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
![**Problem 3: Function Sketching**
Sketch a curve \( y = f(x) \) that satisfies the following conditions:
- \( f(0) = 3 \)
- The derivative \(\frac{dy}{dx} = -2\)
Determine the function \( f(x) \).
**Solution Approach:**
Given that the derivative \(\frac{dy}{dx} = -2\) is constant, this indicates the slope of the curve is constant, suggesting a linear relationship.
1. **Equation of the Line:**
Since the derivative is constant, \( f(x) \) is in the form of a line, \( f(x) = mx + b \), where \( m \) is the slope.
2. **Slope:**
From \(\frac{dy}{dx} = -2\), the slope \( m = -2 \).
3. **Using the Point:**
From the condition \( f(0) = 3 \), we substitute into the equation:
\[
f(0) = -2(0) + b = 3 \implies b = 3
\]
4. **Function:**
Therefore, the function is:
\[
f(x) = -2x + 3
\]
The graph of \( f(x) \) would be a straight line with a slope of \(-2\) passing through the point \((0, 3)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4eaca33a-bcfa-4336-aecf-f320ab80f63b%2F6f685b21-dd48-427d-aff4-bb5fcf82e403%2Ftzsndia_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Function Sketching**
Sketch a curve \( y = f(x) \) that satisfies the following conditions:
- \( f(0) = 3 \)
- The derivative \(\frac{dy}{dx} = -2\)
Determine the function \( f(x) \).
**Solution Approach:**
Given that the derivative \(\frac{dy}{dx} = -2\) is constant, this indicates the slope of the curve is constant, suggesting a linear relationship.
1. **Equation of the Line:**
Since the derivative is constant, \( f(x) \) is in the form of a line, \( f(x) = mx + b \), where \( m \) is the slope.
2. **Slope:**
From \(\frac{dy}{dx} = -2\), the slope \( m = -2 \).
3. **Using the Point:**
From the condition \( f(0) = 3 \), we substitute into the equation:
\[
f(0) = -2(0) + b = 3 \implies b = 3
\]
4. **Function:**
Therefore, the function is:
\[
f(x) = -2x + 3
\]
The graph of \( f(x) \) would be a straight line with a slope of \(-2\) passing through the point \((0, 3)\).
Expert Solution
Step 1
The function that satisfies .
We have to find and sketch its graph.
Step by step
Solved in 3 steps with 1 images
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