Problem 39 - Calculus: Early Transcendental Functions

Question

Find the function \(f(x)\) satisfying the given conditions. $$f^{\prime}(x)=3 e^{x}+x, f(0)=4$$

Step-by-Step Solution

Answer

The function satisfying the given conditions is \(f(x) = 3e^x + \frac{1}{2}x^2 + 1\).

Step 1: Identify given information

Before proceeding with solving, it's important to acknowledge and write down the information that is given in the exercise. We are given that the derivative \(f^{\prime}(x)=3 e^{x}+x\), and that \(f(0)=4\).

Step 2: Find the Antiderivative

The antiderivative of \(f^{\prime}(x)\) will give us the original function \(f(x)\). The antiderivative, also known as the indefinite integral, of \(f^{\prime}(x)\) is \(f(x)\). To find this, we perform the integration of \(f^{\prime}(x)\) w.r.t. \(x\), \(\int f^{\prime}(x) dx = \int (3 e^{x}+x) dx\). The integral of \(3e^x\) is \(3e^x\) and the integral of \(x\) is \(\frac{1}{2}x^2\). Remember to add the constant of integration, \(C\), which represents the family of potential solutions. So, we have \(f(x) = 3e^x + \frac{1}{2}x^2 + C\).

Step 3: Find the Specific Solution

Now we use the given point \(f(0)=4\) to find our specific solution. Substituting this point into our derived function from Step 2, we get \(4 = 3e^0 + \frac{1}{2}(0)^2 + C\). This simplifies to \(4 = 3 + C\), and further simplifies to \(C = 1\). Thus, the function that satisfies the given condition is \(f(x) = 3e^x + \frac{1}{2}x^2 + 1\).

Key Concepts

Indefinite Integral

When we talk about indefinite integrals, we're discussing integration without limits of integration. This process can be thought of as the reverse of differentiation.
An indefinite integral is often described as finding the antiderivative of a function. This is because, through indefinite integration, we find a function whose derivative is the given function.
For example, if you have a function like the derivative of another function, say, \( f'(x) = 3e^x + x \), taking the indefinite integral of \( f'(x) \) with respect to \( x \) involves integrating each term separately.
- The integral of \( 3e^x \) remains \( 3e^x \), since the derivative of \( e^x \) is itself.- The integral of \( x \) becomes \( \frac{1}{2}x^2 \), as the power rule for integration states we increase the power by one and divide by the new power. Additionally, because there are no set boundaries for our integral, we add a constant \( C \).
This represents that there is not just one solution, but a family of functions differing by a constant value.

Initial Conditions

Initial conditions are critical in determining a specific solution from a family of solutions offered by an indefinite integral.
Once we have an antiderivative (or general solution), an initial condition helps us pinpoint the precise function that satisfies the conditions of the problem.
In our example, after we found the indefinite integral \( f(x) = 3e^x + \frac{1}{2}x^2 + C \), we used the initial condition \( f(0) = 4 \) to find the specific solution.By substituting \( x = 0 \) into the expression for \( f(x) \), we replace \( x \) with 0, and \( f(x) \) with 4, thus forming an equation to solve for \( C \).
Initial conditions help ensure that the solution we determine is not just any antiderivative, but the one fitting within the context of the problem's requirements.

Integration Constant

The integration constant, often represented as \( C \), is fundamental in indefinite integration as it signifies the family of possible functions that could derive into the same original function.
Whenever we perform an indefinite integral, the result includes an arbitrary constant \( C \), because differentiation of a constant yields zero, it remains an unknown during integration.
In our example, when determining the indefinite integral \( f(x) = 3e^x + \frac{1}{2}x^2 + C \), \( C \) represents all the possible vertical shifts of the antiderivative function.
To find the value of \( C \), the initial condition \( f(0) = 4 \) comes into play. This condition allows us to substitute and solve for \( C \) specifically, turning our general solution into the unique function that fits the initial problem setup, thus, \( C = 1 \) in this context. By knowing \( C \), we're able to nail down the exact version of the function needed.