Problem 25 - Calculus for Biology and Medicine

Question

Let $$f(x, y)=4-x^{2}-y^{2}$$ Compute \(f_{x}(1,1)\) and \(f_{y}(1,1)\), and interpret these partial derivatives geometrically.

Step-by-Step Solution

Answer

\(f_{x}(1,1) = -2\) and \(f_{y}(1,1) = -2\); both indicate slopes in respective directions.

Step 1: Identify the Function

The exercise gives you the function \( f(x, y) = 4 - x^2 - y^2 \). This is a function of two variables: \(x\) and \(y\).

Step 2: Calculate the Partial Derivative with respect to x

To find \(f_{x}\), the partial derivative of \( f(x, y)\) with respect to \(x\), treat \(y\) as a constant and differentiate in terms of \(x\). This yields: \(f_{x} = \frac{\partial}{\partial x}(4 - x^2 - y^2) = -2x\).

Step 3: Evaluate \( f_{x}(1, 1) \)

Substitute \(x = 1\) into the partial derivative function \(f_{x} = -2x\): \( f_{x}(1, 1) = -2(1) = -2\).

Step 4: Calculate the Partial Derivative with respect to y

Next, compute the partial derivative of \(f(x, y)\) with respect to \(y\), treating \(x\) as a constant. This is: \(f_{y} = \frac{\partial}{\partial y}(4 - x^2 - y^2) = -2y\).

Step 5: Evaluate \( f_{y}(1, 1) \)

Substitute \(y = 1\) into the partial derivative function \(f_{y} = -2y\): \( f_{y}(1, 1) = -2(1) = -2\).

Step 6: Geometric Interpretation of Partial Derivatives

The partial derivative \(f_{x}(1, 1) = -2\) indicates the slope of the tangent line to the surface \(z = f(x, y)\) in the direction of the positive \(x \)-axis at point \((1, 1)\). Similarly, \(f_{y}(1, 1) = -2\) represents the slope of the tangent line to the surface in the direction of the positive \(y\)-axis at the same point.

Key Concepts

Geometric Interpretation

Partial derivatives serve as a bridge connecting functions of two or more variables to their geometric properties. When we compute the partial derivatives of a function like \( f(x, y) = 4 - x^2 - y^2 \), we gain insights into its shape and behavior. These derivatives essentially describe the slope of the surface, or graph, formed by \( f(x, y) \) in three-dimensional space.

• The partial derivative \( f_x(1, 1) = -2 \) tells us about the slope of the slice of the surface when moving along the \( x \)-axis while keeping all other variables constant.
• Similarly, \( f_y(1, 1) = -2 \) indicates the slope of the surface slice along the \( y \)-axis.

These slopes are instrumental in understanding how the function changes at any given point, such as \( (1, 1) \), providing a snapshot prediction of the surface’s tilt or orientation. Through partial derivatives, the surface seemingly transforms into simpler, linear segments when viewed from each axis, making it easier to visualize and comprehend the function’s intricacies.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. Rather than dealing with single-variable functions, where the domain is a line, multivariable functions have a domain that could encompass a plane or even higher-dimensional spaces. In this context, functions take the form \( f(x, y,...) \), where \( x \), \( y \), and so on are independent variables.

• The concept of multivariable calculus encompasses both differentiation and integration of functions with more than one variable.
• Partial derivatives, such as \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \), are essential types of derivatives in this field.

Partial derivatives show how each variable independently affects the function’s output. For example, while dealing with \( f(x, y) \), altering \( x \) while keeping \( y \) constant explores how the function changes in relation to \( x \). This is crucial for analyzing complex systems where multiple factors influence outcomes. Through multivariable calculus, we can tackle diverse problems, from physics to economics, offering insights into natural and theoretical phenomena.

Tangent Planes

Tangent planes represent a key concept that emerges from studying the partial derivatives of multivariable functions. Just as a tangent line approximates a curve at a point, a tangent plane approximates a surface at a given point in space. This is particularly useful when dealing with functions of two variables, such as \( f(x, y) \).

• The equation of a tangent plane at a point \( (x_0, y_0) \) is derived using partial derivatives.
• The general formula is: \( z = f(x_0, y_0) + f_x(x_0, y_0)(x-x_0) + f_y(x_0, y_0)(y-y_0) \).

For the function \( f(x, y) = 4 - x^2 - y^2 \), calculated partial derivatives at any point allow us to construct the tangent plane equation precisely at that point. At \( (1, 1) \), knowing \( f_x(1, 1) = -2 \) and \( f_y(1, 1) = -2 \), the tangent plane simplifies to \( z = 1 - 2(x-1) - 2(y-1) \). This plane provides a versatile and straightforward view of the surface's local behavior, promoting better understanding and analysis of the multivariable function.