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Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t3 + 1, y = t10 + t; t = -1 y =

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t3 + 1, y = t10 + t; t = -1 y =

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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**Problem Statement:** Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. Given: - \( x = t^3 + 1 \) - \( y = t^{10} + t \) - \( t = -1 \) **Task:** Find the equation of the tangent line to the curve at the specified parameter value. **Solution Outline:** 1. **Calculate the derivatives**: - \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) to find the slope of the tangent line, \(\frac{dy}{dx}\). 2. **Evaluate at \( t = -1 \)**: - Substitute \( t = -1 \) into the derivatives to find the slope at this point. 3. **Find the coordinates of the point**: - Evaluate \( x \) and \( y \) at \( t = -1 \). 4. **Equation of the tangent line**: - Use the point-slope form of the equation of a line. This problem involves using differentiation and substitution for parameterized curves.
Transcribed Image Text:**Problem Statement:** Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. Given: - \( x = t^3 + 1 \) - \( y = t^{10} + t \) - \( t = -1 \) **Task:** Find the equation of the tangent line to the curve at the specified parameter value. **Solution Outline:** 1. **Calculate the derivatives**: - \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) to find the slope of the tangent line, \(\frac{dy}{dx}\). 2. **Evaluate at \( t = -1 \)**: - Substitute \( t = -1 \) into the derivatives to find the slope at this point. 3. **Find the coordinates of the point**: - Evaluate \( x \) and \( y \) at \( t = -1 \). 4. **Equation of the tangent line**: - Use the point-slope form of the equation of a line. This problem involves using differentiation and substitution for parameterized curves.
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