Solve the given information Let f(x) = sin x. Determine an error bound for the approxiamtion f(0.34) = sin 0.34. and compare it to the actual error.

Question

Question

Answers (1)

Relevant Questions

asked 2021-01-19

Solve the given information Let \(\displaystyle{f{{\left({x}\right)}}}={\sin{{x}}}.\) Use the following values and five-digit rounding arithmetic to construct the Hermite interpolating polynomial to approximate \(\displaystyle{f{{\left({0.34}\right)}}}={\sin{{0.34}}}.\)
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}{\mid}{c}\right\rbrace}{x}&{f{{\left({x}\right)}}}&{f}'{\left({x}\right)}\backslash{h}{l}\in{e}{0.30}&{0.29552}&{0.95534}\backslash{0.32}&{0.31457}&{0.94924}\backslash{0.35}&{0.34290}&{0.93937}\backslash{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)

See answers (1)

asked 2020-12-16

c. Let \(f(x) = \frac{e^{x}-e^{-x}}{x}\).
The actual values values is \(f(0.1) = 2.003335000\). Find the relative error for the values obtained in parts (b) and (c)

See answers (1)

asked 2020-12-28

An experiment on the probability is carried out, in which the sample space of the experiment is
\(S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.\)
Let event \(E={2, 3, 4, 5, 6, 7}, event\)
\(F={5, 6, 7, 8, 9}, event G={9, 10, 11, 12}, and event H={2, 3, 4}\).
Assume that each outcome is equally likely. List the outcome s in For G.
Now find P( For G) by counting the number of outcomes in For G.
Determine P (For G ) using the General Addition Rule.

See answers (1)

asked 2020-10-18

a) Evaluate the polynomial \(\displaystyle{y}={x}^{{3}}-{7}{x}^{{2}}+{8}{x}-{0.35}\)
at \(\displaystyle{x}={1.37}\) . Use 3-digit arithmetic with chopping. Evaluate the percent relative error. b) Repeat (a) but express y as \(\displaystyle{y}={\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}-{0.35}\) Evaluate the error and compare with part (a).

See answers (1)

asked 2020-12-29

a) Use base b = 10, precision k = 4, idealized, chopping floating-point arithmetic to show that fl(g(1.015)) is inaccurate, where \(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{\frac{{1}}{{4}}}}-{1}}}{{{x}-{1}}}}\) b) Derive the second order (n = 2) quadratic Taylor polynomial approximation for \(\displaystyle{f{{\left({x}\right)}}}={x}^{{\wedge}}\frac{{1}}{{4}},\) expanded about a = 1, and use it to get an accurate approximation to g(x) in part (a). c) Verify that your approximation in (b) is more accurate.

See answers (1)

asked 2020-10-31

The arithmetic mean (average) of two numbers c and d is given by \(\overline{x} = \frac{c+d}{2}\)
The value \(\overline{x}\) is equidistant between c and d, so the sequence c,\(\overline{x}\), d is an arithmetic sequence. inserting k equally spaced values between c and d, yields the arithmetic sequence \(c, \overline{X}_{1}, \overline{X}_{2}, \overline{X}_{3}, \overline{X}_{4}, ..., \overline{X}_{k}, d\). Use this information for Exercise.
Insert four arithmetic means between 19 and 64.

See answers (1)

asked 2021-02-06

First, construct the sixth degree Taylor polynomial \(\displaystyle{P}_{{6}}{\left({x}\right)}\) for function \(\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({x}^{{2}}\right)}}}\) about \(\displaystyle{x}_{{0}}={0}\) The use \(\displaystyle{\int_{{{0}}}^{{{1}}}}{P}_{{6}}{\left({x}\right)}{\left.{d}{x}\right.}\) to approximate the integral \(\displaystyle\ {\int_{{{0}}}^{{{1}}}}{\sin{{\left({x}^{{2}}\right)}}}{\left.{d}{x}\right.}.\) Use 4-digit rounding arithmetic in all calculations. What is the approximate value?

See answers (1)

asked 2020-12-09

Determine whether the following state-ments are true and give an explanation or counterexample.
a) All polynomials are rational functions, but not all rational functions are polynomials.
b) If f is a linear polynomial, then \(\displaystyle{f}\times{f}\) is a quadratic polynomial.
c) If f and g are polynomials, then the degrees of \(\displaystyle{f}\times{g}\) and \(\displaystyle{g}\times{f}\) are equal.
d) To graph \(\displaystyle{g{{\left({x}\right)}}}={f{{\left({x}+{2}\right)}}}\), shift the graph of f 2 units to the right.

See answers (1)

asked 2020-10-28

Determine the algebraic modeling a.
One type of Iodine disintegrates continuously at a constant rate of \(\displaystyle{8.6}\%\) per day.
Suppose the original amount, \(\displaystyle{P}_{{0}}\), is 10 grams, and let t be measured in days.
Because the Iodine is decaying continuously at a constant rate, we use the model \(\displaystyle{P}={P}_{{0}}{e}^{k}{t}\) for the decay equation, where k is the rate of continuous decay.
Using the given information, write the decay equation for this type of Iodine.
b.
Use your equation to determine the half-life ofthis type of Fodine, That is, find ‘out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs.

See answers (1)

asked 2020-12-05

b. Let \(f(x) = \frac{e^{x}-e^{-x}}{x}\).
Use three-digit rounding arithmetic to evaluate f(0.1).

See answers (1)

Answer 1

We to find an error bound for the approximation and compare it to the actual error. The exact value of \(\displaystyle{\sin{{\left({0.34}\right)}}}={0.33348709214081}.\) So, the actual error is \(\displaystyle{\left({0.33348709214081}-{0.33349}\right)}={2.91}\times{10}^{{-{6}}}\) The bound for the error on [0.30, 0.35] is given by, \(\displaystyle{\left|{f{{\left({x}\right)}}}-{H}_{{5}}{\left({x}\right)}\right|}={\left|{\frac{{{{f}^{{6}}{\left(\xi\right)}}}}{{{6}!}}}{\left({x}={0.30}\right)}^{{2}}{\left({x}-{0.32}\right)}^{{2}}{\left({x}-{0.35}\right)}^{{2}}\right|}{N}{S}{K}={\left|{\frac{{-{\sin{{\left(\xi\right)}}}}}{{{720}}}}{\left({x}-{0.30}\right)}^{{2}}{\left({x}-{0.32}\right)}^{{2}}{\left({x}-{0.35}\right)}^{{2}}\right|}\) When \(\displaystyle\xi\in{\left[{0.30},{0.35}\right]}.\) Evaluating this error term at x = 0.34 yields, |sin(0.34) - H_5 (0.34)| = |\frac{-sin(\xi)}{720}(0.04)^2 (0.02)^2 (-0.01)^2|
\leq |\frac{-sin(0.35)}{720}(0.04)^2 (0.02)^2 (-0.01)^2|
\leq 3.05 \times 10^{-14}ZSK This bound is not inconsistent with the actual error, because the approximation was computed using five-digit rounding arithmetic.

Didn’t find what you are looking for?

Solve the given information Let f(x) = sin x. Determine an error bound for the approxiamtion f(0.34) = sin 0.34. and compare it to the actual error.

Answers (1)

Relevant Questions