Problem 58 - Calculus for Biology and Medicine

Question

Find the length of the subtangent to the curve \(y=\exp \left[x^{2}\right]\) at the point \(\left(2, e^{4}\right)\).

Step-by-Step Solution

Answer

The length of the subtangent is \( \frac{1}{4} \).

Step 1: Recall the Formula for Subtangent

The length of the subtangent of a curve at any point is given by \( y \cdot \left| \frac{dx}{dy} \right| \). To find this, we first need the derivative \( \frac{dy}{dx} \) and then its reciprocal \( \frac{dx}{dy} \).

Step 2: Compute the Derivative \( \frac{dy}{dx} \)

Given \( y = \exp(x^2) \), take the derivative with respect to \( x \). Using the chain rule, we have:\[ \frac{dy}{dx} = \exp(x^2) \cdot 2x = 2x \cdot \exp(x^2). \]

Step 3: Compute the Reciprocal \( \frac{dx}{dy} \) at the Given Point

At the point \( (2, e^4) \), substitute \( x = 2 \) in the derivative:\[ \frac{dy}{dx} = 2 \cdot 2 \cdot e^4 = 4e^4. \]Now calculate \( \frac{dx}{dy} \) as the reciprocal:\[ \frac{dx}{dy} = \frac{1}{4e^4}. \]

Step 4: Substitute into the Subtangent Formula

The length of the subtangent is given by:\[ y \cdot \left| \frac{dx}{dy} \right| = e^4 \cdot \left| \frac{1}{4e^4} \right| = e^4 \cdot \frac{1}{4e^4}. \]Simplify this expression:\[ = \frac{1}{4}. \]

Key Concepts

Understanding Subtangent

In the context of calculus, the subtangent is an intriguing concept linked to tangent lines of curves. Imagine a tangent line touching a curve at a specific point. The subtangent is the horizontal segment of this tangent line, extending from the point of tangency to the x-axis. It's a measure of how fast or slow a curve changes direction near that point.
To find the length of the subtangent, we often use either:

• Rectangular coordinates
• Polar coordinates

In our problem, we use the formula for the subtangent in rectangular coordinates, which states:
\[ \text{length of subtangent} = y \cdot \left| \frac{dx}{dy} \right| \]This requires calculating the derivative \( \frac{dy}{dx} \) and then obtaining its reciprocal \( \frac{dx}{dy} \). Each of these steps are crucial to understanding how curves interact with their axes.

Fundamentals of Differentiation

Differentiation is a core concept in calculus, representing the process of finding the rate at which a function is changing. In simpler terms, it's the mathematical technique used to determine the slope of a tangent line to a curve at any given point. The result of differentiation is called a derivative, and it can be denoted as \( \frac{dy}{dx} \).
In this problem, we're dealing with the function \( y = \exp(x^2) \). Differentiate this function with respect to \( x \) to obtain the slope of the curve at any point:
\[ \frac{dy}{dx} = \exp(x^2) \cdot 2x = 2x \cdot \exp(x^2) \]This expression tells us how steep the curve is at different \( x \) values. The specific calculation for \( x = 2 \) resulted in \( 4e^4 \), helping us better understand the behavior of the function at that particular input, which in turn aids in assessing changes at different points.

Mastering the Chain Rule

The chain rule is an essential tool in calculus used when differentiating composite functions. Composite functions are functions made up of two or more functions nested within each other.
A simple analogy would be like peeling an onion: you handle one layer before moving on to the next. When applying the chain rule to \( y = \exp(x^2) \), notice that it's a composition of the exponential function \( \exp(u) \), where \( u = x^2 \). The chain rule states:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]In this function:

• First, let \( u = x^2 \), making \( y = \exp(u) \)
• Then calculate \( \frac{dy}{du} = \exp(u) \)
• And \( \frac{du}{dx} = 2x \)

Putting it all together, the chain rule provides:\[ \frac{dy}{dx} = \exp(x^2) \cdot 2x \]This method makes handling more complex differentiation tasks simpler by breaking them into manageable parts.