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Magazine Chapter 3: Problem 44
Give an example of: A curve that has two horizontal tangents at the same \(x\) -value, but no vertical tangents.
Short Answer
Expert verified
The function \(f(x) = (x-1)^2(x+1)^2\) satisfies the conditions, having two horizontal tangents at \(x = 1\) and no vertical tangents.
Step by step solution
01
Understand the Problem
We are asked to find a curve with two horizontal tangents at the same \(x\)-value but no vertical tangents. A horizontal tangent implies that the derivative of the function equals zero.
02
Analyze Horizontal Tangents
Horizontal tangents occur where the derivative of the function is zero. For example, in a polynomial function, this can be seen where the slope of the tangent (or simply the derivative) equals zero at specific points.
03
Choose Template for Function
We will choose a quartic polynomial function because it can have more than one extreme point at the same \(x\)-value and three extreme points generally.
04
Consider Specific Example
Consider the function \(f(x) = (x^2 - a^2)^2\), where \(f'(x) = 0\) at \(x = \pm a\). The derivative is \(f'(x) = 4x(x^2 - a^2)\). Horizontal tangents occur at \(x = 0\) and other points depending on the value of \(a\).
05
Evaluate for Two Horizontal Tangents at the Same Point
Set \(f(x) = (x^2 - a^2)^2\) and try \(a = 0\). Then you have \(f(x) = x^4\), which has no two horizontal tangents at the same \(x\). Thus select \(a\) to satisfy intersections such as \((x-1)^2(x+1)^2\).
06
Verify No Vertical Tangents
Check if there are any places where the derivative is undefined. The function \(f(x) = (x-1)^2(x+1)^2\) has derivative \(4x(x^2-1)\), which never becomes undefined, confirming no vertical tangents.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Tangents
When we talk about horizontal tangents, we refer to those parts of a graph where the slope of the function is zero. In simple terms, a horizontal tangent means the curve is flat at that specific point. This scenario occurs where the derivative of the function equals zero.
To illustrate:
To illustrate:
- Imagine driving a car uphill and then reaching the top where it's completely flat for a moment before going down again. That moment of flatness is much like a horizontal tangent on a graph.
- Mathematically, if you want to find where these horizontal tangents occur, you solve for where the derivative of your function equals zero.
Derivative Calculation
Calculating the derivative is a crucial step in finding where horizontal tangents occur. The derivative gives us the slope of the function, or how steep the curve is at any given point.
For a function like a polynomial, the derivative follows specific rules based on power and multiplication. For example:
For a function like a polynomial, the derivative follows specific rules based on power and multiplication. For example:
- The derivative of a simple power function, like \(x^n\), is \(nx^{n-1}\).
- For a product of polynomials, we use the product rule.
Quartic Polynomial Functions
Quartic polynomial functions are those with a degree of four, meaning the highest power of the variable is four. These functions can be a bit challenging because they can twist and turn quite a bit, having up to three points where they might level out to form horizontal tangents.
A typical form of a quartic polynomial might look like \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\). In our example:
A typical form of a quartic polynomial might look like \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\). In our example:
- The choice of quartic polynomial was \(f(x) = (x^2 - a^2)^2\).
- This specific form allows it to potentially have horizontal tangents at the same \(x\)-value.
Vertical Tangents
Vertical tangents are different from horizontal tangents because they describe points where a curve shoots straight up or down, becoming almost vertical. Technically, at these points, the slope approaches infinity, and the derivative of the function does not exist.
In context:
In context:
- A vertical tangent happens when the function's rate of change is undefined.
- With polynomials like the one in our example, vertical tangents are avoided by ensuring the derivative never becomes impossible to calculate.
