Problem 43
Question
In each of Exercises \(41-48\), use the given information to find \(F(c)\). $$ F^{\prime}(x)=4 / x, \quad F\left(e^{2}\right)=7, \quad c=e^{3} $$
Step-by-Step Solution
Verified Answer
F(e^3) = 11.
1Step 1: Understand the Problem
We are given the derivative of a function, \( F'(x) = \frac{4}{x} \), a specific value \( F(e^2) = 7 \), and we need to find \( F(c) \) where \( c = e^3 \). Our goal is to find the antiderivative \( F(x) \) using the given information.
2Step 2: Find the General Form of F(x)
To find \( F(x) \), integrate the derivative: \( F'(x) = \frac{4}{x} \). This gives:\[ F(x) = 4 \ln |x| + C \]where \( C \) is the constant of integration.
3Step 3: Use Initial Condition to Solve for C
We are given \( F(e^2) = 7 \). Substitute \( x = e^2 \) into the equation:\[ 7 = 4 \ln |e^2| + C \]Since \( \ln(e^2) = 2 \cdot \ln(e) = 2 \), the equation becomes:\[ 7 = 4 \cdot 2 + C \]Solving for \( C \), we have:\[ 7 = 8 + C \]\[ C = -1 \]
4Step 4: Write the Specific Form of F(x)
Substitute \( C = -1 \) back into the antiderivative:\[ F(x) = 4 \ln |x| - 1 \]
5Step 5: Evaluate F(c) Where c = e^3
Now substitute \( c = e^3 \) into the specific form of \( F(x) \):\[ F(e^3) = 4 \ln |e^3| - 1 \]\( \ln |e^3| = 3 \ln(e) = 3 \), so:\[ F(e^3) = 4 \cdot 3 - 1 = 12 - 1 = 11 \]
Key Concepts
AntiderivativesConstant of IntegrationNatural Logarithm
Antiderivatives
Antiderivatives, also known as indefinite integrals, are functions that reverse the process of differentiation. If you have a derivative, such as \( F'(x) \), finding its antiderivative means determining the original function \( F(x) \) from which it was derived. Seeing the process as reverse engineering from the calculus world can be a helpful analogy.
To find an antiderivative, you perform integration. Given \( F'(x) = \frac{4}{x} \), the goal is to integrate this expression to recover the function \( F(x) \). This integration leads us to the form \( F(x) = 4 \ln |x| + C \). The integration process involves finding a function whose derivative matches \( \,\frac{4}{x} \) and perhaps understanding the underlying pattern or rule, which in this case involves recognizing the derivative of natural logs.
Antiderivatives are vital because they allow you to understand the surplus of changes an original function underwent, whether in motion, growth, or any variable context, by knowing just the rate of change, \( F'(x) \). Always remember, finding an antiderivative provides a family of functions, each differing by a constant.
To find an antiderivative, you perform integration. Given \( F'(x) = \frac{4}{x} \), the goal is to integrate this expression to recover the function \( F(x) \). This integration leads us to the form \( F(x) = 4 \ln |x| + C \). The integration process involves finding a function whose derivative matches \( \,\frac{4}{x} \) and perhaps understanding the underlying pattern or rule, which in this case involves recognizing the derivative of natural logs.
Antiderivatives are vital because they allow you to understand the surplus of changes an original function underwent, whether in motion, growth, or any variable context, by knowing just the rate of change, \( F'(x) \). Always remember, finding an antiderivative provides a family of functions, each differing by a constant.
Constant of Integration
While finding an antiderivative, it's crucial to account for the constant of integration, denoted as \( C \). This constant accounts for all potential initial conditions since integration produces an entire family of functions.
When we integrate \( F'(x) = \frac{4}{x} \), we find \( F(x) = 4 \ln |x| + C \). The \( C \) symbolizes any constant added to the function that doesn't change the derivative because the derivative of any constant is zero. Without the constant of integration, the solution would lack completeness, providing only one potential function instead of all possibilities.
To determine the specific value of \( C \), we use given conditions, such as \( F(e^2) = 7 \). By substituting and solving, you pinpoint the precise version of the function needed for the problem at hand. In this exercise, we found that \( C = -1 \). Including \( C \) respects all scenarios present in real-world applications, particularly when finding functions from change rates.
When we integrate \( F'(x) = \frac{4}{x} \), we find \( F(x) = 4 \ln |x| + C \). The \( C \) symbolizes any constant added to the function that doesn't change the derivative because the derivative of any constant is zero. Without the constant of integration, the solution would lack completeness, providing only one potential function instead of all possibilities.
To determine the specific value of \( C \), we use given conditions, such as \( F(e^2) = 7 \). By substituting and solving, you pinpoint the precise version of the function needed for the problem at hand. In this exercise, we found that \( C = -1 \). Including \( C \) respects all scenarios present in real-world applications, particularly when finding functions from change rates.
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a fundamental concept often emerging in calculus and especially integration when dealing with functions of the form \( \frac{1}{x} \). This is due to the inherent definition that the derivative of \( \ln(x) \) is \( \frac{1}{x} \).
In our exercise, when integrating \( F'(x) = \frac{4}{x} \), we recognized that the antiderivative involves \( 4 \ln |x| \) because of this rule. Natural logarithms are pivotal because they translate multiplicative processes into additive ones, forming a bridge between algebra and calculus.
Understanding \( \ln(x) \) also involves grasping its properties, such as \( \ln(e^a) = a \), simplifying calculations with exponential and logarithmic expressions. For example, when calculating \( \ln(e^3) \) in our problem, it simplified directly to \( 3 \), streamlining the evaluation of \( F(e^3) \). The interplay between logs and exponents simplifies many complex expressions, making \( \ln(x) \) a powerful tool in calculus, even beyond integration.
In our exercise, when integrating \( F'(x) = \frac{4}{x} \), we recognized that the antiderivative involves \( 4 \ln |x| \) because of this rule. Natural logarithms are pivotal because they translate multiplicative processes into additive ones, forming a bridge between algebra and calculus.
Understanding \( \ln(x) \) also involves grasping its properties, such as \( \ln(e^a) = a \), simplifying calculations with exponential and logarithmic expressions. For example, when calculating \( \ln(e^3) \) in our problem, it simplified directly to \( 3 \), streamlining the evaluation of \( F(e^3) \). The interplay between logs and exponents simplifies many complex expressions, making \( \ln(x) \) a powerful tool in calculus, even beyond integration.
Other exercises in this chapter
Problem 43
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