{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.


(b) Calculate g'(𝓍)


g(𝓍) = ∫₀ˣ sin (πt² ) dt ( a Fresnel integral) 

Verified step by step guidance

1. Step 1: Recognize that the function g(𝓍) is defined as a definite integral with a variable upper limit. This is a classic application of the Fundamental Theorem of Calculus, which states that if g(𝓍) = ∫ₐˣ f(t) dt, then g'(𝓍) = f(𝓍), provided f is continuous.
2. Step 2: Identify the integrand of g(𝓍). In this case, the integrand is sin(πt²). According to the Fundamental Theorem of Calculus, g'(𝓍) will be equal to the integrand evaluated at the upper limit of integration, which is 𝓍.
3. Step 3: Substitute the upper limit 𝓍 into the integrand. This means g'(𝓍) = sin(π𝓍²).
4. Step 4: Confirm that the derivative g'(𝓍) does not require further simplification, as the integrand sin(π𝓍²) is already expressed in terms of 𝓍.
5. Step 5: Note that no additional integration or differentiation is needed, as the problem specifically asks for g'(𝓍), which is directly obtained using the Fundamental Theorem of Calculus.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative can be found by evaluating the integrand at that upper limit. Specifically, if g(x) = ∫ₐˣ f(t) dt, then g'(x) = f(x). This theorem is essential for calculating the derivative of functions defined by integrals.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, between two specified limits. In the context of the given function g(x) = ∫₀ˣ sin(πt²) dt, the integral computes the area under the curve of sin(πt²) from 0 to x. Understanding how to evaluate definite integrals is crucial for applying the Fundamental Theorem of Calculus.

Fresnel Integral

The Fresnel integral is a specific type of integral that arises in wave optics and is defined as g(x) = ∫₀ˣ sin(πt²) dt. It is important in various applications, including diffraction and interference patterns. Recognizing the properties and behavior of Fresnel integrals helps in understanding the function g(x) and its derivative.