{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?

Verified step by step guidance

1. Recall that the Fresnel integral \( S(x) = \int_0^x \sin(t^2) \, dt \) can be expanded into a Maclaurin series by first expanding \( \sin(t^2) \) as a power series and then integrating term-by-term.
2. Write the Maclaurin series for \( \sin(z) \), which is \( \sin(z) = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \). Substitute \( z = t^2 \) to get \( \sin(t^2) = \sum_{n=0}^\infty (-1)^n \frac{t^{4n+2}}{(2n+1)!} \).
3. Integrate the series term-by-term from 0 to \( x \) to find the Maclaurin series for \( S(x) \): \[ S(x) = \sum_{n=0}^\infty (-1)^n \frac{1}{(2n+1)!} \int_0^x t^{4n+2} \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+3}}{(2n+1)! (4n+3)}. \]
4. To approximate \( S(0.05) \) with an error no greater than \( 10^{-4} \), use the alternating series remainder estimation. The error is less than or equal to the absolute value of the first omitted term in the series.
5. Calculate the terms of the series for increasing \( n \) until the absolute value of the next term is less than or equal to \( 10^{-4} \). The number of terms used up to that point is the number required for the desired accuracy.

Key Concepts

Fresnel Integrals

Fresnel integrals S(x) and C(x) are defined as integrals of sine and cosine functions with quadratic arguments, specifically S(x) = ∫₀ˣ sin(t²) dt and C(x) = ∫₀ˣ cos(t²) dt. They arise in optics and wave theory, representing diffraction patterns. Understanding their integral form is essential for approximating their values using series expansions.

Maclaurin Series Expansion

A Maclaurin series is a Taylor series expansion of a function about zero, expressing it as an infinite sum of derivatives at zero multiplied by powers of x. For Fresnel integrals, the integrand sin(t²) can be expanded into a power series, which is then integrated term-by-term to find the series for S(x). This allows approximation of S(x) by summing a finite number of terms.

Error Estimation in Series Approximation

When approximating functions by partial sums of their series, the remainder or error term measures the difference from the exact value. To ensure the error is below a given tolerance (e.g., 10⁻⁴), one must estimate the size of the next term or use bounds on the remainder. This guides how many terms are needed for the desired accuracy.