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'(𝓍)
g(𝓍) = ∫₀ˣ sin (πt² ) dt ( a Fresnel integral)
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8. Definite Integrals
Fundamental Theorem of Calculus
2:49 minutesProblem 5.R.30
Textbook Question
Evaluate the following derivatives.
d/d𝓍 ∫₃ᵉˣ cos t² dt
Verified step by step guidance1
Step 1: Recognize that the problem involves differentiating an integral with variable limits. This is a classic application of the Leibniz rule for differentiation under the integral sign.
Step 2: Recall the Fundamental Theorem of Calculus, which states that if you have an integral of the form ∫ₐᵇ f(t) dt, where the limits of integration are functions of x, the derivative with respect to x is given by: d/d𝓍 ∫ₐᵇ f(t) dt = f(b) * (db/d𝓍) - f(a) * (da/d𝓍).
Step 3: Identify the limits of integration in the given problem. The lower limit is a constant (3), and the upper limit is eˣ, which is a function of x. This means da/d𝓍 = 0 and db/d𝓍 = d/d𝓍(eˣ) = eˣ.
Step 4: Substitute the limits into the formula. The derivative becomes: cos((eˣ)²) * eˣ - cos(3²) * 0. Note that the second term vanishes because the derivative of a constant lower limit is zero.
Step 5: Simplify the expression. The final derivative is cos((eˣ)²) * eˣ. This is the result after applying the Leibniz rule and simplifying.
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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 links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the integral of f from a to b can be computed using F. This theorem allows us to evaluate the derivative of an integral function, which is essential for solving the given problem.
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Differentiation Under the Integral Sign
Differentiation under the integral sign is a technique that allows us to differentiate an integral with respect to a parameter. In this case, we differentiate the integral of cos(t²) with respect to x, treating the limits of integration as constants. This method is crucial for evaluating the derivative of the given integral expression.
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Chain Rule
The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. When applying the Chain Rule, we differentiate the outer function and multiply it by the derivative of the inner function. In the context of the given problem, it helps in evaluating the derivative of the integral with respect to x, especially when the limits of integration are functions of x.
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