Problem 15 - Calculus

Question

Draw a graph of \(y=x^{3}-x .\) Where is the slope zero?

Step-by-Step Solution

Answer

The slope is zero at \(x = \pm \sqrt{\frac{1}{3}}\).

Step 1: Differentiate the function

To determine where the slope of the function is zero, we need to find the derivative of the function. Differentiate the function \(y = x^3 - x\) with respect to \(x\): \[ \frac{dy}{dx} = 3x^2 - 1. \]

Step 2: Set the derivative equal to zero

To find where the slope is zero, set the derivative equal to zero: \[ 3x^2 - 1 = 0. \] This will help us identify the critical points where the slope of the tangent to the curve is zero.

Step 3: Solve for \(x\)

Solve the equation \(3x^2 - 1 = 0\) for \(x\). Add 1 to both sides to obtain \(3x^2 = 1\). Divide both sides by 3: \[ x^2 = \frac{1}{3}. \] Take the square root of both sides to find \(x\): \[ x = \pm \sqrt{\frac{1}{3}}. \]

Step 4: Identify the points with zero slope

The points where the slope of the graph is zero are at \(x = \sqrt{\frac{1}{3}}\) and \(x = -\sqrt{\frac{1}{3}}\). These values of \(x\) correspond to the locations on the curve where the slope is zero.

Key Concepts

Derivative

In calculus, a derivative represents the rate of change of a function with respect to one of its variables. For the function given, the derivative helps us determine how the function behaves as the variable changes. The derivative of a function at a certain point tells us the slope of the tangent line at that point.

• To find the derivative, we differentiate the function.
• The differentiation turns the function into another expression that gives us the slope for any point on the curve.

For the function \( y = x^3 - x \), the derivative is calculated as \( \frac{dy}{dx} = 3x^2 - 1 \). This expression tells us how the y-values change as x changes. Understanding derivatives is crucial as it provides insights into the rate at which one variable affects another and is fundamental for finding slopes and critical points.

Slope

The slope of a function at any given point is the value of the derivative at that point. It indicates the steepness and direction of the function's graph.

• A positive slope means the function is increasing.
• A negative slope indicates the function is decreasing.
• A zero slope points to a horizontal tangent, often a critical point.

For the function \( y = x^3 - x \), to find where the slope is zero, we set \( 3x^2 - 1 \) to zero and solve for \( x \). The points where the slope is zero are \( x = \pm \sqrt{\frac{1}{3}} \). At these points, the graph of the function flattens out, indicating potential maximum or minimum points or points of inflection.

Critical Points

Critical points in a function occur where the derivative is zero or undefined. They are essential for understanding the function's behavior, as they can indicate local maxima, minima, or points of inflection.

• To find critical points, set the derivative equal to zero.
• Solutions to this give the x-values where the graph might change direction.

For \( y = x^3 - x \), setting the derivative \( 3x^2 - 1 = 0 \) gives critical points at \( x = \pm \sqrt{\frac{1}{3}} \). These points are essential for constructing the function's graph since they highlight where the curve might peak, dip or become level.

Graph of a Function

A graph of a function visually displays how the function's values change over a range of inputs. It helps in understanding the overall behavior of the function, including trends and key characteristics like maxima, minima, and inflection points.

• The function \( y = x^3 - x \) shows both growth and decline because it is a cubic function.
• Where the derivative equals zero, the graph either reaches a peak, a dip, or changes concavity.

The critical points at \( x = \pm \sqrt{\frac{1}{3}} \) indicate where the graph of this function has horizontal tangents. Because it is cubic, the graph will likely have an S-shape, with these critical points signifying the points of inflection or local turning points. Drawing the graph with this information in mind gives a complete picture of the function's behavior over its domain.