Question

Find all points on the graph of the function at which the curvature is zero. $$ y=1-x^{3} $$

Short answer

The point on the graph where the curvature is zero is at (0,1).

Step-by-step solution

Find the first derivative

The first step is to find the first derivative \(y'\) of the function. The derivative of the given function \(y = 1 - x^{3}\) can be obtained using the power rule. By applying the power rule, we get \(y' = 0 - 3x^{2} = -3x^{2}\).

Find the second derivative

Next, we need to find the second derivative \(y''\) from the first derivative found in Step 1. Again using the power rule, the derivative of \(-3x^{2}\) becomes \(y'' = -6x\).

Solve the second derivative for zero

Solve for \(x\) when \(y'' = 0\). Setting \(-6x = 0\), we find \(x = 0\).

Find the corresponding \(y\) value

Substitute \(x = 0\) into the original function \(y = 1 - x^{3}\) to find the \(y\) value. Therefore \(y = 1 - (0)^{3} = 1\).

Key concepts

Derivatives

Understanding the concept of a derivative is crucial for tackling a broad array of problems in calculus. Essentially, a derivative represents the rate at which a function's value changes as its input changes. When we say we are taking the derivative of a function, like in the original exercise \(y = 1 - x^{3}\), we are looking for a new function (denoted as \(y'\) or \(\frac{dy}{dx}\)) that gives the slope of the tangent to the graph of the function at any point \(x\).

The process of finding \(y'\) involves using rules for differentiation; among them, the power rule is particularly helpful. For instance, when we apply the power rule to each term of the given function, we efficiently tackle the first step of finding where the curvature of the graph is zero.

Second Derivative

Once we've obtained the first derivative, entering the realm of the second derivative \(y''\) unveils information about the concavity of a graph and points of inflection. The second derivative is simply the derivative of the derivative and it tells us how the slope of the tangent line \(y'\) is changing as \(x\) changes.

In our given function, by taking the derivative of \(y' = -3x^{2}\), we obtain \(y'' = -6x\). Solving for when \(y'' = 0\) reveals where the concavity of the graph changes—a vital step in determining where the curvature is zero. Essentially, a zero second derivative indicates the point where the function changes from concave up to concave down, or vice versa, thus potentially affecting the curvature of the graph.

Power Rule

The power rule is a quick and simple method to differentiate functions with exponents. It states that if we have a function of the form \(x^n\), its derivative is \(nx^{n-1}\). This rule is used twice in our exercise; first, to find \(y'\) from the original function \(y = 1 - x^{3}\), and second, to find \(y''\) from \(y'\).

When applying the power rule to \(x^{3}\), the derivative is \(3x^{2}\), and we include the minus sign from the original function to get \(y' = -3x^{2}\). We apply the power rule again to this derivative to find the second derivative \(y'' = -6x\), an essential step for completing the curvature analysis.

Calculus

Calculus is the mathematical study dealing with continuous change. It is divided into two main branches: differential calculus and integral calculus. Differential calculus, where our exercise falls under, focuses on finding the rate at which quantities change, while integral calculus is about the accumulation of quantities.

Understanding the nuances of calculus is not only about solving equations but interpreting their geometric implications – for example, analyzing the curvature of a graph. The insight provided by the calculations we perform when determining the curvature at certain points on a graph bridges the abstract computations of derivatives with the tangible shape and behavior of the graph.