Lin In 3D

Introduction: It is easy to graph ancestrys and hear their x-intercepts. You will be command through the basic ideas of Newtons Method, which uses x-intercepts of appropriate lines to approximate x-intercepts of more ambitious functions. Note: We need zippoes of a function y to find its x-intercepts; zeroes of y to find stationary points of y; and zeroes of y to find possible points of inflexion of y. Sometimes we just need to find where two functions cross. many another(prenominal) calculators use Newtons Method with y=x2-a and an initial guess of 1 to find the square root of a.

Elements of this lab were adapted from Solows encyclopaedism by Discovery, Edwards & Penneys Single Variable Calculus, and Harvey & Kenellys Explorations with the TI-85. more(prenominal) information can be found in the annotated Bibliography at http://www.southwestern.edu/~shelton/Files/ in the list of Word files.

THEORY          allow y = f(x) be a function. On the figure below, graph the tangent 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 close to the zero of the function? Give a reason for your answer.

What is the equation of the line T1(x) tangent to the graph of f at (x0,f(x0))? Show that the x-intercept of T1(x), x1, is abandoned by x1= x0-f(x0)/f(x0) .

We repeat the process, using x1 as our new range 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. Show x1 , and x2. Show x3, if possible.

save a formula for x2 in terms of x1.

Write a formula for xn+1 in terms of xn.

MATHEMATICA Define f[x_]:=x3 - 4 x2 - 1 .

spot it with x in the interval [-10,10].

expenditure the mouse to estimate the x value of the root.



Define x[0] to be 5 the first time.

Find the derivative of f[x] = x3 - 4 x2 - 1.

here(predicate) are the two steps for a single looping: Calculate the next x: x[n+1]=x[n] - f[ x[n] ] / f[ x[n] ] Increment n.

Perform some(prenominal) iterations.

Newtons Method does not always work well. It is sensitive to your initial guess.

Use Newtons Method on the same function with x[0] = 2. abide by that the Method does not converge to the root. What seems to be happening? 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, coiffure Newtons method with f[x_]:=y4[x]-y5[x].

Begin with x[0]=3.

Choose several other x[0].

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