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Chapter 2: Problem 114

Determine the point(s) at which the graph of \(f(x)=\frac{x}{\sqrt{2 x-1}}\) has a horizontal tangent line.

Short Answer

Expert verified
The function \(f(x) = \frac{x}{\sqrt{2x-1}}\) has no point at which it has a horizontal tangent line.

Step by step solution

01

Find the derivative of \(f(x)\)

Using the quotient rule, the derivative of the function \(f(x) = \frac{x}{\sqrt{2x-1}}\) becomes \(f'(x) = \frac{\sqrt{2x-1} - (x \cdot \frac{1}{2 \sqrt{2x-1}})}{(2x-1)}\). Simplifying, we get \(f'(x) = \frac{1}{2 \sqrt{2x-1}}\).
02

Set the derivative equal to 0 and solve for \(x\)

Setting \(f'(x) = 0\), we get \(\frac{1}{2 \sqrt{2x-1}} = 0\). This equation doesn't have a real root, hence there are no values of \(x\) for which \(f(x)\) has a horizontal tangent.
03

Check for points of undefined derivative as potential horizontal tangents

The derivative is undefined when the denominator equals zero. But solving \(2x-1 = 0\) for \(x\) gives \(x = 0.5\), which is not in the domain of the original function \(f(x)\) because for \(x = 0.5\), \(f(x)\) is undefined. Therefore, it is not considered as a point of horizontal tangent line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Derivatives
The concept of a derivative is essential in calculus, as it represents how a function's value changes in relation to changes in the independent variable. In simpler terms, the derivative of a function at a given point can be thought of as the slope of the tangent line to the graph of the function at that point.
To find where the function has a horizontal tangent line, we look for points where the derivative is equal to zero. This is because a horizontal line has a slope of zero. Calculating derivatives often requires rules, such as the product rule, chain rule, or the quotient rule, depending on the type of function you're working with.
  • Horizontal Tangent Line: Where the derivative of the function is zero, indicating a zero slope.
  • Slope: The measure of how steep a line is; calculated as the change in the y-coordinate divided by the change in the x-coordinate.
Quotient Rule Essentials
When dealing with functions that are ratios of two other functions, we use the quotient rule to find the derivative. The quotient rule is a handy tool that simplifies the differentiation of a function that is a fraction.
For a function in the form of \( f(x) = \frac{u(x)}{v(x)} \), the derivative, using the quotient rule, is given by:
\[ f'(x) = \frac{u'(x) \, v(x) - u(x) \, v'(x)}{(v(x))^2} \]
This formula is crucial when aiming to calculate derivatives of functions like \( f(x) = \frac{x}{\sqrt{2x-1}} \), as it involves both a numerator and a denominator that are themselves functions of \( x \).
  • Numerator: The top part of the fraction, which in this formula is \( u(x) \).
  • Denominator: The bottom part of the fraction, or \( v(x) \) in the quotient rule formula.
Exploring Function Domain
The domain of a function is the set of all possible input values (\( x \)-values) that will produce a valid output from the function. Therefore, determining this domain is fundamental and must be considered when analyzing the function's behavior.
For instance, consider the function \( f(x) = \frac{x}{\sqrt{2x-1}} \). The expression inside the square root, \( 2x-1 \), must be non-negative for the function to remain real and defined, implying that \( 2x-1 \geq 0 \). Solving for \( x \), we get \( x \geq 0.5 \).
So, the domain for this function is all real numbers \( x \) such that \( x \geq 0.5 \).
  • Domain: All the possible input values for which the function is defined.
  • Illumination for Function Analysis: Knowing the domain helps avoid potential pitfalls in the function's behavior, such as undefined regions.

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Most popular questions from this chapter

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