Problem 36
Question
In each of Exercises \(35-40\), an expression \(f(x)\) is given. Find all functions \(F\) such that \(F^{\prime}(x)=f(x)\). $$ 3 x $$
Step-by-Step Solution
Verified Answer
The functions are \( F(x) = \frac{3}{2}x^2 + C \), where \( C \) is any constant.
1Step 1: Understand the Problem
We are given a function \( f(x) = 3x \) and we need to find a function \( F(x) \) such that when it is differentiated, we get \( f(x) = 3x \). In mathematical terms, we need to find an antiderivative or indefinite integral of \( f(x) = 3x \).
2Step 2: Set Up the Integral
To find the function \( F(x) \), we need to integrate \( f(x) = 3x \). Set up the integral: \[ F(x) = \int 3x \, dx. \] This step involves recognizing \( F(x) \) as an integral of \( 3x \).
3Step 3: Solve the Integral
Integrate the function \( 3x \). The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), so the integral of \( 3x \) is \( 3\cdot\frac{x^2}{2} = \frac{3}{2}x^2 \). Don't forget to add the constant of integration \( C \). Thus, \( F(x) = \frac{3}{2}x^2 + C \).
4Step 4: Verify the Solution
Differentiate \( F(x) = \frac{3}{2}x^2 + C \) to verify it gives \( f(x) = 3x \). The derivative of \( \frac{3}{2}x^2 \) with respect to \( x \) is \( 3x \), and the derivative of the constant \( C \) is zero. Therefore, \( F'(x) = 3x \), confirming our solution is correct.
Key Concepts
AntiderivativeIndefinite IntegralVerification of Solutions
Antiderivative
The concept of an antiderivative is crucial in calculus, especially in understanding how to go from the rate of change of a function back to the original function itself. If a function \( F(x) \) is an antiderivative of \( f(x) \), it means that when you take the derivative of \( F(x) \), you end up with \( f(x) \). In this exercise, \( f(x) = 3x \) is the rate at which some function is changing, and our task is to find \( F(x) \), the original function before differentiation.
It's important to note that antiderivatives are not unique. They contain a family of functions differing by a constant, because the derivative of a constant is zero. Therefore, if \( F(x) \) is an antiderivative of \( f(x) \), then \( F(x) + C \), where \( C \) is any constant, is also an antiderivative of \( f(x) \).
For example, in this problem, we found that \( F(x) = \frac{3}{2}x^2 + C \) is an antiderivative of \( f(x) = 3x \). This means differentiating \( \frac{3}{2}x^2 + C \) will give us back \( 3x \). Every function of the form \( \frac{3}{2}x^2 + C \) will satisfy this requirement, confirming the importance of the constant in defining the whole family of antiderivatives.
It's important to note that antiderivatives are not unique. They contain a family of functions differing by a constant, because the derivative of a constant is zero. Therefore, if \( F(x) \) is an antiderivative of \( f(x) \), then \( F(x) + C \), where \( C \) is any constant, is also an antiderivative of \( f(x) \).
For example, in this problem, we found that \( F(x) = \frac{3}{2}x^2 + C \) is an antiderivative of \( f(x) = 3x \). This means differentiating \( \frac{3}{2}x^2 + C \) will give us back \( 3x \). Every function of the form \( \frac{3}{2}x^2 + C \) will satisfy this requirement, confirming the importance of the constant in defining the whole family of antiderivatives.
Indefinite Integral
Related to the concept of an antiderivative is the indefinite integral. This term is often used interchangeably with antiderivative, though it emphasizes the process of integration over differentiation.
The indefinite integral of a function \( f(x) \) is denoted by \( \int f(x) \, dx \). It represents the collection of all functions whose derivative is \( f(x) \). This is why, when calculating an indefinite integral, we add the constant of integration \( C \).
In this exercise, we set up the indefinite integral as \( \int 3x \, dx \). Calculating this involves finding the function \( F(x) \) such that \( F'(x) = 3x \). Using the power rule for integration, which states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), we determine that \( \int 3x \, dx = \frac{3}{2}x^2 + C \). The constant \( C \) again represents the family of functions that make up the indefinite integral.
Thus, the indefinite integral serves as a bridge between the derivative and the original function, capturing all possible original functions by accounting for constant shifts.
The indefinite integral of a function \( f(x) \) is denoted by \( \int f(x) \, dx \). It represents the collection of all functions whose derivative is \( f(x) \). This is why, when calculating an indefinite integral, we add the constant of integration \( C \).
In this exercise, we set up the indefinite integral as \( \int 3x \, dx \). Calculating this involves finding the function \( F(x) \) such that \( F'(x) = 3x \). Using the power rule for integration, which states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), we determine that \( \int 3x \, dx = \frac{3}{2}x^2 + C \). The constant \( C \) again represents the family of functions that make up the indefinite integral.
Thus, the indefinite integral serves as a bridge between the derivative and the original function, capturing all possible original functions by accounting for constant shifts.
Verification of Solutions
Once an antiderivative or indefinite integral is found, it's essential to verify that it is indeed the correct solution. Verification involves differentiating your result to ensure it reproduces the original function \( f(x) \). This step confirms the mathematical integrity of your integration process.
In our exercise, we found \( F(x) = \frac{3}{2}x^2 + C \) as the antiderivative of \( f(x) = 3x \). To verify, we take the derivative of \( F(x) \):
\[ \frac{d}{dx} \left( \frac{3}{2}x^2 + C \right) = 3x + 0, \]
which simplifies to \( 3x \), exactly matching our given function \( f(x) \). This confirms that our antiderivative is correct.
Verification is a crucial step because it ensures accuracy and builds confidence in mathematical work, especially in more complex integrations. Always remember to differentiate your result when checking your integration to make sure nothing has been overlooked.
In our exercise, we found \( F(x) = \frac{3}{2}x^2 + C \) as the antiderivative of \( f(x) = 3x \). To verify, we take the derivative of \( F(x) \):
\[ \frac{d}{dx} \left( \frac{3}{2}x^2 + C \right) = 3x + 0, \]
which simplifies to \( 3x \), exactly matching our given function \( f(x) \). This confirms that our antiderivative is correct.
Verification is a crucial step because it ensures accuracy and builds confidence in mathematical work, especially in more complex integrations. Always remember to differentiate your result when checking your integration to make sure nothing has been overlooked.
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