Problem 20 - Calculus for Biology and Medicine

Question

In what direction does \(f(x, y)=e^{x} \cos y\) increase most rapidly at \((0, \pi / 2) ?\)

Step-by-Step Solution

Answer

The function increases most rapidly in the negative y-direction at (0, π/2).

Step 1: Find the Gradient

The direction in which a function increases the fastest at any point is given by the gradient of the function at that point. To find the gradient, we need to compute the partial derivatives of the function with respect to each variable. Here, the function is given by \( f(x, y) = e^x \cos y \). So, the partial derivatives are as follows:\[ \frac{\partial f}{\partial x} = e^x \cos y \]\[ \frac{\partial f}{\partial y} = -e^x \sin y \]The gradient is then:\[ abla f(x, y) = \left( e^x \cos y, -e^x \sin y \right) \]

Step 2: Evaluate the Gradient at the Given Point

To determine the direction of fastest increase at the point \((0, \frac{\pi}{2})\), we need to evaluate the gradient at this point.Substitute \(x = 0\) and \(y = \frac{\pi}{2}\) into the gradient:\[ abla f(0, \frac{\pi}{2}) = \left( e^0 \cos \frac{\pi}{2}, -e^0 \sin \frac{\pi}{2} \right) \]This simplifies to:\[ abla f(0, \frac{\pi}{2}) = \left( 1 \cdot 0, -1 \cdot 1 \right) = (0, -1) \]

Step 3: Interpret the Result

The gradient vector \( (0, -1) \) indicates the direction in which the function \( f(x, y) \) increases most rapidly at the point \((0, \frac{\pi}{2})\). Specifically, the function increases most rapidly in the negative \( y \)-direction.

Key Concepts

Partial Derivatives Demystified

Partial derivatives are an essential concept in multivariable calculus, responsible for the detailed analysis of how a function changes in relation to each of its variables. Imagine a function such as \( f(x, y) = e^x \cos y \). Here, partial derivatives allow us to decipher how the function \( f \) changes as only one of the variables \( x \) or \( y \) changes, while the other remains constant.

• Notation: Partial derivatives are denoted as \( \frac{\partial f}{\partial x} \) or \( \frac{\partial f}{\partial y} \) for variations with respect to \( x \) or \( y \), respectively.
• Application: For the function \( f(x, y) = e^x \cos y \), the partial derivative with respect to \( x \) is \( e^x \cos y \), highlighting how \( f \) changes as \( x \) varies.
• Significance: This method becomes crucial when desiring to understand complex functions in higher dimensions, aiding in the construction of the function’s gradient.

Partial derivatives provide the foundational information necessary for constructing the gradient vector, which further enlightens us about the function's behavior.

Understanding Directional Derivatives

Directional derivatives open up a world where changes in a function can be observed in specific directions, not just aligned with the coordinate axes. They allow you to measure how a function, like \( f(x, y) = e^x \cos y \), changes if you were to move in any direction from a given point.

• Definition: A directional derivative is the rate at which the function \( f \) changes in a specified direction, given by a vector \( \mathbf{v} \).
• Calculation: The directional derivative of \( f \) along \( \mathbf{v} \) is computed as the dot product of the gradient of \( f \) at a point and the unit vector in the direction of \( \mathbf{v} \).
• Utility: This is invaluable for understanding the behavior of a function when moving in non-standard directions, especially those dictated by real-world constraints.

Thus, directional derivatives expand our toolkit for analyzing functions beyond their basic partial derivatives, offering insights into changes not limited to standard axis-aligned directions.

Unraveling Function Optimization

Function optimization is the art of finding the maximum or minimum values of a function, key to numerous applications in science and engineering. When dealing with a function like \( f(x, y) = e^x \cos y \), the gradient plays a pivotal role.

• Purpose: By finding where the gradient is zero, typically through setting each component of the gradient to zero, critical points can be identified.
• Process: At these points, further analysis, often using second derivatives, helps determine if these are maxima, minima, or saddle points.
• Outcome: This process allows us to optimize the function by identifying points where it reaches its highest or lowest value.

Optimization involves leveraging mathematical tools, like gradients, to make the best possible decision or prediction based on a function’s behavior, guiding everything from economic strategies to engineering designs.