Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Sketch the graph of \(y=\sqrt[3]{x}\) for \(-1 \leq x \leq 1\). Does the graph have a tangent at (0,0) ? Remember, Your vote counts.

Step-by-Step Solution

Answer

The graph has no tangent at (0,0) because the derivative is undefined at this point.

Step 1: Understanding the Function

The function given is a cube root function represented as \( y = \sqrt[3]{x} \). This is an odd function that includes negative, positive, and zero values of \( x \) in its domain.

Step 2: Identify Key Points

First, identify points where you know the output. For \( x = 0 \), \( y = \sqrt[3]{0} = 0 \). For \( x = 1 \), \( y = \sqrt[3]{1} = 1 \). For \( x = -1 \), \( y = \sqrt[3]{-1} = -1 \). These points are: (0,0), (1,1), (-1,-1).

Step 3: Sketch the Graph

Using the identified points, sketch the graph. It will be symmetric with respect to the origin due to its odd nature. Begin at the point (-1, -1), pass smoothly through (0,0), and continue to (1,1). The graph will appear as an S-like curve.

Step 4: Assess Tangent at (0,0)

The tangent is assessed by finding the derivative. The derivative of \( y = \sqrt[3]{x} \) is \( y' = \frac{1}{3}x^{-2/3} \), which becomes undefined at \( x = 0 \) due to the negative exponent making the expression undefined. Therefore, there cannot be a tangent line because the slope is undefined at \( x = 0 \).

Key Concepts

Graph Sketching

When it comes to sketching the graph of the cube root function, you start by understanding its core characteristics. The function in question, given as \( y = \sqrt[3]{x} \), is a classic example of an odd function due to its symmetry relative to the origin. This means that if you flip it 180 degrees around the origin, it mirrors itself perfectly. This odd function property impacts how the graph looks and behaves.
Next, you identify key points to help plot the curve. For instance:

• When \( x = 0 \), \( y = \sqrt[3]{0} = 0 \), resulting in the point (0,0).
• When \( x = 1 \), \( y = \sqrt[3]{1} = 1 \), giving you the point (1,1).
• When \( x = -1 \), \( y = \sqrt[3]{-1} = -1 \), which is point (-1,-1).

With these key locations in place, you can sketch the graph as an 'S' shaped curve lightly winding through them. Beginning at (-1, -1), passing smoothly through (0,0), and continuing to (1,1). This sweeping motion reflects the continuous nature of the cube root function across both negative and positive values.

Tangent Line

In mathematics, a tangent line refers to a straight line that touches a curve at only one point, reflecting the instantaneous direction of the curve at that point. For the cube root function \( y = \sqrt[3]{x} \), we look specifically at the point (0,0), curious as to whether a tangent line exists there.
To decide, you first must determine the slope of the tangent line using derivatives. However, when the derivative lacks definition at a certain point, the standard tangent does not exist. This phenomenon may seem perplexing, but it primarily stems from the curve's nature at that exact spot.

Derivative Calculation

The derivative of a function provides critical insight into its behavior, specifically showing the rate of change or the slope of the function at any given point. When tackling \( y = \sqrt[3]{x} \), you calculate its derivative to assist in understanding the tangent's presence or absence at (0,0).
Beginning with the derivative \( y' = \frac{1}{3}x^{-2/3} \), you notice it consists of a negative fractional exponent. This is crucial because, at \( x = 0 \), the function becomes undefined. A negative exponent, particularly in a fraction, suggests division by zero, which is an undefined operation in mathematics. Hence, the derivative cannot be computed at \( x = 0 \), confirming the absence of a defined tangent line at this point. The curve is simply too steep and changes direction too rapidly at the origin.

Odd Function Properties

Functions exhibit varying symmetries, and knowing this helps in sketching graphs and solving equations. Odd functions, like \( y = \sqrt[3]{x} \), display specific properties that contribute to their symmetrical nature. Such functions fulfill the equation \( f(-x) = -f(x) \) for any \( x \) in their domain.
The cube root function, for instance, beautifully illustrates this attribute. As depicted by the graph symmetrically arranged around the origin, each positive \( x \) and \( y \) pair mirrors a corresponding negative pair. The elegant 'S' shaped trajectory confirms this symmetry, enhancing our understanding of the concept of odd functions.
Understanding symmetry not only simplifies graphing but also enables solutions to complex calculus problems, as these properties frequently unveil hidden solutions or shortcuts. Whether exploring functions from a theoretical or practical perspective, recognizing these patterns furthers insights into their overall behavior.