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1. (a) We want to develop a method for calculating the function sint dt f)-inf t...
Q1 2016 a) We want to develop a method for calculating the function f(x) = sin(t)/t...
Q1 2016
a) We want to develop a method for calculating the function f(x)
= sin(t)/t
dt
for small or moderately small values of x. this is a special
function called the sine integral, and it is related to another
special function called the exponential integral. it rises in
diffraction problems.
Derive a Taylor-series expression for f(x), and give an upper
bound for the error when the series is terminated after the n-th
order term. sint = see image
b)we...
(b) In diffraction theory, it is sometimes necessary to evaluate the function sine f (x) = for small to moderate (posit...
(b) In diffraction theory, it is sometimes necessary to evaluate the function sine f (x) = for small to moderate (positive) values of the variable x. One way to do this is to make use of the Taylor-MacLaurin series θ2n-1 θ-31+5!-...+ (-1)"-1 sin θ = (2n-1+ Rr) with remainder term θ2n I lere, ξ is some number in the interval 0 < ξ < θ. Derive a Taylor-series expression for f(x), and give an upper bound for the crror when...
f(x) = | (1 point) Consider the function cos(t) - 1 dt. Which of the following...
f(x) = | (1 point) Consider the function cos(t) - 1 dt. Which of the following is the Taylor Series for f(x) centred at 2-07 A (-1)"(2n - 2). 2-3 (2n) B. (-1)" (2n - 1)(2n)! 2-1 + C. no ho 2n-1 (-1)" (2n - 1)(2n)! (-1)"_222 D. (2n +1)!
(1 point) Consider the function f(x) = f* cos(t) – 1 dt. t2 Which of the...
(1 point) Consider the function f(x) = f* cos(t) – 1 dt. t2 Which of the following is the Taylor Series for f(x) centred at x = 0? w A. (-1)" (2n – 1)(2n)! -x2n- +C. n=0 (-1)"(2n – 2) 2n–3. B. (2n)! n=1 c. Σ (-1)" (2n + 1)! -x2n-2 n=1 D. Š (-1)" -X2n-1 (2n – 1)(2n)! n=1
(1 point) Consider the function cos(t) f(x) = dt. Which of the following is the Taylor...
(1 point) Consider the function cos(t) f(x) = dt. Which of the following is the Taylor Series for f(2) centred at x = 0? O (-1)" A. 2n-1 (2n - 1)(2n)! O B. (-1)" (2n – 1)(2n)! 2n-1 +C n0 O C. (-1)" 220-2 (2n +1)! (-1)"(2n - 2) (2n)! D. n=1 2n 3
Problem 12. (1 point) Consider the function f(0) = %,* cos(t) – 1 dt. +2 Which...
Problem 12. (1 point) Consider the function f(0) = %,* cos(t) – 1 dt. +2 Which of the following is the Taylor Series for f(2) centred at x = 0? O (-1)" A. n1 (2n – 1)(2n); 22n-1 (-1)"(2n - 2) B. n1 22n-3 (2n)! (-1) C. n0 -22n-1 +C (2n-1)(2n)! D. (-1) 2n-2 1 (2n +1)!
(1 point) Consider the function f(x) = Es cos(t) – 1 t2 dt. Which of the...
(1 point) Consider the function f(x) = Es cos(t) – 1 t2 dt. Which of the following is the Taylor Series for f(x) centred at x = 0? 2n-1 Α.Σ (-1)" (2n – 1)(2n)! X +C. n=0 oo 2n-1 B. (-1)" (2n – 1)(2n)!" X n=1 (-1)" X20-2 (2n + 1)! M n=1 D. iM: (-1)"(2n – 2), 2n–3 (2n)! X n=1
1. (a) We need to calculate accurate values of the function for very large values of x. However, it is found that ju...
1. (a) We need to calculate accurate values of the function for very large values of x. However, it is found that just programming this formula into a computer gives very poor accuracy for large x Explain why this happens, and show how to re-write the function so that it can be used reliably, even when x is large. [6 points] (b) In diffraction theory, it is sometimes necessary to evaluate the function sin θ f(x) for small to moderate...
(1 point) Given the function f(t) = {sine if 0 <t< 61 if 61 <t. sint...
