Problem 15 - Essential Calculus Early Transcendentals

Question

Find the antiderivative \(F\) of \(f\) that satisfies the given condition. Check your answer by comparing the graphs of \(f\) and \(F .\) $$f(x)=5 x^{4}-2 x^{5}, \quad F(0)=4$$

Step-by-Step Solution

Answer

The antiderivative is \(F(x) = x^5 - \frac{1}{3}x^6 + 4\).

Step 1: Find the General Antiderivative

To find the antiderivative of the function \(f(x) = 5x^4 - 2x^5\), we use the power rule for integration. The power rule is given by \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\). Thus, \(\int 5x^4 \, dx = \frac{5}{5}x^{5} = x^{5}\) and \(\int -2x^5 \, dx = -\frac{2}{6}x^{6} = -\frac{1}{3}x^{6}\). So, the general antiderivative is:\[F(x) = x^5 - \frac{1}{3}x^{6} + C\] where \(C\) is the constant of integration.

Step 2: Use the Initial Condition

Apply the initial condition \(F(0) = 4\) to determine the constant \(C\). Substitute \(x = 0\) in the antiderivative:\[F(0) = 0^5 - \frac{1}{3} \cdot 0^6 + C = 4\], which simplifies to \(C = 4\).

Step 3: Write the Particular Antiderivative

Substitute the value of \(C\) back into the antiderivative expression to get the particular solution:\[F(x) = x^5 - \frac{1}{3}x^{6} + 4\].

Step 4: Verify by Graphing

To verify, graph both functions \(f(x) = 5x^4 - 2x^5\) and the antiderivative \(F(x) = x^5 - \frac{1}{3}x^{6} + 4\). The graph of \(F(x)\) should be an antiderivative of \(f(x)\), indicating that the slope of \(F(x)\) corresponds to the value of \(f(x)\) at each point. Check that when \(x = 0\), \(F(x) = 4\) as given.

Key Concepts

Power Rule for Integration

The power rule for integration is a handy tool that helps us find antiderivatives or indefinite integrals. When you look at a polynomial like \( f(x) = 5x^4 - 2x^5 \), you break it down into individual terms which are easier to integrate. The general formula is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). This means that you add one to the exponent of each term and divide by the new exponent, plus a constant of integration, \( C \).
Let's break down the integration of each term from the exercise:

• For the term \( 5x^4 \), you have to increase the exponent by one to get \( 5x^5 \). Then divide by 5 to get \( x^5 \).
• For the term \( -2x^5 \), you increase the exponent to get \( -2x^6 \) and divide by 6, which simplifies to \( -\frac{1}{3}x^6 \).

So, the antiderivative of \( f(x) \) becomes \( F(x) = x^5 - \frac{1}{3}x^6 + C \), where \( C \) is yet to be determined.

Initial Conditions in Integration

When you integrate a function, you often end up with a constant of integration, \( C \). To determine \( C \), we use initial conditions. An initial condition is a specific point on the curve of the antiderivative that you already know. This is important because it helps you find the particular solution to the indefinite integral.
In the exercise, we are given \( F(0) = 4 \). This tells us that when \( x = 0 \), the value of \( F(x) \) should be 4. By substituting \( x = 0 \) into the equation \( F(x) = x^5 - \frac{1}{3}x^6 + C \), we simplify:

• \( 0^5 - \frac{1}{3} \cdot 0^6 + C = 4 \)
• Which results in \( C = 4 \).

So, our particular antiderivative solution is \( F(x) = x^5 - \frac{1}{3}x^6 + 4 \). This specific function exactly matches the curve at the given initial condition, ensuring that the solution is precisely tailored to the problem.

Graphical Verification of Antiderivatives

Graphing is a fantastic way to verify the antiderivative you computed by alignment of visual cues. It ensures the derived function fits well with the original function's behavior. In this exercise, you want to graph both the function \( f(x) = 5x^4 - 2x^5 \) and its antiderivative \( F(x) = x^5 - \frac{1}{3}x^6 + 4 \).
Upon graphing:

• Look to see if the slope of \( F(x) \) matches the values of \( f(x) \) at each point. This is crucial because the derivative of \( F(x) \) should equal \( f(x) \).
• Verify that the value of \( F(x) \) at \( x = 0 \) is 4, consistent with the initial condition.

Through the graphs, you see that as \( x \) changes, the rate at which \( F(x) \) rises or falls should mimic the values of \( f(x) \). These graphical insights serve as a confirmation, reassuring you that the obtained antiderivative fulfills all the conditions effectively.