Problem 28 - Calculus Early Transcendentals

Question

Find all functions \(f\) whose derivative is \(f^{\prime}(x)=x+1\).

Step-by-Step Solution

Answer

Answer: The original function is \(f(x) = \frac{x^2}{2} + x + C\), where \(C\) is an integration constant.

Step 1: Identify the derivative function

We are given the derivative function \(f'(x) = x + 1\). Our goal is to find the original function, \(f(x)\).

Step 2: Integrate the derivative function

Now, we need to integrate the given derivative function to find the original function. The integration of \(f'(x)\) will give us \(f(x) + C\), where \(C\) is an integration constant. So, perform the integration: \(\int (x+1) dx\)

Step 3: Integrate each term

To integrate \((x+1)\), we can integrate each term separately. The integration of \(x\) is \(\frac{x^2}{2}\), and the integration of \(1\) is \(x\). Hence, we have: \(\int x dx + \int 1 dx\) Now, perform the integrations: \(\frac{x^2}{2} + x\)

Step 4: Add the integration constant

After performing the integration, we need to add an integration constant, which we denote as \(C\). This constant represents any constant term that may have been in the original function. So, the final integrated function is: \(f(x) = \frac{x^2}{2} + x + C\) This is the result of integrating the given derivative function, and it represents all possible functions whose derivative is \(f'(x) = x + 1\).