{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)

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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.

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.

Step 3: Substitute the upper limit 𝓍 into the integrand. This means g'(𝓍) = sin(π𝓍²).

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

{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)