Problem 50 - Thomas Calculus

Question

Does the curve \(y=2 \sqrt{x}\) have any horizontal tangents? If so, where? Give reasons for your answer.

Step-by-Step Solution

Answer

The curve has no horizontal tangents, as its derivative is never zero.

Step 1: Understand Horizontal Tangents

A tangent is horizontal if the slope (derivative) of the curve at that point is zero. We need to find the derivative of the given function and set it to zero to determine where the slope is horizontal.

Step 2: Differentiate the Function

Find the derivative of the function \(y=2\sqrt{x}\). Using the power rule, rewrite \(\sqrt{x}\) as \(x^{1/2}\) and differentiate: \(\frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}}\). So, the derivative is \(y' = \frac{1}{\sqrt{x}}\).

Step 3: Set the Derivative to Zero

Set the derivative \(y' = \frac{1}{\sqrt{x}}\) equal to zero: \(\frac{1}{\sqrt{x}} = 0\). This equation has no solutions because \(\frac{1}{\sqrt{x}}\) is never zero; it is undefined for \(x = 0\) and positive elsewhere.

Step 4: Conclude About Horizontal Tangents

Since the derivative \(\frac{1}{\sqrt{x}}\) is never zero, the curve \(y=2\sqrt{x}\) does not have horizontal tangents at any point.

Key Concepts

Derivatives

Derivatives are a cornerstone of calculus that help us understand how functions change. They measure the rate at which a function is changing at any given point. This is crucial because it tells us the slope of the tangent line to the curve of that function.When calculating a derivative, you're essentially finding a formula that gives the slope of the tangent line at any point on the curve. For example, if you have a function like \( y=2\sqrt{x} \), you would use derivative rules like the power rule to find this derivative, expressed as \( y' = \frac{1}{\sqrt{x}} \).Here are some key points about derivatives:

• They tell us whether the function is increasing or decreasing at a point.
• A positive derivative indicates the function is going up, and a negative derivative shows it's going down.
• A zero derivative at a point often indicates a local maximum or minimum, but not in the case of horizontal asymptotes like our example function.

Understanding derivatives helps us answer many questions about functions and their behaviors.

Tangents

Tangents are lines that touch a curve at exactly one point without crossing it. Around this point, the tangent line has the same slope as the curve itself. This concept is important for determining things like instantaneous rate of change and curve behavior.For the curve described by \( y=2\sqrt{x} \), a horizontal tangent would imply that the slope of the tangent (the derivative) at that point is zero. However, when you compute the derivative \( y' = \frac{1}{\sqrt{x}} \), it turns out that it never equals zero. This means there are no points where a horizontal tangent exists.Consider these characteristics of tangents:

• Tangents can be either horizontal, sloped, or vertical.
• Horizontal tangents have a derivative of zero, while vertical ones can have an undefined slope.
• Tangent lines provide a linear approximation of the curve at small intervals nearby the point of tangency.

Tangents give us immediate insights into the nature of curves and allow us to estimate values and directions effectively.

Mathematical Functions

Mathematical functions are equations that relate an input to a single output. In calculus, functions commonly involve variables raised to powers, roots, exponentials, and other operations.For instance, the function \( y=2\sqrt{x} \) is a basic form, where the output \( y \) depends on the input \( x \mid \), transformed by taking the square root and multiplying by 2. This particular function produces a curve that starts from the origin and increases smoothly as \( x \) increases.Here are some core aspects of mathematical functions:

• They can be continuous, which means no breaks or jumps in their graphs.
• They help model real-world phenomena, such as physics problems and economic trends.
• Functions are defined by their domain, the set of all input values, and their range, the set of all possible outputs.

Understanding functions is fundamental in math because they describe how quantities depend on each other, forming the basis for both simple and complex mathematical models.