), there is no simple formula for any g(x) that has the property g'() f() The functions f(x) = e-z , f(x) = 012, f(x) = V1+14 are examples of f(x) for which you will never find a simple g(x) with the property g' (x)- f(x). It is not that we have not found a formula yet. Rather, we can prove that there is no formula that involves a finite combination of exponential functions, trigonometric functions, nth root functions, and the other basic functions that we are familiar with. On the other hand, there are many integration methods that will produce a g() for many common choices of f (x). Two of these methods are called substitution (which you will see later in Chapter 5) and integration by parts (which is taught in Math 152). These two methods can be used to do all of the problems that you see below. Since you have not seen substitution and integration by parts, you will have to use a different method, as indicated roblem 1. Find constants A, B, C, D, E such that re drAee + Ce + Dre" + Ee +constant. Hint: Differentiate the right side of the equation and solve for A, B, C, D, E Problem 2. Using the same hint, find constants A, B, C, D, E such that Problem 3. Using a similar hint, find constants A, B, C such that x2 cos(32) dz = Az? sin(3x) + Bx cos(3r) + C sin(3x) + constant. Problem A. Using a similar hint, find constants A, B, C such that z? sin(32) dz = Az? cos(3x) + Bx sin(3z) + Ccos(3x) + constant. Problem 5. Using a similar hint, find constants A, B such that Â®3eÂ® dr = Ax2ez" + Bez? + constant. Problem 6. Using similar methods, find r cos(ar) dz sin(az) de da where a is a constant