Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due

Math 1172 Remainders Project - Written Portion

Due: Monday, April 7 by 11:59 PM EST

Problem 1 [55 pts]: Many integrals that arise in applications cannot be evaluated in terms of the common functions from calculus (like exponentials, trigonometric functions, logarithms, polynomials, etc) that we have studied. One important type of such integrals are called Fresnel integrals, which arise in the study of optics. There are two such integrals:

A. [10 pts] Use the Taylor series for cos(t) and the rules discussed in class to show that the Taylor series for C(x) =  dt centered at x = 0 is · In order to earn full credit, show your work in summation notation and perform all calculations by hand.

B. [10 pts] Suppose now that 0 ≤ x ≤ 1. Explain (1) why the series  converges and (2) why the remainder results for alternating series can be used to approximate the value of the series.

C. [25 pts] The series from the previous page and the remainder results for alternating series can be used to approximate the following integrals to within .00001 of their exact values. For each integral below, do the following.

– Find the smallest N guaranteed by he alternating series remainder results so  approximates  dt to within .00001 of its exact value.

Hint: You will have to use technology to do this; there is no general way to solve equations with factorials analytically. Make sure to justify why the

n

-value you found is the smallest!

– Justify why the N you found is the smallest such N. Do not just use technology to compute the integral and find N that way.

– Evaluate  for the N you find and the integral  dt to verify that the sum approximates the integral to within .00001.

– Write down the results to 6 decimal places.

• The alternating series remainder results show that to within .00001,  dx ≈ .

• I used to find to 6 decimal places,  dx = .

• The alternating series remainder results show that to within .00001,  dx ≈ .

• I used to find to 6 decimal places,  dx = .

D. [10 pts] If you worked Part C correctly, you will notice that the number of terms you need to use in the series to obtain your approximations is different. Without referencing your calculations, explain why you should expect why you need more terms in series for  dt than you do in the series for  dt.

In your explanation, you should explicitly reference (1) the relationship between the partial sums and the Taylor polynomials and (2) the relationship of the x-values where you approximate both integrals and the center of the Taylor series.

Problem 2 [45 pts] Bessel functions arise in various engineering and physics applications. They are solutions to the differential equation

and it can be shown that there is no “nice” formula for them. However, it is possible to use the equation to find power series representations for them. When α = 1, it can be shown that the power series is

is a solution to this equation.

A. [5 pts] Use the internet to find four different instances or applications of Bessel’s function.

B. [10 pts] Use the third order Taylor polynomial for J1(x) to approximate J1(3). Report your final answer to 6 decimal places.

C. [10 pts] Explain why the alternating series remainder results can be used to find the maximum possible error made by this approximation, and calculate this maximum possible error to 6 places.

D. [10 pts] Explain whether the higher order Taylor polynomials should approximate J1(3) than the lower order ones.

E. [10 pts] Suppose that we want to approximate J1(3) to within 1 × 10−10 of its exact value. According to the alternating series remainder results, what is the lowest order Taylor polynomial that should be used to perform this approximation? Explain your response.