In this problem we will see how to use Riemann Sums to calculate In(2).(a) According to the Fundamental Theorem of Calculus which bound b is needed to get1dx =In(2)? Show your work.(b) Draw the rectangular figure corresponding to a left endpoint Riemann sum with n = 5rectangles with equal bases. Is this an over or an under estimate?1(c) Give a clear explanation of how to estimate/calculate In(2) to within an accuracy of ɛ,for example, e = 0.01. Specifically how many subdivisions are needed?(d) With n = 5 compare the left endpoint approximation L5, the right endpointapproximation Rz and their average to ln(2) (use a calculator.) Which is best?

a. Given: ∫1b1xdx=ln2 By fundamental theorem of Calculus, we have…

In this problem we will see how to use Riemann Sums to calculate ln(2). (a) According to the Fundamental Theorem of Calculus which bound b is needed to get 1 dx In(2)? Show your work. (b) Draw the rectangular figure corresponding to a left endpoint Riemann sum with n = 5 rectangles with equal bases. Is this an over or an under estimate? 1 Y = 1 (c) Give a clear explanation of how to estimate/calculate In(2) to within an accuracy of ɛ, for example, ɛ = 0.01. Specifically how many subdivisions are needed?

In this problem we will see how to use Riemann Sums to calculate ln(2). (a) According to the Fundamental Theorem of Calculus which bound b is needed to get 1 dx In(2)? Show your work. (b) Draw the rectangular figure corresponding to a left endpoint Riemann sum with n = 5 rectangles with equal bases. Is this an over or an under estimate? 1 Y = 1 (c) Give a clear explanation of how to estimate/calculate In(2) to within an accuracy of ɛ, for example, ɛ = 0.01. Specifically how many subdivisions are needed?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Question

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In this problem we will see how to use Riemann Sums to calculate In(2).
(a) According to the Fundamental Theorem of Calculus which bound b is needed to get
1
dx =
In(2)? Show your work.
(b) Draw the rectangular figure corresponding to a left endpoint Riemann sum with n = 5
rectangles with equal bases. Is this an over or an under estimate?
1
(c) Give a clear explanation of how to estimate/calculate In(2) to within an accuracy of ɛ,
for example, e = 0.01. Specifically how many subdivisions are needed?
(d) With n = 5 compare the left endpoint approximation L5, the right endpoint
approximation Rz and their average to ln(2) (use a calculator.) Which is best?

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= 1) The function f(x) periodic on the interval [0, 2л] has complex Fourier series f(x): Σ(1/n²) einx where the sum over n goes from - infinity to infinity. Convert this to cosine and sine Fourier Series by finding the values of A's and B's in the expression Ao + ΣAn cos(nx) + Σ Bn sin(nx) where each sum goes from 1 to infinity. Hint: consider the n and -n term together in the complex Fourier Series or use Euler's identity.
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