10. Given that y = sin-1 2x. Show that (1 - 4x2) d²y = 4x dy dx? dr Find the Maclaurin series for sin 2x in ascending powers of x up to the term in x. Using the series expansion above, where x = estimate the value of 4 correct to four decimal places.

Q: (b) Obtain 3 non zero terms of the Maclaurins series for sin²x. Hence, evaluate sin² x 0.5 1.0.5 5…

A: The given problem is to evaluate the given definite integral using Maclaurin series of sin2x upto 3…

Q: Compute the value of the improper integral. (If the integral diverges to o, type 00; if the integral…

A: Integral Test : Suppose f(x) is continuous , positive and decreasing function on the interval [k,∞)…

Q: 2. Obtain a half range cosine series for kx, for 0 < x <5 f(x) = (k(l – x), for 글드 xS 1 1 1 Deduce…

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Q: Use a power series to approximate the definite integral to within 0.001. 1/2 a² In(1+ x) dx 0.

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Q: d Evaluate the differential dx of the series at x = 1rad. (x2 sin knx) where k> 0. Determine also…

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Q: (a) Write the first 5 terms of a power series approximation for f(x) = e centered atx 0. (b)Use the…

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Q: b) Obtain a Taylors series of degree three for the function f(x) = log (1+2x) near x = 1 hence…

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Q: Find the Maclaurin Series of the following functions. ex 7 sinh(x) 8. 1- x 9 ) x² arctan(x

A: The Taylor series of a function, f(x) about x=a is defined as: f(x)=∑n=0∞f(n)(a)n!(x−a)n The…

Q: It is known that +00 sin x = Σ (-1)" (2n + 1)! n=0 1. Find the Maclaurin series for x sin(x²).…

A: Solution (1) Maclaurin SeriesTaylor series of function fx at a is defined…

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Q: The value of ao for cosine series on interval [0,]for ƒ(x) = x is Select one: a. b. 0 C. d. NOT e. 1

A: The Fourier cosine series of a function f(x) over an interval [a,b] is given by;

Q: 1. Using a Maclaurin series, evaluate the integral shown below to an accuracy of 0.0001: sin x – x…

A: To evaluate , I = ∫01sinx - xx2dx

Q: If the kth partial sum, SŁ of the telescoping series E [sin(v4n) – sin(/4(n+ 1))] n=1 is given by…

A: We will use formula for kth partial sum and compare the result with given Sk

Q: Develop sin t in the Half-Range cosine series in the interval 0<t<L. L :(0)\ novo ms 10l

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Q: 3. (a) What is the slope of the curve y = 2x* – x³at x = 1? sin x (b) If y = dy find dx 2 (c) Find…

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Q: (b) Obtain 3 non zero terms of the Maclaurins series for sin? r. Hence, evaluate 0.5 sin² x dx. Give…

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Q: By replacing cos(z") with the partial Maclaurin series you found in part (b), compute the exact…

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Q: The Maclaurin series of sin.x is given by sinx = 1! 3! ' 5! 7! Use the above expansion to check if…

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Q: 5. Approximate the definite integral sin(x?)dx correct to within 10-3. Hint: First find the…

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Q: Find the sum of the series Σ >x sin" (x). n=0

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Q: Let the improper integrals " sin(-)dx and sin?()dx (i) Write down the first 3 terms of Maclaurin…

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Q: The Maclaurin series of sin.x is given by * x7 +. 7! sinx = - 1! 3! 5! Use the above expansion to…

A: Given: Differential equation xy''+sinxy=0. Maclaurin series of sinx,…

Q: compute the values up to four decimal places of the following using maclaurin's series: Sinh^2 (0.5)

A: We have to find value of the sinh20.5 using Maclaurin's series.

Q: /3 In (x+ 1) The series representing the integral dx is given by

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Q: 00 2k f(x) = *, kk W!

A: In this question, we solve the limit and nth derivative from power series as well as the power…

Q: - 5 Use a series to estimate the following integral's value with an error of magnitude less than 10…

A: To use a series to estimate the value of the integral with an error of magnitude less than 10-5 Let…

Q: Q5. Let the improper integrals, sin()dx and sin² ()dx (i) (ii) Write down the first 3 terms of…

A: Introduction: Sometimes the Maclaurin series is very efficient to integrate a function. The power…

Q: d E A. By power series, prove that: -cosx = -sinx dx

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Q: • Derive the machurin series for f(x) = cesx. Then write down the Equation of the parabola that best…

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Q: Use the identity 2 sin x cOs x = sin(2x) to find the power series expansion of sin x at x = 0.…

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Q: If f(x) = x for 0 < x < n/2 and f(x) = a – x for 1/2 < x < n, show that the cosine series for this…

A: Given that, fx=x0≤x≤π2π-xπ2&lt;x≤π Here, fx is defined on the interval, 0, π. In this case L=π and…

Q: Prove that for x < 1, x5 dx = A + x - x4 +1 9. Use the first two terms to approximate 1/2 4 dx/ (x*…

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Q: We know from integral calculus that by using the substitution method: 2x dx = arctan(x²) + C 1+x*…

A: We know form the Maclaurin series representation formula:

Q: a) Evaluate the given integral x + 3x – 3 (x + 1)(x² + 6x + 10) b) sin'x cosx+sinx c) Find the…

A: According to the Bartleby Guidelines only one question can be solved at once, post the remaining…

Q: (i) Find the Maclaurin series for sin x and prove that it represents sin x for all x E R.

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: -8 Use a series to estimate the following integral's value with an error of magnitude less than 10…

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Q: For the Maclaurin series expansion of cos x xºx² COS X = terms 1 to + 0! 2! 4! 6! Complete the table…

A: Given: cos(x)=x00!-x22!+x44!-x66!+⋯ To find: Relative error up to 6 terms.

Q: I. Find a power series for the given functions. 1. f(x) = 2 centered at 0. %D 2-х2

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Q: 4. Consider the following function: So, sin rx, 0 16k2 k=1 1 - -

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Q: Compute the value of the following improper integral. If it converges, enter its value. Enter…

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Q: Q4 compute the Fourier sine and Fourier cosine series for the given function. Sketch for a few…

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Q: Compute accurate to 3 decimal places using binomial series the value of the definite integral. dx a.…

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Q: Q1 Find Maclaurim series of the function f(x)= sinh(x).

A: Given function isf(x)=sinh(x)The Maclaurin series of the functionis determined as shown below

Q: (b) The Maclaurin series of sinx is given by sin x=| 1! 3! 5! 7! Use the above expansion to check if…

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Q: 1. Use the Integral Test to show that diverges. x2 +1 n= 2. What does the Integral Test say about…

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Q: ex-e-* Find Maclaurin series for: 1. sinh x = 2

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Q: Given the Maclaurin series for sin x is sinx E=o(-1)" (2n+1)! x2n+1 %3| find the Maclaurin series…

A: The given function is:   f(x)=x sin(3x)

Q: Use a series representation of sin 3x to find values of r and s for which (sin 3x lim + s) = 0. x2

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Q: For x > 0, the power series 1 - + * +... + (-1)". x2n + (2n+1)! converges to which of the following?…

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Q: Compute the value of the improper integral. (If the integral diverges to co, type oo; if the…

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