Problem 29 - Applied Calculus

Question

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=2+4 x+5 x^{2} $$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = 2x + 2x^2 + \frac{5}{3}x^3 \), and it is unique.

Step 1: Identify the Given Function

The given function is \( f(x) = 2 + 4x + 5x^2 \). We need to find an antiderivative \( F(x) \) of this function.

Step 2: Apply the Power Rule of Integration

Integrate each term of \( f(x) \) separately. According to the power rule of integration, \( \int x^n \cdot dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the integration constant.

Step 3: Integrate Each Term

1. For the constant \( 2 \), the integral is \( 2x.\) 2. For \( 4x \), the integral is \( 4 \cdot \frac{x^2}{2} = 2x^2.\) 3. For \( 5x^2 \), the integral is \( 5 \cdot \frac{x^3}{3} = \frac{5}{3}x^3. \)

Step 4: Combine the Integrated Terms

Combine the integrations: \[ F(x) = 2x + 2x^2 + \frac{5}{3}x^3 + C \] Here, \( C \) is an arbitrary constant.

Step 5: Apply the Initial Condition

Use the initial condition \( F(0) = 0 \) to solve for \( C \). Substitute \( x = 0 \) into the antiderivative:\[ F(0) = 2(0) + 2(0)^2 + \frac{5}{3}(0)^3 + C = 0 \] This simplifies to \( C = 0 \).

Step 6: Write the Final Antiderivative

Substituting \( C = 0 \) into the antiderivative, we have: \[ F(x) = 2x + 2x^2 + \frac{5}{3}x^3 \]

Step 7: Evaluate Uniqueness

Since \( F(x) = 2x + 2x^2 + \frac{5}{3}x^3 \) satisfies the initial condition \( F(0) = 0 \) and there are no additional constants possible to satisfy this condition, this antiderivative is unique.

Key Concepts

Power Rule of Integration

Antiderivatives are functions that reverse the process of differentiation. To find the antiderivative of a function, we often rely on a powerful technique known as the Power Rule of Integration. This rule simplifies the process of integration by providing a straightforward formula: - If you have a term in the form of \(x^n\), you can find its integral by using \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. - Apply this rule to each term separately.
In our example, the function \(f(x) = 2 + 4x + 5x^2\) contains three terms. We integrate each one: - For the constant \(2\), it becomes a linear term \(2x\). - For \(4x\), use the power rule to get \(2x^2\). - For \(5x^2\), applying the rule results in \(\frac{5}{3}x^3\).
Combining these results gives us the antiderivative \(F(x) = 2x + 2x^2 + \frac{5}{3}x^3 + C\), demonstrating the ease and consistency of the Power Rule.

Initial Conditions

Initial conditions give specific values that an antiderivative must satisfy. In many cases, they allow us to determine the constant of integration \(C\) and, therefore, identify a particular solution. Consider the antiderivative \(F(x) = 2x + 2x^2 + \frac{5}{3}x^3 + C\), derived for the function \(f(x) = 2 + 4x + 5x^2\).
Given the initial condition \(F(0) = 0\), we substitute \(x = 0\) into the antiderivative: - \(0 = 2(0) + 2(0)^2 + \frac{5}{3}(0)^3 + C\) simplifies to \(C = 0\). This step is crucial; it transforms the general antiderivative into a unique solution: \(F(x) = 2x + 2x^2 + \frac{5}{3}x^3\).
Initial conditions ensure that the antiderivative aligns precisely with certain values, providing specificity and accuracy in solutions.

Uniqueness of Solutions

The uniqueness of a solution for an antiderivative derived from an initial condition is an important concept. When an initial condition is applied to solve for the constant \(C\), any further variation leading to a different antiderivative is prevented. With \(F(x) = 2x + 2x^2 + \frac{5}{3}x^3\) obtained earlier, the condition \(F(0) = 0\) was satisfied with \(C = 0\) through a straightforward substitution.
Because this value of \(C\) was calculated to fulfill the initial condition, the solution becomes distinct. No other constant can make \(F(x)\) meet the requirement \(F(0) = 0\).

• The established antiderivative fits the initial condition perfectly.
• There is no room for alteration in \(F(x)\).

Thus, given the initial condition, the antiderivative \(F(x) = 2x + 2x^2 + \frac{5}{3}x^3\) is unique, highlighting the critical role of initial conditions in determining and solidifying solutions.