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d'y Find for the curve given parametrically dx² by x(t) = 4+t², y(t) = 1² +2+³.

d'y Find for the curve given parametrically dx² by x(t) = 4+t², y(t) = 1² +2+³.

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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Question
Find \(\frac{d^2y}{dx^2}\) for the curve given parametrically by \[ x(t) = 4 + t^2, \quad y(t) = t^2 + 2t^3. \]
Transcribed Image Text:Find \(\frac{d^2y}{dx^2}\) for the curve given parametrically by \[ x(t) = 4 + t^2, \quad y(t) = t^2 + 2t^3. \]
### Differential Equations Below are a series of second-order differential equations, each represented in the form \(\frac{d^2y}{dx^2}\), which denotes the second derivative of \(y\) with respect to \(x\). 1. \(\frac{d^2y}{dx^2} = \frac{3}{2t}\) 2. \(\frac{d^2y}{dx^2} = \frac{3}{t}\) 3. \(\frac{d^2y}{dx^2} = \frac{5}{6t}\) 4. \(\frac{d^2y}{dx^2} = \frac{5t}{6}\) 5. \(\frac{d^2y}{dx^2} = \frac{3t}{2}\) 6. \(\frac{d^2y}{dx^2} = \frac{2t}{3}\) These equations can be used to evaluate the behavior of a function \(y\) in relation to changes in \(x\) and \(t\). Each equation demonstrates a unique relationship between the second derivative and variables \(t\) and constants, offering a varied perspective on how \(y\) changes as \(x\) changes. Understanding and solving such equations is foundational in calculus, with applications across physics, engineering, and other scientific domains.
Transcribed Image Text:### Differential Equations Below are a series of second-order differential equations, each represented in the form \(\frac{d^2y}{dx^2}\), which denotes the second derivative of \(y\) with respect to \(x\). 1. \(\frac{d^2y}{dx^2} = \frac{3}{2t}\) 2. \(\frac{d^2y}{dx^2} = \frac{3}{t}\) 3. \(\frac{d^2y}{dx^2} = \frac{5}{6t}\) 4. \(\frac{d^2y}{dx^2} = \frac{5t}{6}\) 5. \(\frac{d^2y}{dx^2} = \frac{3t}{2}\) 6. \(\frac{d^2y}{dx^2} = \frac{2t}{3}\) These equations can be used to evaluate the behavior of a function \(y\) in relation to changes in \(x\) and \(t\). Each equation demonstrates a unique relationship between the second derivative and variables \(t\) and constants, offering a varied perspective on how \(y\) changes as \(x\) changes. Understanding and solving such equations is foundational in calculus, with applications across physics, engineering, and other scientific domains.
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