Get answer: Using Newtons Method In Exercises 516, approximate the zero(s) of the

Using Newton’s Method In Exercises 1–4, complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=x^{2}-5, \quad x_{1}=2.2\) Text Transcription: f(x)=x^2-5, x_1=2.2

Using Newton’s Method In Exercises 1–4, complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=x^{3}-3, \quad x_{1}=1.4\) Text Transcription: f(x)=x^3-3, x_1=1.4

Using Newton’s Method In Exercises 1–4, complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=\cos x, \quad x_{1}=1.6\) Text Transcription: f(x)=cos x, x_1=1.6

Using Newton’s Method In Exercises 1–4, complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=\tan x, \quad x_{1}=0.1\) Text Transcription: f(x)=tan x, x_1=0.1

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}+4\) Text Transcription: f(x)=x^3+4

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=2-x^{3}\) Text Transcription: f(x)=2-x^3

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}+x-1\) Text Transcription: f(x)=x^3+x-1

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{5}+x-1\) Text Transcription: f(x)=x^5+x-1

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=5 \sqrt{x-1}-2 x\) Text Transcription: f(x)=5 sqrt x-1-2x

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x-2 \sqrt{x+1}\) Text Transcription: f(x)=x-2 sqrt x+1

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x-e^{-x}\) Text Transcription: f(x)=x-e^-x

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x-3+\ln x\) Text Transcription: f(x)=x-3+ln x

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}-3.9 x^{2}+4.79 x-1.881\) Text Transcription: f(x)=x^3-3.9x^2+4.79x-1.881

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{4}+x^{3}-1\) Text Transcription: f(x)=x^4+x^3-1

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=1-x+\sin x\) Text Transcription: f(x)=1-x+sin x

Using Newton’s Method In Exercises 5–16, approximate the zero(s) of the function. Use Newton’s Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}-\cos x\) Text Transcription: f(x)=x^3-cos x

Finding Point(s) of Intersection In Exercises 17–20, apply Newton’s Method to approximate the x-value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. [Hint:Let \(h(x)=f(x)-g(x)\).] \(\begin{array}{l}f(x)=2 x+1 \\ g(x)=\sqrt{x+4}\end{array}\) Text Transcription: h(x)=f(x)-g(x) f(x)=2 x+1 g(x)=sqrt x+4

Finding Point(s) of Intersection In Exercises 17–20, apply Newton’s Method to approximate the -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. [Hint:Let \(h(x)=f(x)-g(x)\).] \(\begin{array}{l} f(x)=2-x^{2} \\ g(x)=e^{x / 2} \end{array}\) Text Transcription: h(x)=f(x)-g(x) f(x)=2-x^2 g(x)=e^x/2

Finding Point(s) of Intersection In Exercises 17–20, apply Newton’s Method to approximate the -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. [Hint:Let \(h(x)=f(x)-g(x)\).] \(\begin{array}{l} f(x)=x \\ g(x)=\tan x \end{array}\) Text Transcription: h(x)=f(x)-g(x) f(x)=x g(x)=tan x

Finding Point(s) of Intersection In Exercises 17–20, apply Newton’s Method to approximate the -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. [Hint:Let \(h(x)=f(x)-g(x)\).] \(\begin{array}{l} f(x)=\arccos x \\ g(x)=\arctan x \end{array}\) Text Transcription: h(x)=f(x)-g(x) f(x)=arccos x g(x)=arctan x

Mechanic’s Rule The Mechanic’s Rule for approximating \(\sqrt{a}, a>0\), is \(x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right), \quad n=1,2,3 \ldots\) where x1 is an approximation of \(\sqrt{a}\). (a) Use Newton’s Method and the function \(f(x)=x^{2}-a\) to derive the Mechanic’s Rule. (b) Use the Mechanic’s Rule to approximate \(\sqrt{5}\) and \(\sqrt{7}\) to three decimal places. Text Transcription: sqrt a, a>0 x_n+1=1/2(x_n+a/x_n), n=1,2,3 ldots g(x)=arctan x sqrt a f(x)=x^2-a sqrt 5 sqrt 7

Approximating Radicals (a) Use Newton’s Method and the function \(f(x)=x^{n}-a\) to obtain a general rule for approximating \(x=\sqrt[n]{a}\) (b) Use the general rule found in part (a) to approximate \(\sqrt[4]{6}\) and \(\sqrt[3]{15}\) to three decimal places. Text Transcription: f(x)=x^n-a x=sqrt[n]a sqrt[4]6 sqrt[3]15

Failure of Newton’s Method In Exercises 23 and 24, apply Newton’s Method using the given initial guess, and explain why the method fails. \(y=2 x^{3}-6 x^{2}+6 x-1, \quad x_{1}=1\) Text Transcription: y=2x^3-6 x^2+6x-1, x_1=1

Failure of Newton’s Method In Exercises 23 and 24, apply Newton’s Method using the given initial guess, and explain why the method fails. \(y=x^{3}-2 x-2, \quad x_{1}=0\) Text Transcription: y=x^3-2x-2, x_1=0

Fixed Point In Exercises 25–28, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function f is a value of x such that \(f\left(x_{0}\right)=x_{0}\).] \(f(x)=\cos x\) Text Transcription: x_0 ft(x_0)=x_0 f(x)=cos x

