Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ z=3 x+5 y-1 $$

Step-by-Step Solution

Answer

The first partial derivative of \(z = 3x + 5y - 1\) with respect to \(x\) is \(\frac{\partial z}{\partial x} = 3\) and with respect to \(y\) is \(\frac{\partial z}{\partial y} = 5\).

Step 1: Identify the Function and Variables

Identify the function and the variables to which you are expected to differentiate. In this problem, the function is \(z = 3x + 5y - 1\) and the variables are \(x\) and \(y\).

Step 2: Find the Partial Derivative with Respect to \(x\)

To find the partial derivative of a function with respect to \(x\), we differentiate the function with respect to \(x\) while treating all other variables as constants. Here, when taking the partial derivative of \(z\) with respect to \(x\), the term \(5y\) is a constant. Thus, \(\frac{\partial z}{\partial x} = 3\).

Step 3: Find the Partial Derivative with Respect to \(y\)

Similarly, to find the partial derivative with respect to \(y\), we differentiate the function with respect to \(y\) while keeping \(x\) as a constant. Here, when taking the partial derivative with respect to \(y\), the term \(3x\) is a constant. Thus, \(\frac{\partial z}{\partial y} = 5\).

Key Concepts

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions that depend on more than one variable. Rather than just one dimension, we deal with functions that live within a multidimensional space. For example, a function might depend on both the variables \(x\) and \(y\), as seen in our original scenario with the function \(z = 3x + 5y - 1\).

In multivariable calculus, we explore concepts like partial derivatives, which are essential in understanding how functions change in response to changing variables. These derivatives help us analyze surfaces and curves in higher dimensions.

Key concepts in multivariable calculus include:

• Understanding vectors and vector fields, representing quantities having both direction and magnitude.
• The extension of derivatives to more than one dimension, exploring rates of change in multiple directions.
• Integration over regions of higher dimensions than just intervals.

Grasping these concepts can provide powerful insight into applications in engineering, physics, computer graphics, and economics.

Differentiation

Differentiation involves finding the derivative of a function, which essentially measures how the function value changes as its input changes. In single-variable calculus, this is straightforward as the derivative gives the rate of change concerning that one variable.

In multivariable calculus, this notion extends to multiple variables. This is important because in real-world applications, especially those involving complex systems, functions are usually dependent on more than one variable.

A few important points about differentiation in the context of multiple variables:

• Partial Derivatives: These are derivatives of functions with respect to one variable while keeping the others constant. For example, finding \(\frac{\partial z}{\partial x}\) of \(z = 3x + 5y - 1\) treats \(y\) as a constant while differentiating with respect to \(x\).
• Notations for Partial Derivatives: The notations \(\frac{\partial}{\partial x}\) and \(\frac{\partial}{\partial y}\) are common and help in distinguishing from ordinary derivatives.

This extension makes differentiation a powerful tool in analyzing multidimensional systems.

Functions of Multiple Variables

When dealing with functions of multiple variables, it's crucial to understand how changes in one variable impact the overall function, while other variables remain constant. This is foundational in fields like physics, engineering, and economics. For example, the function \(z = 3x + 5y - 1\) implies that \(z\) changes as either \(x\) or \(y\) changes.

Functions of multiple variables can describe surfaces in a three-dimensional space. Consider mapping weather data across geography: temperature might depend on both latitude and longitude.

Crucial elements include:

• Domain and Range: Just like functions of a single variable, functions of multiple variables have domains (where they exist) and ranges (outputs or results).
• Graphical Representation: These functions can be represented using 3D graphs or contour plots, which provide insights into how the function behaves over a plane or space.
• Applications and Implications: Used in optimization problems, understanding inter-dependencies in complex systems, and modeling real-world scenarios.

Mastering functions of multiple variables allows students to tackle complex real-world problems with many influencing factors.