Find equations of the tangent line and the normal line to the graph of the given function at the specified point. f(x) = 8xe*, (0, 0) tangent line normal line y = y =
Find equations of the tangent line and the normal line to the graph of the given function at the specified point. f(x) = 8xe*, (0, 0) tangent line normal line y = y =
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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Q7. Please answer all the parts to this question
![**Problem Statement**
Find equations of the tangent line and the normal line to the graph of the given function at the specified point.
\[ f(x) = 8xe^x \]
Specified Point: \( (0, 0) \)
**Solution**
**1. Tangent Line**
\[ y = \]
**2. Normal Line**
\[ y = \]
**Explanation**
The problem involves finding the equations of the lines tangent and normal to the function \( f(x) = 8xe^x \) at the point \( (0, 0) \).
- The **tangent line** at a point on a curve is the straight line that just "touches" the curve at that point. Its slope is equal to the derivative of the function at that point.
- The **normal line** is perpendicular to the tangent line and thus has a slope that is the negative reciprocal of the tangent line's slope.
To solve, differentiate the function to find the slope of the tangent line, and use the point-slope form of a straight line equation. Then calculate the normal line using the negative reciprocal of the tangent slope.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87311036-785b-48b7-ad0a-ffb7f8402d00%2F487f3893-a539-47c2-a96d-396d777b88a2%2Fcdqmgd_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find equations of the tangent line and the normal line to the graph of the given function at the specified point.
\[ f(x) = 8xe^x \]
Specified Point: \( (0, 0) \)
**Solution**
**1. Tangent Line**
\[ y = \]
**2. Normal Line**
\[ y = \]
**Explanation**
The problem involves finding the equations of the lines tangent and normal to the function \( f(x) = 8xe^x \) at the point \( (0, 0) \).
- The **tangent line** at a point on a curve is the straight line that just "touches" the curve at that point. Its slope is equal to the derivative of the function at that point.
- The **normal line** is perpendicular to the tangent line and thus has a slope that is the negative reciprocal of the tangent line's slope.
To solve, differentiate the function to find the slope of the tangent line, and use the point-slope form of a straight line equation. Then calculate the normal line using the negative reciprocal of the tangent slope.
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