Problem 8 - 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 \(f^{\prime}(-1) .\) $$ f(x)=\pi x-\sqrt{3} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \pi x - \sqrt{3}\) is \(f'(x) = \pi\). When evaluated at \(x = 0, 2, -1\), the derivative \(f'(x)\) is \(\pi\).

Step 1: Compute the Derivative

First, find the derivative of the function \(f(x) = \pi x - \sqrt{3}\). In this case, the derivative of \(f(x)\), denoted as \(f'(x)\), of any constant times \(x\) is the constant itself and the derivative of any constant is equal to zero. So, the derivative of the function \(f(x)\) is simply \(\pi\).

Step 2: Evaluate the Derivative at \(x = 0\)

Substitute \(x = 0\) into \(f'(x)\). Considering that the derivative is a constant function \(\pi\), \(f'(x)\) evaluated at \(x = 0\) is also \(\pi\).

Step 3: Evaluate the Derivative at \(x = 2\)

Substitute \(x = 2\) into \(f'(x)\). Similar to the previous step, \(f'(x)\) evaluated at \(x = 2\) is \(\pi\).

Step 4: Evaluate the Derivative at \(x = -1\)

Substitute \(x = -1\) into \(f'(x)\). Again, \(f'(x)\) evaluated at \(x = -1\) is \(\pi\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It's traditionally divided into two main parts: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. Integral calculus, on the other hand, involves the process of integration, which is essentially the reverse of differentiation and is used to calculate things like area under a curve and the accumulated value of a function over an interval.

Studying calculus is essential for understanding complex systems in physics, engineering, economics, and more, as it provides the tools to model and analyze dynamic processes. The problem from the textbook solutions involves differential calculus, specifically the computation of derivatives, which is a fundamental skill in this field.

Derivative Computation

The computation of a derivative is central to differential calculus and involves finding the rate at which a function's output value changes with respect to changes in its input value. When computing derivatives, certain rules apply, such as the power rule, product rule, quotient rule, and chain rule.

For linear functions, which are functions of the form mx + b, where m and b are constants, the derivative is particularly simple to find—it's just the coefficient of the x-term, which is m. This means for the function f(x) = \(\pi x - \sqrt{3}\), the derivative with respect to x is the constant \(\pi\). There is no x in the derivative because the rate of change of a linear function is constant, and hence the function's graph is a straight line with slope m.

Constant Function Derivative

When we talk about the derivative of a constant function, we're referring to a function that always returns the same value, no matter what the input is. The graph of a constant function is a horizontal line. According to the rules of differentiation, the derivative of a constant is always zero because a constant does not vary or change as the independent variable changes.

However, it's crucial to distinguish between a constant function and a function with a constant multiplied by the variable, like the one in our exercise f(x) = \(\pi x - \sqrt{3}\). Here, \sqrt{3}\ is a constant and its derivative is zero, but \(\pi x\) is not a constant function; it's a linear function with a constant rate of change represented by \(\pi\).

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input. In the context of derivatives, evaluating a function often involves substituting a specific value for the input variable and simplifying. Since the derivative of the function in our exercise is a constant \(\pi\), evaluating it at any point, such as x = 0, x = 2, or x = -1, will always yield the same result, which is \(\pi\).

This is an essential concept because evaluating derivatives at specific points can tell us the rate of change of the function at those points. For functions that aren't constant, the derivative can vary, and evaluating it at different points is crucial to understanding the function's behavior.