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d (a) / ₁ (t²-1)²dt. -10 -10 (b)/(t²-1)²dt. (c) d dx 32² -10 3x² (t²-1)² dt. d (d) — (t²-1)²dt. dx 4x-1

d (a) / ₁ (t²-1)²dt. -10 -10 (b)/(t²-1)²dt. (c) d dx 32² -10 3x² (t²-1)² dt. d (d) — (t²-1)²dt. dx 4x-1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Calculate the following derivatives using the first part of the fundamental theorem of calculus. You do not need to simplify your answer.

Here are the mathematical expressions that explore concepts in calculus involving derivatives and integrals: (a) \(\frac{d}{dx} \int_{-10}^{x} (t^2 - 1)^2 \, dt.\) This expression involves differentiating an integral with respect to \(x\), where the variable \(x\) is the upper limit of the integral. The integrand is \((t^2 - 1)^2\). (b) \(\frac{d}{dx} \int_{x}^{-10} (t^2 - 1)^2 \, dt.\) This is similar to the first expression, but here, the variable \(x\) is the lower limit of the integral, and the limit of integration is reversed. (c) \(\frac{d}{dx} \int_{-10}^{3x^2} (t^2 - 1)^2 \, dt.\) In this expression, the upper limit of the integral is given by \(3x^2\), indicating that the upper limit is a function of \(x\). (d) \(\frac{d}{dx} \int_{4x-1}^{3x^2} (t^2 - 1)^2 \, dt.\) Here, both the lower and upper limits of integration are functions of \(x\), with the lower limit as \(4x-1\) and the upper limit as \(3x^2\). Each expression requires understanding the Fundamental Theorem of Calculus, especially when dealing with variable limits of integration.
Transcribed Image Text:Here are the mathematical expressions that explore concepts in calculus involving derivatives and integrals: (a) \(\frac{d}{dx} \int_{-10}^{x} (t^2 - 1)^2 \, dt.\) This expression involves differentiating an integral with respect to \(x\), where the variable \(x\) is the upper limit of the integral. The integrand is \((t^2 - 1)^2\). (b) \(\frac{d}{dx} \int_{x}^{-10} (t^2 - 1)^2 \, dt.\) This is similar to the first expression, but here, the variable \(x\) is the lower limit of the integral, and the limit of integration is reversed. (c) \(\frac{d}{dx} \int_{-10}^{3x^2} (t^2 - 1)^2 \, dt.\) In this expression, the upper limit of the integral is given by \(3x^2\), indicating that the upper limit is a function of \(x\). (d) \(\frac{d}{dx} \int_{4x-1}^{3x^2} (t^2 - 1)^2 \, dt.\) Here, both the lower and upper limits of integration are functions of \(x\), with the lower limit as \(4x-1\) and the upper limit as \(3x^2\). Each expression requires understanding the Fundamental Theorem of Calculus, especially when dealing with variable limits of integration.
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