Fixed Point In Exercises 25–28, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function f is a value of x such that \(f\left(x_{0}\right)=x_{0}\).] \(f(x)=\cot x, \quad 0<x<\pi\) Text Transcription: x_0 ft(x_0)=x_0 f(x)=cot x, 0<x<pi

Fixed Point In Exercises 25–28, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function f is a value of x such that \(f\left(x_{0}\right)=x_{0}\).] \(f(x)=e^{x / 10}\) Text Transcription: x_0 ft(x_0)=x_0 f(x)=e^x/10

Fixed Point In Exercises 25–28, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function f is a value of x such that \(f\left(x_{0}\right)=x_{0}\).] \(f(x)=-\ln x\) Text Transcription: ft(x_0)=x_0 f(x)=-ln x

Approximating Reciprocals Use Newton’s Method to show that the equation \(x_{n+1}=x_{n}\left(2-a x_{n}\right)\) can be used to approximate 1/a when \(x_{1}\) is an initial guess of the reciprocal of a. Note that this method of approximating reciprocals uses only the operations of multiplication and subtraction. (Hint: Consider \(f(x)=\frac{1}{x}-a\).) Text Transcription: x_1 x_n+1=x_n(2-a x_n) f(x)=1/x-a

Approximating Reciprocals Use the result of Exercise 29 to approximate (a) \(\frac{1}{3}\) and \(\frac{1}{11}\) to three decimal places. Text Transcription: 1/3 1/11

Using Newton’s Method Consider the function \(f(x)=x^{3}-3 x^{2}+3\). (a) Use a graphing utility to graph f. (b) Use Newton’s Method to approximate a zero with x1 = 1 as an initial guess. (c) Repeat part (b) using \(x_{1}=\frac{1}{4}\) as an initial guess and observe that the result is different. d) To understand why the results in parts (b) and (c) are different, sketch the tangent lines to the graph of f at the points \((1, f(1)) \text { and }\left(\frac{1}{4}, f\left(\frac{1}{4}\right)\right\). Find the x-intercept of each tangent line and compare the intercepts with the first iteration of Newton’s Method using the respective initial guesses. (e) Write a short paragraph summarizing how Newton’s Method works. Use the results of this exercise to describe why it is important to select the initial guess carefully. Text Transcription: f(x)=x^3-3x^2+3 x_{1}=1/4 (1, f(1)) and (1/4, f(1/4)

Using Newton’s Method Repeat the steps in Exercise 31 for the function \(f(x)=\sin x\) with initial guesses of x1 = 1.8 and x1 = 3. Text Transcription: f(x)=sin x

Newtons Method In your own words and using a sketch, describe Newtons Method for approximating the zeros of a function.

How Do You See It? For what value(s) will Newton’s Method fail to converge for the function shown in the graph? Explain your reasoning. Text Transcription:

Using Newton’s Method Exercises 35–37 present problems similar to exercises from the previous sections of this chapter. In each case, use Newton’s Method to approximate the solution. Minimum Distance Find the point on the graph of \(f(x)=4-x^{2}\) that is closest to the point (1, 0). Text Transcription: f(x)=4-x^2

Using Newton’s Method Exercises 35–37 present problems similar to exercises from the previous sections of this chapter. In each case, use Newton’s Method to approximate the solution. Medicine The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is given by \(C=\frac{3 t^{2}+t}{50+t^{3}}\). When is concentration the greatest? Text Transcription: C=3t^2+t/50+t^3

Using Newton’s Method Exercises 35–37 present problems similar to exercises from the previous sections of this chapter. In each case, use Newton’s Method to approximate the solution. Minimum Time You are in a boat 2 miles from the nearest point on the coast (see figure). You are to go to a point Q that is 3 miles down the coast and 1 mile inland. You can row at 3 miles per hour and walk at 4 miles per hour. Toward what point on the coast should you row in order to reach Q in the least time? Text Transcription:

Crime The total number of arrests T (in thousands) for all males ages 15 to 24 in 2010 is approximated by the model \(T=0.2988 x^{4}-22.625 x^{3}+628.49 x^{2}-7565.9 x+33,478\) for \(15 \leq x \leq 24\) where x is the age in years (see figure). Approximate the two ages that had total arrests of 300 thousand. (Source: U.S. Department of Justice) Text Transcription: T=0.2988x^4-22.625x^3+628.49x^2-7565.9x+33,478 15 leq x leq 24

True or False? In Exercises 39–42, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The zeros of \(f(x)=\frac{p(x)}{q(x)}\) coincide with the zeros of \(p(x)\). Text Transcription: f(x)=p(x)/q(x) p(x)

If the coefficients of a polynomial function are all positive, then the polynomial has no positive zeros.

True or False? In Exercises 39–42, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)\) is a cubic polynomial such that \(f^{\prime}(x)\) is never zero, then any initial guess will force Newton’s Method to converge to the zero of f. Text Transcription: f(x) f^prime(x)

True or False? In Exercises 39–42, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The roots of \(\sqrt{f(x)}=0\) coincide with the roots of \(f(x)=0\). Text Transcription: sqrtf(x)=0 f(x)=0

Tangent Lines The graph of \(f(x)=-\sin x\) has infinitely many tangent lines that pass through the origin. Use Newton’s Method to approximate to three decimal places the slope of the tangent line having the greatest slope. Text Transcription: f(x)=-sin x

Point of Tangency The graph of \(f(x)=\cos x\) and a tangent line to f through the origin are shown. Find the coordinates of the point of tangency to three decimal places. Text Transcription: f(x)=cos x