Textbook Question
Working with area functions Consider the function Ζ and its graph.
(b) Estimate the points (if any) at which A has a local maximum or minimum.
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8. Definite Integrals
Fundamental Theorem of Calculus
1:44 minutesProblem 5.3.102b
Textbook Question
{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'(π)
g(π) = β«βΛ£ sin (ΟtΒ² ) dt ( a Fresnel integral)
Verified step by step guidance1
Step 1: Recognize that the function g(π) 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(π) = β«βΛ£ f(t) dt, then g'(π) = f(π), provided f is continuous.
Step 2: Identify the integrand of g(π). In this case, the integrand is sin(ΟtΒ²). According to the Fundamental Theorem of Calculus, g'(π) will be equal to the integrand evaluated at the upper limit of integration, which is π.
Step 3: Substitute the upper limit π into the integrand. This means g'(π) = sin(ΟπΒ²).
Step 4: Confirm that the derivative g'(π) does not require further simplification, as the integrand sin(ΟπΒ²) is already expressed in terms of π.
Step 5: Note that no additional integration or differentiation is needed, as the problem specifically asks for g'(π), which is directly obtained using the Fundamental Theorem of Calculus.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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) = β«βΛ£ f(t) dt, then g'(x) = f(x). This theorem is essential for calculating the derivative of functions defined by integrals.
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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) = β«βΛ£ sin(ΟtΒ²) dt, the integral computes the area under the curve of sin(ΟtΒ²) from 0 to x. Understanding how to evaluate definite integrals is crucial for applying the Fundamental Theorem of Calculus.
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Fresnel Integral
The Fresnel integral is a specific type of integral that arises in wave optics and is defined as g(x) = β«βΛ£ sin(ΟtΒ²) 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.
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