{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
c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

Verified step by step guidance

1. Recall that the Fresnel integrals are defined as \(S(x) = \int_0^x \sin(t^2) \, dt\) and \(C(x) = \int_0^x \cos(t^2) \, dt\). To approximate these integrals, we use polynomial approximations (Taylor series) of the integrands \(\sin(t^2)\) and \(\cos(t^2)\) from part (b).
2. Write down the Taylor series expansions for \(\sin(t^2)\) and \(\cos(t^2)\) around \(t=0\). For example, \(\sin(t^2)\) can be expanded as \(t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} - \cdots\), and \(\cos(t^2)\) as \$1 - \frac{t^4}{2!} + \frac{t^8}{4!} - \cdots$.
3. Substitute these polynomial approximations into the integrals for \(S(x)\) and \(C(x)\), so that \(S(x) \approx \int_0^x \left( t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} - \cdots \right) dt\) and \(C(x) \approx \int_0^x \left( 1 - \frac{t^4}{2!} + \frac{t^8}{4!} - \cdots \right) dt\).
4. Integrate each term of the polynomial inside the integral with respect to \(t\) from 0 to \(x\). For example, \(\int_0^x t^2 dt = \frac{x^3}{3}\), \(\int_0^x t^6 dt = \frac{x^7}{7}\), and so on. This will give you polynomial expressions in terms of \(x\).
5. Finally, plug in the values \(x=0.05\) for \(S(0.05)\) and \(x=-0.25\) for \(C(-0.25)\) into the resulting polynomial expressions to get the approximate values of the Fresnel integrals.

Key Concepts

Fresnel Integrals

Fresnel integrals S(x) and C(x) are defined as the integrals of sin(t²) and cos(t²) from 0 to x, respectively. They arise in wave optics and describe diffraction patterns. Understanding their definition and properties is essential for approximating their values.

Polynomial Approximation of Functions

Polynomial approximations, such as Taylor or Maclaurin series, express complex functions as sums of polynomial terms. These approximations simplify evaluating integrals or functions at specific points, especially when exact integration is difficult or impossible.

Evaluating Definite Integrals Using Approximations

Approximating definite integrals like S(x) and C(x) often involves substituting polynomial approximations of the integrand and integrating term-by-term. This method provides numerical estimates for integral values at given points, useful when closed-form solutions are unavailable.