5. High accuracy Differentiation formulas. Using Taylor series.
5. High accuracy Differentiation formulas. Using Taylor series.
(a) Prove the following centered finite difference formula that is \(\mathrm{O}\left(\mathrm{h}^{4}\right)\) for the first derivative
$$ f^{\prime}\left(x_{i}\right)=\frac{-f\left(x_{i+2}\right)+8 f\left(x_{i+1}\right)-8 f\left(x_{i}-1\right)+f\left(x_{i-2}\right)}{12 h}+O\left(h^{4}\right) $$
(b) Compute the centered finite difference approximation of \(\mathrm{O}\left(\mathrm{h}^{2}\right)\) and \(\mathrm{O}\left(\mathrm{h}^{4}\right)\) for the first derivative of \(y=\sin x\) at \(x=\pi / 4\) using the value of \(h=\pi / 12\). Calculate the true percent relative error in both cases.
Homework Answers
Approximating derivatives
Approximating derivatives$$ \begin{aligned} f^{\prime}(x) & \approx\left(\delta_{+} f\right)(x)=\frac{f(x+h)-f(x)}{h} & \text { Forward difference } \\ f^{\prime}(x) & \approx\left(\delta_{-} f\right)(x)=\frac{f(x)-f(x-h)}{h} & \text { Backward difference } \\ f^{\prime}(x) & \approx(\delta f)(x)=\frac{f(x+h)-f(x-h)}{2 h} & \text { Centered difference for 1st derivative } \\ f^{\prime \prime}(x) & \approx\left(\delta^{2} f\right)(x)=\frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}} & \text { Centered difference for 2nd derivative } \end{aligned} $$1. a. Use Taylor's polynomials to derive the centered difference approximation for the first derivative:$$ f^{\prime}(x) \approx \delta f(x)=\frac{f(x+h)-f(x-h)}{2 h}, $$include the error in...
(a) Use the following data to find the velocity and acceleration at t = 10 seconds:
Numerical methods(a) Use the following data to find the velocity and acceleration at t = 10 seconds:Time (s):0246810121416Position (m):00.71.83.45.16.37.38.08.4Use second-order correct (i) centered finite-difference, and (ii) backward finite-difference methods. (b) Use the Taylor expansions for f(x +h), f(x+2h), f(x +3h) and derive the following forward finite-difference formulas for the second derivative. Write down the error term$$ f^{\prime \prime}(x) \approx \frac{-f(x+3 h)+4 f(x+2 h)-5 f(x+h)+2 f(x)}{h^{2}} $$
PLEASE ANSWER ALL FIVE PROBLEMS
Find \(\mathrm{dy} / \mathrm{dt}\).12) \(y=\cos ^{5}(\pi t-8)\)A) \(-5 \pi \cos ^{4}(\pi t-8) \sin (\pi t-8)\)B) \(-5 \cos ^{4}(\pi \mathrm{t}-8) \sin (\pi \mathrm{t}-8)\)C) \(5 \cos ^{4}(\pi t-8)\)D) \(-5 \pi \sin ^{4}(\pi t-8)\)Use implicit differentiation to find dy/dx.13) \(x y+x=2\)A) \(-\frac{1+y}{x}\)B) \(\frac{1+y}{x}\)C) \(\frac{1+x}{y}\)D) \(-\frac{1+x}{y}\)Find the derivative of \(y\) with respect to \(x, t\), or \(\theta\), as appropriate.14) \(y=\ln 8 x^{2}\)A) \(\frac{2}{x}\)B) \(\frac{1}{2 x+8}\)C) \(\frac{2 x}{x^{2}+8}\)D) \(\frac{16}{x}\)Find the derivative of \(\mathrm{y}\) with respect to \(\mathrm{x}, \mathrm{t}\), or \(\theta\), as appropriate.15) \(y=\left(x^{2}-2 x+6\right) e^{x}\)A)...
Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+...
Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+ (+0) (1.1) 4! show that the central finite difference formula for the first derivative can be written as: f'(x)= f(x+h)-f(x-1) + ch" +0(hº) (1.2) 2h Determine cp and of the derived equation. [4 marks] Consider the function: f(x) = sin +COS (1.3) 2 2 Let x =ih with n=0.25, give your answer in 3 decimals for (ii) to (vi): (ii) Evaluate f(x) for i...
