Problem 57 - Calculus for Biology and Medicine

Question

Find the length of the subtangent to the curve \(y=2^{x}\) at the point \((1,2)\).

Step-by-Step Solution

Answer

The length of the subtangent is \(\frac{1}{\ln(2)}\).

Step 1: Differentiate to Find the Slope

First, find the derivative of the function \(y = 2^x\) with respect to \(x\), which represents the slope of the tangent at any point \(x\). Using the chain rule, the derivative is \(\frac{dy}{dx} = 2^x \ln(2)\).

Step 2: Evaluate the Slope at the Given Point

Substitute \(x = 1\) into the derivative to find the slope of the tangent at the point \((1, 2)\). \[ \frac{dy}{dx}\bigg|_{x=1} = 2^1 \ln(2) = 2 \ln(2) \].

Step 3: Determine the Subtangent Formula

The length of the subtangent for a curve is given by \(\frac{y}{\frac{dy}{dx}}\). In this case, substitute \(y = 2\) and \(\frac{dy}{dx} = 2 \ln(2)\) into the formula.

Step 4: Calculate the Length of the Subtangent

Using the subtangent formula, we have: \[ \text{Length of subtangent} = \frac{y}{\frac{dy}{dx}} = \frac{2}{2 \ln(2)} = \frac{1}{\ln(2)} \]

Key Concepts

Derivative of Exponential Function

To understand the concept of the derivative of an exponential function, let's break it down. An exponential function can be generally expressed as \(y = a^x\), where \(a\) is a constant base. The derivative involves applying rules of differentiation to find the rate at which the function \(y\) changes concerning \(x\).
In our example, we have the function \(y = 2^x\). To find its derivative, we use the chain rule, which helps with functions composed of other functions. The chain rule states that the derivative of \(a^x\) is \(a^x \, \ln(a)\). For our function, when \(a = 2\), this results in:

• \(\frac{dy}{dx} = 2^x \, \ln(2)\)

This equation gives us the slope of the tangent line at any point on the curve \(y = 2^x\). Calculating it comprehensively helps us further in finding subtangents and other properties of the curve.

Tangent Line Slope

The slope of the tangent line is a crucial concept because it describes the steepness of the curve at any given point.If you're picturing a curve and you place a straight line tangent to this curve at a specific point, you're looking at how steep that line is.
For the function \(y = 2^x\) at the point \((1, 2)\), we determine the slope by evaluating the derivative we calculated earlier:

• \(\frac{dy}{dx}\bigg|_{x=1} = 2^1 \, \ln(2) = 2 \, \ln(2)\)

Here, \(2 \, \ln(2)\) is the slope of the tangent line at the point \((1,2)\). This figure represents how quickly the function value \(y\) changes for a small change in \(x\). Understanding this concept helps inform us how a curve behaves locally, right at that particular point. The larger the slope, the steeper the tangent.

Subtangent Formula

After grasping the derivative and slope, the concept of a subtangent becomes clearer.A subtangent is the projection of the tangent line on the x-axis, essentially measuring how much horizontal distance it covers.
The subtangent formula is an elegant result given by the fraction \(\frac{y}{\frac{dy}{dx}}\). This formula connects the height of the curve \(y\) and the slope of the tangent, \(\frac{dy}{dx}\). In our specific example:

• At \((1, 2)\), \(y = 2\) and \(\frac{dy}{dx} = 2 \, \ln(2)\).

Plug these into the formula:

• \[ \text{Length of subtangent} = \frac{y}{\frac{dy}{dx}} = \frac{2}{2 \, \ln(2)} = \frac{1}{\ln(2)} \]

This expression allows you to calculate how far horizontally the tangent line extends at the specific point where \(x = 1\).Knowing the subtangent can offer insights into the behavior of the curve in terms of its slope and position relative to the axes.