Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For what value(s) of \(x\) is the slope of the tangent line to \(f(x)=\frac{1}{3} x^{3}\) equal to 1 ?

Step-by-Step Solution

Answer

The values of \(x\) for which the slope of the tangent line to \(f(x)=\frac{1}{3} x^{3}\) is equal to 1 are \(x = 1\) and \(x = -1\).

Step 1: Calculate the derivative

To begin with, find the derivative of \(f(x)=\frac{1}{3} x^{3}\) using the power rule, which states that the derivative of \(x^n\) where \(n\) is any real number, is \(nx^{(n-1)}\). In this case, \(f'(x) = x^{2}\).

Step 2: Solve for \(x\)

Next, set this derivative equal to 1, and solve for \(x\). This forms the equation \(x^{2} = 1\). Solving for \(x\) yields two solutions, \(x = 1\) and \(x = -1\).

Key Concepts

Derivative of a Function

When we talk about the derivative of a function, we refer to a fundamental tool in calculus that measures how a function changes as its input changes. It's like taking a snapshot of the growth or decline rate of the function at any point. The derivative is often represented as f'(x), pronounced as 'f prime of x', and can be thought of as the slope of the tangent line to the function's graph at a certain point.

The concept of a derivative is crucial because it allows us to grasp the behavior of functions, such as their increasing or decreasing nature, their maximum and minimum values, and the concavity of their graphs. Finding the derivative also helps in real-world applications like physics, where it can represent velocity or acceleration.

Power Rule

The power rule is a quick method in calculus for taking the derivative of a function of the form x^n, where n is a real number. The rule states that to find the derivative of x^n, one simply multiplies n by x to the power of (n-1).

Mathematically, we write this as:
\[ \frac{d}{dx}(x^n) = nx^{(n-1)} \.\] The power rule greatly simplifies the process of differentiation, especially when dealing with polynomials. For instance, in our exercise example, f(x)=\frac{1}{3}x^3, applying the power rule gives us the derivative as f'(x) = x^2. This elegant shortcut saves time and effort, bypassing the need for more complex limit definitions of derivatives.

Solving Equations

Solving equations lies at the heart of algebra and many other areas of mathematics. When a problem asks to 'solve for x', it means to find the value or values of x that make the equation true. To solve an equation, we perform operations that will isolate the variable, which often involves moving terms from one side of the equation to the other and simplifying.

In the context of derivatives, like in our example, once we find f'(x), we set it equal to a certain value (in the problem, this value is 1) and solve for x. This translates to finding the points where the slope of the tangent line to the graph of the function is equal to a specific value. In our example, solving x^2 = 1 yields x=1 or x=-1, indicating two points on the curve of f(x) where the slope of the tangent is 1.