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Magazine Chapter 3: Problem 27
Exer. 25-30: Determine whether \(f\) has (a) a vertical tangent line at (0,0) and \(\mid\) b) a cusp at (0,0) . $$ f(x)=x^{2 / 5} $$
Short Answer
Expert verified
The function has a cusp at (0,0) but no vertical tangent line.
Step by step solution
01
Understand the Graphical Characteristics
The problem asks us to identify a vertical tangent line or a cusp at the point \((0,0)\) for the function \(f(x) = x^{2/5}\). A vertical tangent line occurs when the slope (derivative) approaches infinity, and a cusp is a point where the derivative approaches \(+fty\) from one side and \(-\infty\) from the other.
02
Find the Derivative
Compute the derivative of the function \(f(x) = x^{2/5}\). Use the power rule for derivatives:\[ \frac{d}{dx}(x^n) = nx^{n-1} \] So, the derivative \(f'(x)\) is \[ f'(x) = \frac{2}{5}x^{-3/5} \] which simplifies to \[ f'(x) = \frac{2}{5} \cdot \frac{1}{x^{3/5}} \].
03
Evaluate the Derivative at x=0
Evaluate \(f'(x)\) as \(x\) approaches 0. In \(f'(x) = \frac{2}{5}x^{-3/5}\), substitute \(x = 0\): \[ f'(0) = \lim_{x \to 0} \frac{2}{5}x^{-3/5} \] Since \(x^{-3/5}\) tends to \(+\infty\) as \(x\) approaches \(0^+\) and to \(-\infty\) as \(x\) approaches \(0^-\), the derivative does not exist at \(x=0\) and reflects a cusp.
04
Conclusion About Vertical Tangent and Cusp
A vertical tangent requires the derivative to approach \(\pm \infty\) from both sides, while a cusp requires it to approach \(+\infty\) from one side and \(-\infty\) from the other side. Thus, the function \(f(x) = x^{2/5}\) has a cusp at \((0,0)\) but no vertical tangent line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertical Tangent Lines: A Steep Exploration
In mathematics, a vertical tangent line is like finding the steepest part of a roller coaster. Imagine a line on a graph that shoots straight up or down. This happens at points where the slope of the tangent line approaches infinity. For a function to have a vertical tangent at a specific point, the slope or the derivative must reach infinite values from both directions as it approaches that point.
- Vertical tangents show extreme changes in the behavior of a function.
- They represent points where a function's graph is very steep.
- It's crucial to examine the slope through derivatives to identify them.
The Derivative: A Tool to Measure Change
The derivative is a fundamental concept in calculus, representing the rate at which a function changes at any given point. Think of it like measuring how fast an object moves at a precise moment. In mathematical terms, the derivative of a function f(x) is often denoted as f'(x).
Here are some key ideas about derivatives:
Here are some key ideas about derivatives:
- They help determine the slope of a line tangent to the curve at any point.
- Derivatives can indicate where a function is increasing or decreasing.
- Finding the derivative uses rules like the power rule, which can simplify calculations.
The Power Rule: Simplifying Derivatives
The power rule is an essential tool for differentiating functions quickly and easily. If you're given a function in the form of x raised to a power, the power rule is a hero here. Expressed as \( \frac{d}{dx}(x^n) = nx^{n-1} \), it allows you to swiftly calculate the derivative.
Applying the power rule involves:
Applying the power rule involves:
- Multiplying the exponent by the coefficient, if any.
- Decreasing the exponent by one to find the new power of x.
- Using this technique, complex derivatives become less daunting.
Graphical Characteristics: Making Sense of Shapes and Curves
Graphical characteristics refer to the features of a function's graph that reveal a deeper understanding of its behavior. When we examine a graph, we're looking at a visual representation of all possible outputs for inputs provided to a function.
Here are elements you might consider:
Here are elements you might consider:
- Identifying the shapes, like curves, linear segments, or sharp points (cusps).
- Looking for symmetrical properties or periodic features.
- Recognizing areas where the graph is steep or flat, indicating rate of change.
