Problem 37 - Calculus for Biology and Medicine

Question

In what direction does \(f(x, y)=\sqrt{x^{2}-y^{2}}\) increase most rapidly at \((5,3)\) ?

Step-by-Step Solution

Answer

The function increases most rapidly in the direction \( \left( \frac{5}{4}, -\frac{3}{4} \right) \).

Step 1: Calculate the Gradient

To understand the direction in which the function increases most rapidly, we need to calculate the gradient vector of the function. The gradient of a function \( f(x, y) \) is a vector formed by the partial derivatives with respect to each variable. So, first compute the partial derivative: \[ \frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 - y^2}} \] and \[ \frac{\partial f}{\partial y} = -\frac{y}{\sqrt{x^2 - y^2}}. \] Thus, the gradient is \( abla f(x,y) = \left( \frac{x}{\sqrt{x^2 - y^2}}, -\frac{y}{\sqrt{x^2 - y^2}} \right). \)

Step 2: Evaluate the Gradient at the Point

Now substitute the point \((5, 3)\) into the gradient to find its value at that specific point. \[ abla f(5, 3) = \left( \frac{5}{\sqrt{5^2 - 3^2}}, -\frac{3}{\sqrt{5^2 - 3^2}} \right). \] Calculate the inner term: \( \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \). Thus, \[ abla f(5, 3) = \left( \frac{5}{4}, -\frac{3}{4} \right). \]

Step 3: Determine the Direction

The gradient vector \( abla f(x, y) \) points in the direction of the greatest increase. At the point \((5, 3)\), this is \( \left( \frac{5}{4}, -\frac{3}{4} \right) \). Therefore, the function increases most rapidly in the direction of this vector: it is directed positive in the x-component and negative in the y-component.

Key Concepts

Partial Derivatives

Partial derivatives are a key concept in understanding how multivariable functions behave. Think of a multivariable function as a landscape, where each point has a specific height. When we talk about partial derivatives, we're asking, "How steep is the slope if I only move in one direction?" Moving only in the x-direction asks what happens to the height if you vary just **x** slightly and keep **y** constant. This is expressed by \( \frac{\partial f}{\partial x} \).
Similarly, moving in the y-direction and inquiring about the slope is explained using \( \frac{\partial f}{\partial y} \).

• If the partial derivative with respect to x is positive, the function increases as x increases.
• If it's negative, the function decreases as x increases.
• The same applies for changes in y and its respective partial derivative.

Partial derivatives give crucial information about how the multivariable function changes with each independent variable. They are foundational for forming gradients and understanding multivariable calculus thoroughly.

Directional Derivatives

Directional derivatives help us understand how a function changes when moving in a specific direction.
Unlike partial derivatives, which look at changes along the standard axes (x and y), directional derivatives consider any direction, even diagonally!
To find a directional derivative, first calculate the gradient vector of the function. The gradient points in the direction of greatest increase of the function.

• The directional derivative in the direction of a vector \( \textbf{u} \) is found by taking the dot product of the gradient vector and \( \textbf{u} \).
• The greater the value of this derivative, the sharper the increase in that direction.
• A directional derivative of zero indicates no increase or decrease in that direction.

Understanding directional derivatives allows us to predict not just steepness, but also how the terrain changes as you walk along any arbitrary path.

Multivariable Functions

Multivariable functions go beyond simple graphs and involve functions with more than one input. Consider the mountain analogy: the function's inputs, such as x and y, determine a specific point or 'position' on the mountain, and the output is the height at that position.

• Each input represents a different dimension, creating a rich, three-dimensional landscape.
• Commonly used in fields such as physics, engineering, and economics to model complex systems.

To study these landscapes, partial and directional derivatives are essential. They help measure how the function changes with small adjustments in each dimension, much like checking the slope of a mountain path.
Understanding multivariable functions involves thinking more three-dimensionally compared to single-dimensional calculus, requiring a solid grasp of how each variable independently and collectively influences the output.