Problem 23 - Essential Calculus Early Transcendentals

Question

Find \(f\) $$f^{\prime}(x)=1+3 \sqrt{x}, \quad f(4)=25$$

Step-by-Step Solution

Answer

The function is \( f(x) = x + 2x^{3/2} + 5 \).

Step 1: Integrate the derivative

Start by finding the original function, \( f(x) \), by integrating the given derivative \( f'(x) = 1 + 3\sqrt{x} \). Use the power rule for integration: \[ \int (1 + 3\sqrt{x}) \, dx = \int 1 \, dx + \int 3x^{1/2} \, dx. \] The integrals of these terms are: \[ \int 1 \, dx = x \] and \[ \int 3x^{1/2} \, dx = 3 \times \frac{2}{3} x^{3/2} = 2x^{3/2}. \] Thus, the integrated function is \[ f(x) = x + 2x^{3/2} + C, \] where \( C \) is the constant of integration.

Step 2: Use initial condition to find C

Given the condition \( f(4) = 25 \), substitute \( x = 4 \) into the integrated function to solve for \( C \):\[ f(4) = 4 + 2(4)^{3/2} + C = 25. \] Calculate \( 4^{3/2} \): \[ 4^{3/2} = (2^2)^{3/2} = 2^3 = 8. \] So, the expression becomes: \[ 4 + 2(8) + C = 25. \] This simplifies to: \[ 4 + 16 + C = 25 \Rightarrow C = 5. \]

Step 3: Write the final function

Now, substitute \( C = 5 \) back into the function:\[ f(x) = x + 2x^{3/2} + 5. \] This is the original function \( f(x) \) that satisfies the given derivative and initial condition.

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus used for integrating functions. If you have a term in the form of \(x^n\), the Power Rule states that to integrate this term, you increase the exponent by one and then divide by the new exponent.
The formula looks like this:

• \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

where \(C\) is the constant of integration.
In simpler terms, if you start with an expression like \(x^{1/2}\), you follow these steps:

• Increase the exponent by 1 to get \(x^{3/2}\).
• Divide by the new exponent \(3/2\), resulting in \(\frac{2}{3}x^{3/2}\).

Multiplying this by the original coefficient, such as 3, yields \(2x^{3/2}\).
This concept allowed us to integrate \(3\sqrt{x}\) in the given exercise.

Constant of Integration

Whenever you integrate a function, there's always a constant added to the result because integration can be thought of as the "reverse" of differentiation.
Since the derivative of a constant is zero, any constant could have been there before you differentiated.
That's why when you find \(\int 1 \, dx = x + C\), the \(C\) represents any constant value.
This constant of integration, \(C\), is crucial. Without it, you won't have all potential antiderivatives of a function.
In our exercise, after integrating \(f'(x)\), we get \(f(x) = x + 2x^{3/2} + C\).
The constant \(C\) is found using an initial condition to make sure the solution is specific to the problem setup.

Initial Condition

An initial condition is extra information used to find a specific solution to a differential equation.
Once you integrate and obtain a general form of the function, the initial condition helps to find the constant of integration \(C\).
In the problem given, the initial condition is \(f(4) = 25\).
We take our general solution \(f(x) = x + 2x^{3/2} + C\) and plug in the known values: \(x = 4\) and \(f(4) = 25\).

• Perform the calculation: \(4 + 2(4^{3/2}) + C = 25\), which simplifies down to determine \(C = 5\).

The initial condition ensures our function specifically aligns with the problem setup, making it unique to the given differentiation and constraint.