Problem 2 - Calculus for Biology and Medicine

Question

The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=x^{2}-3 y^{2} ;(-1,1,-2) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = -2x - 6y + 2 \).

Step 1: Identify the function and the point

Given the function is \( f(x, y) = x^2 - 3y^2 \) and the point \((-1, 1, -2)\). To find the equation of the tangent plane, we need the partial derivatives of the function at the given point.

Step 2: Compute the partial derivatives

First, find the partial derivative of \( f \) with respect to \( x \), denoted \( f_x \). We get \( f_x(x, y) = 2x \).Similarly, compute the partial derivative with respect to \( y \), denoted \( f_y \). We have \( f_y(x, y) = -6y \).

Step 3: Evaluate partial derivatives at the given point

Substitute \( x_0 = -1 \) and \( y_0 = 1 \) into the partial derivatives:\[ f_x(-1, 1) = 2(-1) = -2 \]\[ f_y(-1, 1) = -6(1) = -6 \]

Step 4: Use the formula for the tangent plane

The equation of the tangent plane at a point \((x_0, y_0, z_0)\) is:\[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]Substitute \( x_0 = -1 \), \( y_0 = 1 \), \( z_0 = -2 \), \( f_x(-1, 1) = -2 \), and \( f_y(-1, 1) = -6 \):\[ z + 2 = -2(x + 1) - 6(y - 1) \]

Step 5: Simplify the equation

Expand and simplify the equation:\[ z + 2 = -2x - 2 - 6y + 6 \]Rearrange the terms:\[ z = -2x - 6y + 2 \]

Key Concepts

Partial Derivatives

Partial derivatives are the backbone of understanding how functions behave when we change one of their variables while keeping the others constant. In the context of a function like the one given here, namely, \( f(x, y) = x^2 - 3y^2 \), partial derivatives help us determine the slope of the function in the direction of each variable.
For instance, the partial derivative with respect to \( x \), or \( f_x \), is calculated by treating \( y \) as a constant and differentiating \( f \) with respect to \( x \). Similarly, to find \( f_y \), we treat \( x \) as a constant while differentiating. These derivatives are essential for finding tangent planes because they provide the rates at which the function changes along each axis, which ultimately determines the plane's orientation.

• For this function, \( f_x = 2x \) and \( f_y = -6y \).

With these formulas, we can evaluate how the function behaves locally around any given point.

Function Evaluation

Now that we have derived the partial derivatives \( f_x = 2x \) and \( f_y = -6y \), our next step is to plug in the coordinates of the given point \((-1, 1)\) to evaluate these derivatives. This step is crucial as it tells us the exact slope of the tangent plane at this specific location.

• Evaluate \( f_x(-1, 1) = 2(-1) = -2 \).
• Evaluate \( f_y(-1, 1) = -6(1) = -6 \).

By evaluating these at \( x_0 = -1 \) and \( y_0 = 1 \), we obtain the slopes of the tangent plane along the \( x \) and \( y \) directions respectively. These evaluated slopes are essential for constructing the tangent plane equation.

Equation of Tangent Plane

Having the partial derivatives evaluated, we can proceed to formulate the equation of the tangent plane. The general formula for the tangent plane at a given point \((x_0, y_0, z_0)\) for the function \( z = f(x, y) \) is given by:
\[z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]
Let's substitute the known values, \(f_x(-1, 1) = -2\), \(f_y(-1, 1) = -6\), and the point \((-1, 1, -2)\):
\[z + 2 = -2(x + 1) - 6(y - 1)\]
This step applies the evaluated function values to determine the exact plane expression.

Mathematical Notation

Mathematical notation is a tool that helps us communicate complex mathematical ideas in a clear and standardized way. In this exercise, understanding notation is fundamental:

• Symbols like \( f_x \) and \( f_y \) denote partial derivatives, which represent the rate of change of the function in relation to each axis.
• The `(x_0, y_0, z_0)` notation refers to a specific point in three-dimensional space where the tangent plane is calculated.
• The formula for the tangent plane, \( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \), showcases how these derivatives come together to determine the plane's equation.

Each symbol and number in mathematical notation has a precise meaning, contributing to a systematic approach to solving problems, such as finding tangent planes.