Problem 28 - Calculus for Biology and Medicine

Question

Let $$ f(x, y)=2 x^{3}-3 y x $$ Compute \(f_{x}(1,2)\) and \(f_{y}(1,2)\), and interpret these partial derivatives geometrically.

Step-by-Step Solution

Answer

\( f_x(1, 2) = 0 \) and \( f_y(1, 2) = -3 \). At (1,2), the function is flat in the \(x\)-direction and decreases in the \(y\)-direction.

Step 1: Understand Partial Derivatives

A partial derivative of a function with respect to one of its variables is the derivative while treating the other variables as constants. For function \( f(x, y) \), \( f_x \) is the partial derivative with respect to \( x \), and \( f_y \) is the partial derivative with respect to \( y \).

Step 2: Compute \( f_x(x, y) \)

The partial derivative of \( f(x, y) = 2x^3 - 3yx \) with respect to \( x \) is calculated by differentiating each term in \(x\) and treating \(y\) as a constant. The derivative of \(2x^3\) is \(6x^2\), and the derivative of \(-3yx\) with respect to \(x\) is \(-3y\). Thus, \( f_x(x, y) = 6x^2 - 3y \).

Step 3: Compute \( f_y(x,y) \)

The partial derivative of \( f(x, y) = 2x^3 - 3yx \) with respect to \( y \) is calculated by differentiating each term in \(y\) and treating \(x\) as a constant. Since \(2x^3\) does not contain \(y\), its derivative is \(0\). The derivative of \(-3yx\) with respect to \(y\) is \(-3x\). Thus, \( f_y(x, y) = -3x \).

Step 4: Evaluate \( f_x(1, 2) \)

Substitute \( x = 1 \) and \( y = 2 \) into the expression for \( f_x(x, y) \): \[ f_x(1, 2) = 6(1)^2 - 3(2) = 6 - 6 = 0 \]

Step 5: Evaluate \( f_y(1, 2) \)

Substitute \( x = 1 \) into the expression for \( f_y(x, y) \): \[ f_y(1, 2) = -3(1) = -3 \]

Step 6: Geometric Interpretation

The value \( f_x(1, 2) = 0 \) indicates that at the point \((1,2)\), the slope of the function with respect to the \(x\)-direction is zero, suggesting a possible local minimum or maximum in the \(x\)-direction. The value \( f_y(1, 2) = -3 \) indicates a negative slope in the \(y\)-direction, meaning the function decreases as \(y\) increases at \((1,2)\).

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. Consider the function \( f(x, y) = 2x^3 - 3yx \). Here, \( f \) depends on two variables, \( x \) and \( y \). In multivariable calculus, we study how changes in one of these variables affect the function, while holding the other variable constant. This is where partial derivatives come in.

Partial derivatives work similarly to ordinary derivatives but apply in a multivariable context. For example, the partial derivative \( f_x \) represents the rate of change of the function concerning \( x\), with \( y \) kept constant. Conversely, \( f_y \) measures how the function changes in response to \( y \), with \( x\) being constant.

This approach allows us to analyze the behavior of the function in various "slices" across its domain, giving insights into its overall shape and behavior across each variable. This is especially valuable in understanding real-world phenomena that depend on multiple variables, such as temperature changes with both altitude and latitude.

Geometric Interpretation

The geometric interpretation of partial derivatives provides a visual understanding of how a function behaves around specific points. Imagine the graph of \( f(x, y) = 2x^3 - 3yx \) as a surface in three-dimensional space. The point \((1, 2)\) corresponds to a specific location on this surface.

The partial derivative \( f_x(1, 2) = 0 \) indicates that if you move along the surface by changing \( x \), while keeping \( y \) at 2, the slope of the surface is zero. This means you're at a level point with no inclination, suggesting either a local extremum in the \( x \)-direction.

On the other hand, \( f_y(1, 2) = -3 \) tells us that moving in the \( y \)-direction, while keeping \( x = 1 \), results in a slope of -3. This negative value implies the surface is tilting downward as \( y \) increases, indicating a decreasing behavior of the function in this direction.

These interpretations are crucial in fields like optimization, where one needs to find the highest or lowest points on surfaces defined by multivariable functions.

Function Analysis

Analyzing functions in multivariable calculus involves breaking down the behavior of functions for deeper insights. With \( f(x, y) = 2x^3 - 3yx \), we're interested not only in calculating derivatives but also in understanding the function's nature at specific points, like \((1, 2)\).

Function analysis through partial derivatives provides insight into the local behavior of the function. By evaluating \( f_x \) and \( f_y \) at specific points:

• \( f_x(1, 2) = 0 \) suggests no change in the function as \( x \) increases along the slice where \( y = 2 \).


• \( f_y(1, 2) = -3 \) indicates the function insight decreasing as \( y \) increases along the slice where \( x = 1 \).

These evaluations allow us to predict and analyze tendencies, such as stability at points of interest or response to variable changes. Function analysis is key to understanding underlying phenomena across disciplines ranging from economics to engineering, where multivariable systems are prevalent.