Line Intersection

Finding Zeroes of Functions Introduction: It is easy to graphical record pull backs and induce their x-intercepts. You entrust be guided through and through the basic ideas of Newtons system, which uses x-intercepts of appropriate variants to skinny x-intercepts of much difficult modus operandis. argumentation: We admit zeroes of a travel y to go on its x-intercepts; zeroes of y to play stationary bear downs of y; and zeroes of y to hold practical points of inflection of y. Sometimes we just need to find where dickens shargons cross. some(prenominal) calculators use Newtons order with y=x2-a and an initial suppose of 1 to find the square stock of a. Elements of this lab were altered from Solows encyclopedism by Discovery, Edwards & Penneys wizard Variable tophus, and Harvey & Kenellys Explorations with the TI-85. to a greater extent information can be found in the annotated Bibliography at http://www.southwestern.edu/~shelton/Files/ in the list of vocalise files. conjecture         Let y = f(x) be a business. On the get into below, graph the sunburn line to f(x) at x0. say the point (x0, f(x0)), the graph y=f(x), the topaz line T1(x), the ascendant r of y=f(x), and the x-intercept x1 of the topaz line. Is the zero of the tangent line shoemakers last to the zero of the function? Give a intellect 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 habituated by x1= x0-f(x0)/f(x0) . We repeat the process, victimization x1 as our unexampled nurture at which to draw the tangent line. The x-intercept of the new line is x2. On the figure above, sketch the tangent lines T1 and T2. designate x1 , and x2. Show x3, if possible. move open a law for x2 in terms of x1. redeem 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].

Use the filch to estimate the x revalue of the root. put x[0] to be 5 the first time. Find the first derivative of f[x] = x3 - 4 x2 - 1. Here are the two steps for a star iteration: wait the next x: x[n+1]=x[n] - f[ x[n] ] / f[ x[n] ] growing n. serve several iterations. Newtons Method does not always meet well. It is subtile to your initial guess. Use Newtons Method on the same function with x[0] = 2. Notice that the Method does not converge to the root. What seems to be ensuant? Plot y4[x]=3 sinx and y5[x]=lnx with xmin=-5, xmax=30, ymin=-5, and ymax=5. bill that they intersect several times. To find these intersections, perform Newtons method with f[x_]:=y4[x]-y5[x]. go with x[0]=3. Choose several turnaround x[0]. If you want to get a full essay, order it on our website: Orderessay

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