Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the gradient \(\nabla f\). $$ f(x, y)=x^{2} y+3 x y $$

Step-by-Step Solution

Answer

\( \nabla f = (2xy + 3y, x^2 + 3x) \)

Step 1: Finding Partial Derivative with respect to x

To find the gradient of a function, we first calculate the partial derivative of the function with respect to each variable. The first partial derivative to find is with respect to \(x\). For the function \(f(x, y) = x^2y + 3xy\), treat \(y\) as a constant:\[\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2y) + \frac{\partial}{\partial x}(3xy) = 2xy + 3y\].

Step 2: Finding Partial Derivative with respect to y

Next, we calculate the partial derivative of the function with respect to \(y\). For the function \(f(x, y) = x^2y + 3xy\), treat \(x\) as a constant:\[\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2y) + \frac{\partial}{\partial y}(3xy) = x^2 + 3x\].

Step 3: Writing the Gradient

The gradient of the function \(f(x, y) = x^2y + 3xy\) is a vector consisting of these partial derivatives. For any function \(f(x, y)\), the gradient is given by \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\). Substituting the values we get: \[ abla f = (2xy + 3y, x^2 + 3x) \].

Key Concepts

Partial Derivative

Partial derivatives are essential in calculus when dealing with functions of more than one variable. They allow us to understand how a function changes as only one of its variables changes, keeping the others constant. In the context of the function \( f(x, y) = x^2y + 3xy \), two partial derivatives are considered.

First, the partial derivative with respect to \( x \) is calculated by treating \( y \) as a constant. This results in:

• \( \frac{\partial f}{\partial x} = 2xy + 3y \)

Next, when calculating the partial derivative with respect to \( y \), \( x \) is treated as a constant, giving us:

• \( \frac{\partial f}{\partial y} = x^2 + 3x \)

Each partial derivative provides insight into how the function behaves with small changes in \( x \) and \( y \). This is particularly useful in optimization and understanding surfaces and curves in three or more dimensions.

By calculating partial derivatives, we're effectively "slicing" through the multidimensional space to explore the function's behavior along each axis.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of more than one variable. This becomes essential when working with real-world problems where multiple factors influence outcomes. For example, the function \( f(x, y) = x^2y + 3xy \) is a simple multivariable function, depending on both \( x \) and \( y \).

In multivariable calculus, we often use partial derivatives to analyze these functions, as seen in the process of finding the gradient of \( f(x, y) \). By considering each variable separately—while treating the others as constants—we can systematically understand the influences of these variables.

Multivariable calculus also introduces several key concepts such as:

• Contour diagrams and level surfaces to visualize functions.
• Gradient vectors, like the one calculated: \( abla f = (2xy + 3y, x^2 + 3x) \), providing direction and rate of maximum increase.
• Lagrange multipliers for constrained optimization problems.

These tools allow us to solve complex problems and explore dynamic systems in fields like physics, engineering, and economics.

Vector Calculus

Vector calculus combines ideas from calculus and vector algebra to handle vector fields, which are functions that assign a vector to each point in space. The gradient we found in the exercise is an application of vector calculus principles.

The gradient of a function, represented as \( abla f \), is a vector that points in the direction of the greatest rate of increase of the function. For our function \( f(x, y) = x^2y + 3xy \), the gradient \( abla f = (2xy + 3y, x^2 + 3x) \) shows how changes in \( x \) and \( y \) influence the overall function.

In vector calculus, the gradient has several important properties:

• It's perpendicular to level surfaces or curves of the function.
• It indicates the steepest ascent direction and its magnitude corresponds to the rate of increase.

Moreover, vector calculus isn't limited to gradients. It also covers operations such as divergence and curl, allowing us to analyze aspects like fluid flow and electromagnetic fields. These deeper insights into vector fields make vector calculus a powerful tool in both theoretical and practical applications.