Problem 10 - APEX Calculus

Question

A function \(z=f(x, y)\) is given. Find \(\nabla f\). $$ f(x, y)=-4 x+3 y $$

Step-by-Step Solution

Answer

The gradient \(\nabla f\) is \((-4, 3)\).

Step 1: Recall the Definition of Gradient

The gradient of a function \(f(x, y)\) is a vector composed of the partial derivatives of \(f\) with respect to each of its variables. In gradient notation, it is expressed as \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\).

Step 2: Find the Partial Derivative with respect to x

Calculate the partial derivative of \(f(x, y) = -4x + 3y\) with respect to \(x\). The constant \(3y\) has a derivative of zero since it does not depend on \(x\). Thus, the derivative is \(\frac{\partial f}{\partial x} = -4\).

Step 3: Find the Partial Derivative with respect to y

Calculate the partial derivative of \(f(x, y) = -4x + 3y\) with respect to \(y\). The term \(-4x\) is treated as a constant since it does not depend on \(y\), and thus its derivative is zero. The derivative is \(\frac{\partial f}{\partial y} = 3\).

Step 4: Combine Partial Derivatives into the Gradient

Now, form the gradient vector using the partial derivatives found. This gives the gradient of the function as \(abla f = (-4, 3)\).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept when dealing with functions of multiple variables. To understand this, imagine a function that depends on two or more variables, for example, \(f(x, y)\). The partial derivative measures how the function changes as one of its variables is varied, while all other variables are held constant.

To compute a partial derivative, choose a variable to differentiate with respect to and treat all other variables as constants. Here, the process is similar to regular differentiation but focused on one variable only. For instance, in the function \(f(x, y) = -4x + 3y\), the partial derivative with respect to \(x\) was calculated by treating \(y\) as constant, resulting in \(\frac{\partial f}{\partial x} = -4\). Likewise, \(\frac{\partial f}{\partial y} = 3\) is found by treating \(x\) as constant.

Key facts about partial derivatives:

• They describe the rate of change in one direction.
• Each variable gets its own derivative.
• They're essential in constructing the gradient, an important vector in calculus.

Functions of Multiple Variables

Functions of multiple variables expand the idea of a single-variable function into a multi-dimensional space. In essence, these functions depend on two or more inputs, such as \(x\) and \(y\). A simple example is the function \(f(x, y) = -4x + 3y\).

Why do we use these functions? Many real-world scenarios have multiple dependencies. For instance, the temperature at a certain point could depend on both latitude and altitude. Similarly, businesses might model profit as a function of sales and marketing spend.

When dealing with multiple variables, understanding how changes in each variable affect the function is crucial. This is precisely where partial derivatives play a role. They track changes in the function with respect to one variable while keeping others constant. Efficient use of partial derivatives allows us to construct more complex ideas such as gradients, which provide information on function rates of change in a multi-variable context.

Vector Calculus

Vector calculus is a branch of mathematics focused on multi-dimensional spaces where functions often output vectors rather than just numbers. It becomes crucial when dealing with gradients, which capture the direction and rate of function change with respect to several variables.

The gradient, denoted \(abla f\), is a vector that bundles all the partial derivatives of a function. It gives the direction of the steepest ascent of the function. In our example, the gradient \(abla f = (-4, 3)\) signifies the direction in the flat plane described by \(x\) and \(y\) along which the function increases most rapidly.

Key points about vector calculus include:

• Using gradients to find optimal points where function values are maximized or minimized.
• Helping in understanding complex systems by analyzing directional derivatives.
• Being applicable in fields like physics, engineering, and economics.

Thus, vector calculus, with its robust tools, is indispensable for theoretical and practical work in higher dimensions.