Problem 17 - Calculus for Biology and Medicine

Question

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right) .\) \(f(x, y)=x-3 y ;(3,1)\)

Step-by-Step Solution

Answer

The linearization of \(f(x, y) = x - 3y\) at \((3, 1)\) is \(L(x, y) = x - 3y\).

Step 1: Identify the function and the point

We are given the function \(f(x, y) = x - 3y\) and need to find the linearization at the point \((x_0, y_0) = (3, 1)\).

Step 2: Calculate partial derivatives

Calculate the partial derivative of \(f\) with respect to \(x\), \(f_x(x, y)\), which is \(1\). Now calculate the partial derivative with respect to \(y\), \(f_y(x, y)\), which is \(-3\).

Step 3: Evaluate partial derivatives at the point

Substitute the point \((3, 1)\) into the partial derivatives: \(f_x(3, 1) = 1\) and \(f_y(3, 1) = -3\).

Step 4: Use the linearization formula

The formula for linearization at the point \((x_0, y_0)\) is: \(L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\).

Step 5: Calculate \(f(x_0, y_0)\)

Calculate \(f(3, 1) = 3 - 3 \times 1 = 0\).

Step 6: Substitute into the linearization formula

Substitute everything into the linearization formula: \(L(x, y) = 0 + 1\cdot(x - 3) - 3\cdot(y - 1)\).

Step 7: Simplify the expression

Simplify the expression to get the linearization: \(L(x, y) = x - 3 - 3y + 3 = x - 3y\).

Key Concepts

Partial Derivatives

When dealing with a function in multivariable calculus, understanding the behavior of each variable independently can be crucial. This is where partial derivatives come into play. A partial derivative measures how a function changes as just one of the inputs changes, while all other inputs are held constant.
For example, consider the function \(f(x, y) = x - 3y\). The partial derivative of \(f\) with respect to \(x\), denoted as \(f_x(x, y)\), tells us how \(f\) changes when \(x\) changes, while \(y\) remains the same. In our exercise, \(f_x(x, y) = 1\). This means that for every unit increase in \(x\), \(f\) increases by 1.
Similarly, the partial derivative of \(f\) with respect to \(y\), denoted \(f_y(x, y)\), provides insights about the sensitivity of \(f\) concerning changes in \(y\). We find that \(f_y(x, y) = -3\), indicating that a unit increase in \(y\) results in \(f\) decreasing by 3 units. So, partial derivatives give you a targeted look into how each variable uniquely influences function behavior.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus from one dimension to multiple dimensions. Instead of only considering functions of a single variable, we focus on functions that have two or more variables, like \(f(x, y)\).
This branch of calculus involves multiple levels of differentiation and integration. It also requires tools like partial derivatives to analyze functions of several variables.
Why do we need multivariable calculus? It allows for the modeling and solving of complex problems across different fields such as physics, engineering, economics, and more. With multivariable calculus, you can:

• Calculate rates of change for processes involving several variables.
• Optimize functions to find maximum or minimum values.
• Analyze motion along different planes and in space.

These abilities make multivariable calculus a powerful tool for real-world applications. It goes beyond simple equations and enables you to understand the intricate dance between variables that govern many physical and abstract phenomena.

Linear Approximation

Linear approximation, also known as linearization, is a technique used to approximate a function with a line near a given point. It's like zooming in on a small part of a more complex graph and using a straight line to represent that section.
The core idea is to use the information provided by derivatives to create a linear function that approximates the original function nearby a chosen point \((x_0, y_0)\). The linearization formula is given by \[L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\].
In our example, the linearization process involves calculating \(L(x, y) = x - 3y\), which happens to be the same as the original function due to its linear nature. Generally, for more complex functions, the linearization provides a valuable tool for estimating values and understanding function behavior in a localized region.

• It simplifies complex surfaces to tangent planes.
• Helps in understanding the function's local behavior.
• Useful in solving equations where an exact solution is not easy to attain.

Overall, linear approximation bridges the gap between nonlinear functions and linear analysis, providing a practical approach to problem-solving.