**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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Description: **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 curv

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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.

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