Problem 23 - Calculus an Applied Approach

Question

Evaluate \(f_{x}\) and \(f_{y}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=e^{3 x y} \quad(0,4) $$

Step-by-Step Solution

Answer

The calculated partial derivatives \(f_{x}\) and \(f_{y}\) at point (0,4) are: \(f_{x}(0,4)=12\) and \(f_{y}(0,4)=0\).

Step 1: Find \(f_{x}\)

The partial derivative \(f_{x}\) involves taking the derivative of the function with respect to \(x\), while treating all other variables as constants. Thus, for the function \(f(x, y)=e^{3xy}\), applying the chain rule gives: \(f_{x} = \frac{\partial}{\partial x} e^{3 x y} = 3y e^{3 x y}\) .

Step 2: Find \(f_{y}\)

The partial derivative \(f_{y}\) involves taking the derivative of the function with respect to \(y\), while treating all other variables as constants. Thus, for the function \(f(x, y)=e^{3xy}\), applying the chain rule gives: \(f_{y} = \frac{\partial}{\partial y} e^{3 x y} = 3x e^{3 x y}\) .

Step 3: Evaluate \(f_{x}\) and \(f_{y}\) at point (0,4)

The next step is to calculate \(f_{x}\) and \(f_{y}\) at the given point (0,4). For \(f_{x}(0,4)\), substituting \(x=0\) and \(y=4\) into \(f_{x} = 3y e^{3 x y}\) yields: \(f_{x}(0,4)=3*4*e^{3*0*4}=12*1=12\).Similarly, for \(f_{y}(0,4)\), substituting \(x=0\) and \(y=4\) into \(f_{y} = 3x e^{3 x y}\) yields: \(f_{y}(0,4)=3*0*e^{3*0*4}=0\).

Key Concepts

Chain Rule in Multivariable Calculus

Understanding the chain rule in multivariable calculus is essential for computing partial derivatives of functions with more than one variable. In essence, the chain rule allows you to differentiate composite functions — that is, functions made up by combining other functions. When you're dealing with a function of several variables, such as f(x, y) = e^{3xy}, and you need to find the partial derivative with respect to one variable, you apply the chain rule while treating the other variables as constants.

The application of the chain rule in the context of partial derivatives involves identifying the 'inner function' and the 'outer function' and then taking the derivative of the outer function with respect to the inner function multiplied by the derivative of the inner function with respect to the variable of interest.

Step-by-Step Application

• Identify the variables:
  In the function f(x, y) = e^{3xy}, x and y are the variables.
• Treat the non-differentiated variable as a constant:
  When finding f_x, consider y as a constant and vice versa.
• Apply the differentiation rules:
  For f_x, differentiate e^{3xy} as if it were a function of x only (because y is treated as a constant).
• Calculate the result:
  This yields f_x = 3y e^{3xy}. Similarly, for f_y, the procedure results in f_y = 3x e^{3xy}.

This method ensures a systematic approach to finding partial derivatives and applying the chain rule, simplifying the process for functions of multiple variables.

Multivariable Calculus and Partial Derivatives

Multivariable calculus extends the concepts of single-variable calculus to functions of several variables. A critical component is understanding partial derivatives, which measure how the function changes as each variable is varied while keeping other variables fixed.

For the function f(x, y) = e^{3xy}, the partial derivatives represent the rate of change of the function in the direction of one variable at a time. f_x and f_y reveal the sensitivity of the function to changes in x and y, respectively.

Visualizing Partial Derivatives

• f_x: Imagine you are walking on the surface defined by f(x, y) and you can only move in the direction of the x-axis. f_x tells you how steeply the surface rises or falls as you walk.
• f_y: Conversely, if you walk in the direction of the y-axis, f_y indicates the slope of the surface along that path.

By calculating these derivatives, you gain detailed insight into how the function behaves locally in the vicinity of a particular point, enabling accurate predictions and deeper understanding of the function's geometry.

Function Evaluation at a Point

The process of function evaluation at a specific point provides concrete values for the function's behavior and its derivatives at that point. Once the partial derivatives are calculated using methods from multivariable calculus, such as the chain rule, they can be evaluated at points of interest to assess the function's rate of change in each variable direction at those points.

For our function f(x, y) = e^{3xy}, after finding the expressions for f_x and f_y, we can evaluate these at the point (0,4). Plugging in the coordinates into the derived expressions (f_x(0,4) and f_y(0,4)), we obtain concrete numbers:

• f_x(0,4) is calculated by substituting x=0 and y=4 into the expression for f_x, giving us the result 12.
• f_y(0,4), on the other hand, becomes 0 when we substitute these same values because the expression f_y includes a term 3x, which equals 0 when x=0.

This step is crucial as it transforms the abstract expressions of the derivatives into tangible values that provide insights into the function's behavior at specific locations in its domain. It's particularly useful when analyzing the function's dynamics in relation to real-world problems.