Problem 22 - Calculus for Biology and Medicine

Question

In what direction does \(f(x, y)=\ln \left(x^{2}+y^{2}\right)\) increase most rapidly at \((1,1) ?\)

Step-by-Step Solution

Answer

The function increases most rapidly in the direction of the vector \(\langle 1, 1 \rangle\).

Step 1: Understanding the Function

We are given the function \(f(x, y) = \ln(x^2 + y^2)\). Our goal is to find the direction in which the function increases most rapidly at the point \((1,1)\). To do this, we will use the gradient vector, which points in the direction of the greatest rate of increase of the function.

Step 2: Calculate Partial Derivatives

First, calculate the partial derivatives of the function with respect to \(x\) and \(y\). The function is \(f(x, y) = \ln(x^2 + y^2)\). The partial derivative with respect to \(x\) is \(f_x = \frac{\partial}{\partial x}[\ln(x^2 + y^2)] = \frac{2x}{x^2 + y^2}\). The partial derivative with respect to \(y\) is \(f_y = \frac{\partial}{\partial y}[\ln(x^2 + y^2)] = \frac{2y}{x^2 + y^2}\).

Step 3: Evaluate the Gradient at (1, 1)

Now, evaluate the gradient vector at the point \((1,1)\). Substitute \(x = 1\) and \(y = 1\) into the partial derivatives: \(f_x(1,1) = \frac{2 \times 1}{1^2 + 1^2} = \frac{2}{2} = 1\) and \(f_y(1,1) = \frac{2 \times 1}{1^2 + 1^2} = \frac{2}{2} = 1\). Therefore, the gradient vector at \((1, 1)\) is \(abla f(1, 1) = \langle 1, 1 \rangle\).

Step 4: Determine the Direction

The gradient vector \(abla f(1, 1) = \langle 1, 1 \rangle\) indicates the direction in which the function \(f(x, y)\) increases most rapidly. This vector \(\langle 1, 1 \rangle\) suggests that the function increases most rapidly in the direction of the vector itself, which is diagonally towards the first quadrant.

Key Concepts

Understanding Partial Derivatives

Partial derivatives are a tool used in calculus to understand how a function changes as its variables are varied. For a function of two variables, like our function \[f(x, y) = \ln(x^2 + y^2),\]we can measure the change in two directions: along the x-axis and along the y-axis. The partial derivative with respect to x, written as \(f_x\), tells us how the function changes as we adjust x, keeping y constant. Similarly, the partial derivative with respect to y, denoted \(f_y\), shows how the function changes as we adjust y, keeping x constant.

• \(f_x = \frac{2x}{x^2 + y^2}\): Shows change in the function as x changes.
• \(f_y = \frac{2y}{x^2 + y^2}\): Shows change in the function as y changes.

Calculating partial derivatives is crucial for finding the gradient vector, which helps us locate the direction where the function grows fastest.

The Rate of Increase and Gradient Vector

The rate of increase of a function at a particular point is measured by the gradient vector. The gradient vector consists of the partial derivatives with respect to each variable. It points in the direction where the function increases the most rapidly. For our function, the gradient vector \( abla f \) is given by:

• Gradient vector: \( abla f(x, y) = \langle f_x, f_y \rangle = \left\langle \frac{2x}{x^2 + y^2}, \frac{2y}{x^2 + y^2} \right\rangle \)

To find exactly how fast the function increases in this direction, we calculate the magnitude of the gradient vector. The greater the magnitude, the faster the function rises as we move in that direction.
At the specific point \((1, 1)\), the gradient vector simplifies to \( \langle 1, 1 \rangle \). The length of this vector determines the steepness or speed of the increase. It's like climbing a hill in the direction where it steepens the fastest.

Directional Derivative and Maximum Increase

The directional derivative of a function tells us how much the function changes as we move in any given direction. It is calculated using the gradient vector. Specifically, when we have a unit vector \( \mathbf{u} \) pointing in some direction, the directional derivative \(D_uf\) in that direction is the dot product of the gradient vector and \( \mathbf{u} \).
In essence, it is expressed as:

• \(D_uf = abla f(x, y) \cdot \mathbf{u}\)

To maximize this directional derivative, \( \mathbf{u} \) should be in the same direction as the gradient vector. For our exercise, that direction is given by the vector \( \langle 1, 1 \rangle \). It represents the line where the function grows quickest at \((1, 1)\). The directional derivative here tells us the degree of increase as we travel in the gradient's direction, which means highest when \( \mathbf{u} \) aligns with \( abla f \).
This demonstrates why the gradient is a powerful tool—it not only points us to where the function increases rapidly but also quantifies this change.