Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\frac{x+\pi}{2} $$

Step-by-Step Solution

Answer

The derivative, \(f'(x)\), equals to \(\frac{1}{2}\). The derivative at \(x = 0\), \(f'(0)\), is \(\frac{1}{2}\). The derivative at \(x = 2\), \(f'(2)\), is \(\frac{1}{2}\). The derivative at \(x = -1\), \(f'(-1)\), is \(\frac{1}{2}\).

Step 1: Find the derivative of the function

The derivative of the function \(f(x) = \frac{x+\pi}{2}\) can be obtained using the power rule. The power rule states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). In this case, \(n = 1\), so the derivative of \(x\) is \(1\). The derivative of a constant is \(0\), so the derivative of \(\pi\) is \(0\). The result is the derivative, \(f'(x)\), equal to \(\frac{1}{2}\).

Step 2: Find the derivative at \(x = 0\)

To find \(f'(0)\), substitute \(x = 0\) in the derivative function. So, \(f'(0)\) equals to \(\frac{1}{2}\).

Step 3: Find the derivative at \(x = 2\)

To find \(f'(2)\), substitute \(x = 2\) in the derivative function. So, \(f'(2)\) equals to \(\frac{1}{2}\).

Step 4: Find the derivative at \(x = -1\)

To find \(f'(-1)\), substitute \(x = -1\) in the derivative function. So, \(f'(-1)\) equals to \(\frac{1}{2}\).

Key Concepts

Understanding Derivatives

A derivative represents the rate at which a function changes at any given point. Think of it as a snapshot of how quickly something is happening. By finding the derivative of a function, we can understand how the output of the function responds to changes in its input.
For example, if we know the derivative of a function about speed, we can find out how quickly the speed changes, which is effectively acceleration. When dealing with a straightforward linear function like the one in the exercise, the derivative indicates a consistent rate of change.

• It helps in understanding and analyzing the behavior of functions.
• Calculating the derivative is essential in different fields, from physics to economics.

The Power Rule: A Handy Tool

The power rule is a quick method for finding the derivative of functions that involve exponents. It simplifies the process, especially for polynomial terms. The rule states: if you have a term in the form of \(x^n\), the derivative is \(n \cdot x^{n-1}\).
In the example function \(f(x) = \frac{x+\pi}{2}\), we identify that \(x\) is raised to the power of one. By applying the power rule, the derivative of \(x\) is simply \(1\) since \(1 \cdot x^{1-1} = 1\).

• This makes handling terms with powers very straightforward.
• It's a fundamental rule used extensively in calculus.

The power rule doesn't apply to constants, which brings us to our next topic.

Derivatives of Constant Functions

In calculus, a constant function is a function that does not change, regardless of the input. The derivative of a constant is always zero because a constant doesn’t change; it has no rate of change.
For example, the number \(\pi\) is a constant in the function \(f(x) = \frac{x+\pi}{2}\). When taking its derivative, it results in zero, since constants don't vary and have no slope.

• Every constant's derivative is zero, reflecting no change.
• It's important to remember constants disappear when taking derivatives.

Understanding this helps in computing derivatives efficiently and avoiding mistakes.