1-Given the function:
1-Given the function: \(y=\frac{x^{2}-3 x-4}{x^{2}-5 x+4}\), decide if \(f(x)=y\) is continuous or has a removable discontinuity, and find horizontal tond vertical asymptotes.2 A-Use the definition \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) to prove that derivative of \(f(x)=\sqrt{4-x}\) is \(\frac{-1}{2 \sqrt{4-x}}\)2 B- Evaluate the limit \(\lim _{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\) for the given value of \(x\) and function \(f(x) .\)$$ f(x)=\sin x, \quad x=\frac{\pi}{4} $$3-Given the function: \(y=(x+4)^{3}(x-2)^{2}\), find y' and classify critical numbers very carefully using first derivative tess...
(2 points) The area A of the region S that lies under the graph of the...
(2 points) The area \(A\) of the region \(S\) that lies under the graph of the continuous function \(f\) on the interval \([a, b]\) is the limit of the sum of the areas of approximating rectangles:$$ A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x $$where \(\Delta x=\frac{b-a}{n}\) and \(x_{i}=a+i \Delta x\).The expression$$ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right) $$gives the area of the function \(f(x)=\) on...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the finite difference formula defined by the second row of Table 5.2. Table 5.2. Weights for forward finite difference formulas (p 0 in (5.4.2). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q=0. The term order of accuracy is explained in Section 5.5. Order of Node location 2h 3h 4h accuracy 1 2...
Use logarithmic differentiation to determine the derivative
Let $$ f(x)=\frac{x^{4}(x-3)^{2}}{\left(x^{2}+6\right)^{7}} $$Use logarithmic differentiation to determine the derivative f'(x) = f'(3) =
Question 1 15 Points) It is always desirable to have/ use the finite difference approximation wit...
Question 1 15 Points) It is always desirable to have/ use the finite difference approximation with error term. Please using the Taylor Series: higher order of truncation sw(x) h" +R 2! 3! (I) Derive the following forward difference approximation of the 2nd orde 2) What is the order of error for this case? derivative of f(x). f" derivative off(x) h2
Question 1 15 Points) It is always desirable to have/ use the finite difference approximation with error term. Please using...
Question 1. Compute the derivative of the following functions.
Question 1. Compute the derivative of the following functions.(a) \(f(x)=x^{3}-\frac{2}{\sqrt{x}}+4\)(b) \(f(x)=2^{3 x-1}\)(c) \(f(x)=\ln \left(5 x^{2}+1\right)\)(d) \(f(x)=\frac{\tan (x)}{x^{2}+1}\)(e) \(f(x)=e^{x^{2}} \cdot \arctan (2 x)\)(f) \(f(x)=\sin (x)^{2} \cdot\left(\tan (x)+\cos (x)^{2}\right)\).Question 2. In geometry, the folium of Descartes is a curve given by the equation$$ x^{3}+y^{3}-3 a x y=0 $$Here, \(a\) is a constant.The curve was first proposed by Descartes in 1638 . Its claim to fame lies in an incident in the development of calculus. Descartes challenged Fermat to find the tangent line...
Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+ (+0) (1.1) 4! show that the central finite difference formula for the first derivative can be written as: f'(x)= f(x+h)-f(x-1) + ch" +0(hº) (1.2) 2h Determine cp and of the derived equation. [4 marks] Consider the function: f(x) = sin +COS (1.3) 2 2 Let x =ih with n=0.25, give your answer in 3 decimals for (ii) to (vi): (ii) Evaluate f(x) for i...
(2 points) The area \(A\) of the region \(S\) that lies under the graph of the continuous function \(f\) on the interval \([a, b]\) is the limit of the sum of the areas of approximating rectangles:$$ A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x $$where \(\Delta x=\frac{b-a}{n}\) and \(x_{i}=a+i \Delta x\).The expression$$ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right) $$gives the area of the function \(f(x)=\) on...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the finite difference formula defined by the second row of Table 5.2. Table 5.2. Weights for forward finite difference formulas (p 0 in (5.4.2). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q=0. The term order of accuracy is explained in Section 5.5. Order of Node location 2h 3h 4h accuracy 1 2...
Question 1 15 Points) It is always desirable to have/ use the finite difference approximation with error term. Please using the Taylor Series: higher order of truncation sw(x) h" +R 2! 3! (I) Derive the following forward difference approximation of the 2nd orde 2) What is the order of error for this case? derivative of f(x). f" derivative off(x) h2
Question 1 15 Points) It is always desirable to have/ use the finite difference approximation with error term. Please using...
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medolol Sat, Nov 27, 2021 11:24 AM