Problem 29 - APEX Calculus

Question

Form a function \(z=f(x, y)\) such that \(f_{x}\) and \(f_{y}\) match those given. $$ f_{x}=6 x y-4 y^{2}, \quad f_{y}=3 x^{2}-8 x y+2 $$

Step-by-Step Solution

Answer

The function is \( z = 3x^2y - 4xy^2 + 2y + C \).

Step 1: Integrate with respect to x

Given that \( f_{x} = 6xy - 4y^2 \), integrate this function with respect to \(x\) to obtain an expression for \( f(x, y) \). \[\int (6xy - 4y^2) \, dx = 3x^2y - 4xy^2 + g(y)\]Here, \( g(y) \) is an unknown function of \( y \), representing the 'constant' of integration.

Step 2: Differentiate with respect to y

Now differentiate the result from Step 1 with respect to \(y\) and set it equal to the given \( f_y = 3x^2 - 8xy + 2 \). \[\frac{d}{dy} (3x^2y - 4xy^2 + g(y)) = 3x^2 - 8xy + g'(y)\]Setting this equal to \( f_y \):\[3x^2 - 8xy + g'(y) = 3x^2 - 8xy + 2\]

Step 3: Solve for g(y)

From the equation in Step 2, we can see that \[g'(y) = 2\]Integrate \( g'(y) \) with respect to \(y\) to find \( g(y) \):\[g(y) = 2y + C\]where \( C \) is a constant.

Step 4: Form the final function

Substitute \( g(y) = 2y + C \) back into the expression for \( f(x, y) \) obtained in Step 1:\[f(x, y) = 3x^2y - 4xy^2 + 2y + C\]This gives the function \( z = f(x, y) \) that has the required partial derivatives.

Key Concepts

Partial Derivatives

Partial derivatives are crucial in multivariable calculus, helping us understand how a multivariable function changes as we modify one variable while keeping others constant. In the given exercise, we are finding a function \( f(x, y) \) such that its partial derivatives \( f_x \) and \( f_y \) match specified forms. A partial derivative with respect to \( x \), denoted as \( \frac{\partial f}{\partial x} \), involves treating \( y \) as a constant. Conversely, \( \frac{\partial f}{\partial y} \) is taken by treating \( x \) as constant.

This approach allows us to evaluate the function's behavior along different axes. For example, given \( f_x = 6xy - 4y^2 \), it highlights how \( f \) changes as \( x \) varies, and the role \( y \) plays as a parameter. Similarly, \( f_y = 3x^2 - 8xy + 2 \) captures the change in \( f \) with \( y \), with \( x \) acting as a parameter.

• If you have function \( f(x, y) \), partial derivatives describe the slope in the x-direction or y-direction.
• They are useful in optimization and identifying points where functions have steepest inclines.

Integration

Integration, especially in the multivariable calculus, deals with finding a function whose derivative matches a given function. In our exercise, we begin by integrating the partial derivative \( f_x \) with respect to \( x \). This gives us a function, but it includes an unknown function \( g(y) \), highlighting that integrating with respect to one variable treats others as constants, potentially leaving integrated functions of those variables.

In the exercise, starting with the given \( f_x = 6xy - 4y^2 \), the integral looks like this:\[\int (6xy - 4y^2) \, dx = 3x^2y - 4xy^2 + g(y)\]The constant of integration here becomes a function \( g(y) \), as \( y \) is constant during integration. Only after finding derivative constraints can we fully determine \( g(y) \), as done by taking another derivative and comparing it with \( f_y \).

• Think of integration as a reverse process of differentiation.
• In multivariable contexts, 'constants' of integration can be functions of the other variables.
• Integration helps reconstruct the larger picture from rate of change components.

Function of Multiple Variables

A function of multiple variables, like \( f(x, y) \), represents a scenario where output depends on several inputs. These functions can describe surfaces in three-dimensional space, with \( z = f(x, y) \) representing height given coordinates \( x \) and \( y \).

In the exercise, by forming \( z = f(x, y) = 3x^2y - 4xy^2 + 2y + C \), we create a surface where the partial derivatives \( f_x \) and \( f_y \) provide the local slope along \( x \) and \( y \). This functionality makes multivariable functions key in modeling real-world situations.

• Such functions can model phenomena like heat distribution or elevation maps.
• They extend the concept of a curve (in 2D) into surfaces (in 3D).
• Knowing how to manipulate these gives insights into complex systems involving several influencing factors.