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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![### Finding the Equation of a Curve
#### Problem Statement
Find the equation of the curve that passes through the point \((3, 2)\) if its slope is given by
\[ \frac{dy}{dx} = 2x - 2. \]
#### Solution
To find the equation of the curve, we need to integrate the given slope function. The differential equation \(\frac{dy}{dx} = 2x - 2\) can be integrated with respect to \(x\):
\[ y = \int (2x - 2) \, dx. \]
First, solve the integral:
\[ y = \int 2x \, dx - \int 2 \, dx. \]
\[ y = 2 \int x \, dx - 2 \int 1 \, dx. \]
\[ y = 2 \left( \frac{x^2}{2} \right) - 2x + C. \]
Simplify the expression:
\[ y = x^2 - 2x + C. \]
Since the curve passes through the point \((3, 2)\), substitute \(x = 3\) and \(y = 2\) into the equation to find the constant \(C\):
\[ 2 = (3)^2 - 2(3) + C, \]
\[ 2 = 9 - 6 + C, \]
\[ 2 = 3 + C, \]
\[ C = -1. \]
Therefore, the equation of the curve is:
\[ y = x^2 - 2x - 1. \]
This is the required equation of the curve passing through the point \((3, 2)\) with the given slope function.
#### Final Answer
\[ y = x^2 - 2x - 1. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde3c0008-16a0-466c-9506-622673eccccc%2F6c6270be-b2ed-40dd-8151-654ab62d67b8%2Fzoin9of_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Equation of a Curve
#### Problem Statement
Find the equation of the curve that passes through the point \((3, 2)\) if its slope is given by
\[ \frac{dy}{dx} = 2x - 2. \]
#### Solution
To find the equation of the curve, we need to integrate the given slope function. The differential equation \(\frac{dy}{dx} = 2x - 2\) can be integrated with respect to \(x\):
\[ y = \int (2x - 2) \, dx. \]
First, solve the integral:
\[ y = \int 2x \, dx - \int 2 \, dx. \]
\[ y = 2 \int x \, dx - 2 \int 1 \, dx. \]
\[ y = 2 \left( \frac{x^2}{2} \right) - 2x + C. \]
Simplify the expression:
\[ y = x^2 - 2x + C. \]
Since the curve passes through the point \((3, 2)\), substitute \(x = 3\) and \(y = 2\) into the equation to find the constant \(C\):
\[ 2 = (3)^2 - 2(3) + C, \]
\[ 2 = 9 - 6 + C, \]
\[ 2 = 3 + C, \]
\[ C = -1. \]
Therefore, the equation of the curve is:
\[ y = x^2 - 2x - 1. \]
This is the required equation of the curve passing through the point \((3, 2)\) with the given slope function.
#### Final Answer
\[ y = x^2 - 2x - 1. \]
![### Problem Statement:
**Objective:** Find the function \( f \) given the following conditions:
1. The second derivative of the function, \( f''(x) \), is given by:
\[ f''(x) = 2 + \cos(x) \]
2. The initial condition at \( x = 0 \) is:
\[ f(0) = -1 \]
3. The function value at \( x = \frac{\pi}{2} \) is:
\[ f\left(\frac{\pi}{2}\right) = 2 \]
**Find:** \( f(x) = \) ☐](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde3c0008-16a0-466c-9506-622673eccccc%2F6c6270be-b2ed-40dd-8151-654ab62d67b8%2F30k95ax_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**Objective:** Find the function \( f \) given the following conditions:
1. The second derivative of the function, \( f''(x) \), is given by:
\[ f''(x) = 2 + \cos(x) \]
2. The initial condition at \( x = 0 \) is:
\[ f(0) = -1 \]
3. The function value at \( x = \frac{\pi}{2} \) is:
\[ f\left(\frac{\pi}{2}\right) = 2 \]
**Find:** \( f(x) = \) ☐
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