{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
a. Compute S′(x) and C′(x).

Verified step by step guidance

1. Identify the given functions: \( S(x) = \int_0^x \sin(t^2) \, dt \) and \( C(x) = \int_0^x \cos(t^2) \, dt \). These are defined as definite integrals with variable upper limits.
2. Recall the Fundamental Theorem of Calculus Part 1, which states that if \( F(x) = \int_a^x f(t) \, dt \), then \( F'(x) = f(x) \), provided \( f \) is continuous.
3. Apply the theorem to \( S(x) \): since \( S(x) = \int_0^x \sin(t^2) \, dt \), its derivative is \( S'(x) = \sin(x^2) \).
4. Similarly, apply the theorem to \( C(x) \): since \( C(x) = \int_0^x \cos(t^2) \, dt \), its derivative is \( C'(x) = \cos(x^2) \).
5. Summarize the results: \( S'(x) = \sin(x^2) \) and \( C'(x) = \cos(x^2) \). These derivatives express the rate of change of the Fresnel integrals with respect to \( x \).

Key Concepts

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative is the integrand evaluated at that limit. For example, if F(x) = ∫₀ˣ f(t) dt, then F'(x) = f(x). This principle is essential for finding derivatives of integral-defined functions like Fresnel integrals.

Fresnel Integrals

Fresnel integrals S(x) and C(x) are special functions defined as integrals of sine and cosine of t squared, respectively. They arise in optics and wave theory, representing integrals that cannot be expressed in elementary functions. Understanding their definitions is key to differentiating them correctly.

Differentiation of Integral Functions with Variable Limits

When differentiating an integral with a variable upper limit, the derivative is the integrand evaluated at that upper limit, assuming continuity. This concept allows direct computation of S'(x) and C'(x) by substituting x into sin(t²) and cos(t²), respectively.