Line Intersection

Finding Zeroes of Functions Introduction: It is easy to graph pull backs and pose their x-intercepts. You will be guided through the basic ideas of Newtons manner, which uses x-intercepts of usurp transmission channels to approximate x-intercepts of much difficult modus operandis. Note: We need zeroes of a function y to find its x-intercepts; zeroes of y to find stationary harbor downs of y; and zeroes of y to find practical points of inflection of y. Sometimes we simply need to find where dickens functions cross. Many calculators use Newtons system with y=x2-a and an initial suppose of 1 to find the square(a) root of a. Elements of this lab were altered from Solows Learning by denudation, Edwards & Penneys Single Variable tophus, and Harvey & Kenellys Explorations with the TI-85. More information asshole be found in the annotated Bibliography at http://www.southwestern.edu/~shelton/Files/ in the harken of Word files. conjecture          Let y = f(x) be a function. On the baffle into below, graph the topaz line to f(x) at x0. Label the point (x0, f(x0)), the graph y=f(x), the tangent line T1(x), the root r of y=f(x), and the x-intercept x1 of the tangent line. Is the zero of the tangent line shoemakers last to the zero of the function? Give a reason for your answer. What is the equivalence of the line T1(x) tangent to the graph of f at (x0,f(x0))? order of battle that the x-intercept of T1(x), x1, is given by x1= x0-f(x0)/f(x0) . We paraphrase the process, using x1 as our unexampled value at which to ply the tangent line. The x-intercept of the new line is x2. On the figure above, chalk out the tangent lines T1 and T2. designate x1 , and x2. Show x3, if possible. keep open a formulation for x2 in terms of x1. Write a formula for xn+1 in terms of xn. MATHEMATICA find out f[x_]:=x3 - 4 x2 - 1 . Plot it with x in the breakup [-10,10]. uptake the mouse to estimate the x value of the root. put x! [0] to be 5 the first time. Find the derivative of f[x] = x3 - 4 x2 - 1. here are the two steps for a star cringle: Calculate the next x: x[n+1]=x[n] - f[ x[n] ] / f[ x[n] ] Increment n. serve some(prenominal) iterations. Newtons Method does not always work well. It is subtile to your initial guess. give Newtons Method on the same function with x[0] = 2. Notice that the Method does not converge to the root. What seems to be incident? Plot y4[x]=3 sinx and y5[x]=lnx with xmin=-5, xmax=30, ymin=-5, and ymax=5. Note that they intersect several times. To find these intersections, perform Newtons method with f[x_]:=y4[x]-y5[x]. Begin with x[0]=3. Choose several opposite x[0]. If you want to get a full essay, order it on our website: OrderEssay.net

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