(1 point) Given the function f(t) = {sine if 0 <t< 61 if 61 <t. sint 61) Express f(t) in terms of the shifted unit step function uſt – a) f(t) = Now find the Laplace transform F(s) of f(t) F(s) =
point) Consider a function f(x) that has a Taylor Series centred at x = 5 given...
point) Consider a function f(x) that has a Taylor Series centred at x = 5 given by ſan(x – 5)" n=0 he radius of convergence for this Taylor series is R= 4, then what can we say about the radius of convergence of the Power Series an ( 5)"? nons A. R= 20 B.R= 8 C. R=4 D. R= E. R= 2 F. It is impossible to know what R is given this information. point) Consider the function f(x) =...
Q1 2016
a) We want to develop a method for calculating the function f(x)
= sin(t)/t
dt
for small or moderately small values of x. this is a special
function called the sine integral, and it is related to another
special function called the exponential integral. it rises in
diffraction problems.
Derive a Taylor-series expression for f(x), and give an upper
bound for the error when the series is terminated after the n-th
order term. sint = see image
b)we...
(b) In diffraction theory, it is sometimes necessary to evaluate the function sine f (x) = for small to moderate (positive) values of the variable x. One way to do this is to make use of the Taylor-MacLaurin series θ2n-1 θ-31+5!-...+ (-1)"-1 sin θ = (2n-1+ Rr) with remainder term θ2n I lere, ξ is some number in the interval 0 < ξ < θ. Derive a Taylor-series expression for f(x), and give an upper bound for the crror when...
f(x) = | (1 point) Consider the function cos(t) - 1 dt. Which of the following is the Taylor Series for f(x) centred at 2-07 A (-1)"(2n - 2). 2-3 (2n) B. (-1)" (2n - 1)(2n)! 2-1 + C. no ho 2n-1 (-1)" (2n - 1)(2n)! (-1)"_222 D. (2n +1)!
(1 point) Consider the function f(x) = f* cos(t) – 1 dt. t2 Which of the following is the Taylor Series for f(x) centred at x = 0? w A. (-1)" (2n – 1)(2n)! -x2n- +C. n=0 (-1)"(2n – 2) 2n–3. B. (2n)! n=1 c. Σ (-1)" (2n + 1)! -x2n-2 n=1 D. Š (-1)" -X2n-1 (2n – 1)(2n)! n=1
(1 point) Consider the function cos(t) f(x) = dt. Which of the following is the Taylor Series for f(2) centred at x = 0? O (-1)" A. 2n-1 (2n - 1)(2n)! O B. (-1)" (2n – 1)(2n)! 2n-1 +C n0 O C. (-1)" 220-2 (2n +1)! (-1)"(2n - 2) (2n)! D. n=1 2n 3
Problem 12. (1 point) Consider the function f(0) = %,* cos(t) – 1 dt. +2 Which of the following is the Taylor Series for f(2) centred at x = 0? O (-1)" A. n1 (2n – 1)(2n); 22n-1 (-1)"(2n - 2) B. n1 22n-3 (2n)! (-1) C. n0 -22n-1 +C (2n-1)(2n)! D. (-1) 2n-2 1 (2n +1)!
(1 point) Consider the function f(x) = Es cos(t) – 1 t2 dt. Which of the following is the Taylor Series for f(x) centred at x = 0? 2n-1 Α.Σ (-1)" (2n – 1)(2n)! X +C. n=0 oo 2n-1 B. (-1)" (2n – 1)(2n)!" X n=1 (-1)" X20-2 (2n + 1)! M n=1 D. iM: (-1)"(2n – 2), 2n–3 (2n)! X n=1
1. (a) We need to calculate accurate values of the function for very large values of x. However, it is found that just programming this formula into a computer gives very poor accuracy for large x Explain why this happens, and show how to re-write the function so that it can be used reliably, even when x is large. [6 points] (b) In diffraction theory, it is sometimes necessary to evaluate the function sin θ f(x) for small to moderate...
(1 point) Given the function f(t) = {sine if 0 <t< 61 if 61 <t. sint 61) Express f(t) in terms of the shifted unit step function uſt – a) f(t) = Now find the Laplace transform F(s) of f(t) F(s) =
point) Consider a function f(x) that has a Taylor Series centred at x = 5 given by ſan(x – 5)" n=0 he radius of convergence for this Taylor series is R= 4, then what can we say about the radius of convergence of the Power Series an ( 5)"? nons A. R= 20 B.R= 8 C. R=4 D. R= E. R= 2 F. It is impossible to know what R is given this information. point) Consider the function f(x) =...
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