Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Find the partial derivatives, \(F_{1}, F_{2}, F_{1,1}, F_{1,2}, F_{2,1}\) and \(F_{2,2}\) of the following functions. a. \(\quad F(x, y)=3 x-5 y+7\) b. \(\quad F(x, y)=x^{2}+4 x y+3 y^{2}\) c. \(\quad F(x, y)=x^{3} y^{5}\) d. \(\quad F(x, y)=\sqrt{x y}\) e. \(\quad F(x, y)=\ln (x \times y) \quad\) f. \(\quad F(x, y)=\frac{x}{y}\) g. \(\quad F(x, y)=e^{x+y}\) h. \(\quad F(x, y)=x^{2} e^{-y}\) i. \(\quad F(x, y)=\sin (2 x+3 y)\) j. \(F(x, y)=e^{-x} \cos y\)

Step-by-Step Solution

Answer

Compute first and second order partial derivatives for each function using definitions; apply symmetry for mixed derivatives and summarize results.

Step 1: Understanding Partial Derivatives

Before solving, recognize that partial derivatives measure the function's rate of change with respect to one variable while keeping other variables constant. Notations: For a function \( F(x,y) \), \( F_1 = \frac{\partial F}{\partial x} \), \( F_2 = \frac{\partial F}{\partial y} \), and second-order derivatives are \( F_{1,1} = \frac{\partial^2 F}{\partial x^2} \), \( F_{1,2} = \frac{\partial^2 F}{\partial x \partial y} \), and similarly for other combinations.

Step 2: Compute First Order Partial Derivatives for each function

For each function, find \( F_1 \) and \( F_2 \). For instance, take \( F(x,y) = 3x - 5y + 7 \). Differentiate \( F \) with respect to \( x \) to find \( F_1 = 3 \), and differentiate with respect to \( y \) to find \( F_2 = -5 \). Repeat the process for functions (b) to (j).

Step 3: Compute Second Order Partial Derivatives for each function

For each function, compute \( F_{1,1} \), \( F_{1,2} \), \( F_{2,1} \), and \( F_{2,2} \). Continuing from (a), since \( F_1 = 3 \) is constant, \( F_{1,1} = 0 \). Similarly, \( F_2 = -5 \) being constant implies \( F_{2,2} = 0 \), and \( F_{1,2} = F_{2,1} = 0 \) since cross-partial derivatives of constants are zero. Repeat similar operations for functions (b) to (j).

Step 4: Check Symmetry of Mixed Partial Derivatives

Verify that \( F_{1,2} = F_{2,1} \) if the function is continuously differentiable according to Clairaut's theorem. This implies that the mixed partials should be equal unless the function's continuity assumptions are violated.

Step 5: Present Calculations Succinctly

Summarize the computed partial derivatives for each function. Each function's results should present \( F_1, F_2, F_{1,1}, F_{1,2}, F_{2,1}, F_{2,2} \) in sequence for clarity.

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of the concepts of single-variable calculus to functions of more than one variable. In multivariable calculus, we often work with functions like \( F(x, y) \), where changes can occur in any of the variables—independently or simultaneously. This field is essential as many real-world problems involve multiple variables. Understanding multivariable calculus enables us to:

• Analyze surfaces and curves in higher dimensions.
• Examine the behavior of functions across variables.
• Apply calculus concepts to physics and engineering.

A fundamental part of multivariable calculus is the use of partial derivatives, which tell us the rate of change of the function with respect to one variable while keeping others constant. This is key in determining how functions behave in multidimensional space.

Second-order Derivatives

Second-order derivatives are an extension of the concept of first-order derivatives. They measure how the rate of change of a function—the derivative—changes. For a function \( F(x,y) \), second-order derivatives include:

• \( F_{1,1} = \frac{\partial^2 F}{\partial x^2} \): the second derivative of \( F \) with respect to \( x \).
• \( F_{2,2} = \frac{\partial^2 F}{\partial y^2} \): the second derivative of \( F \) with respect to \( y \).
• \( F_{1,2} = \frac{\partial^2 F}{\partial x \partial y} \) or \( F_{2,1} = \frac{\partial^2 F}{\partial y \partial x} \): mixed partial derivatives.

These are crucial for analyzing the curvature of surfaces and understanding the nature of critical points, such as determining if they are minima, maxima, or saddle points.

Clairaut's Theorem

Clairaut's theorem, also known as the equality of mixed partials, provides that if a function \( F(x,y) \) is continuously differentiable in a neighborhood of a point, then the mixed partial derivatives at that point are equal. Mathematically, if the mixed partials:

• \( F_{1,2} = \frac{\partial^2 F}{\partial x \partial y} \)
• \( F_{2,1} = \frac{\partial^2 F}{\partial y \partial x} \)

exist and are continuous, then \( F_{1,2} = F_{2,1} \). This property simplifies the computation of derivatives since it assures us that it doesn't matter the order we differentiate. Clairaut's Theorem is extremely important in ensuring the consistency of partial derivative calculations by confirming the equality of these interchanged derivatives.

Rate of Change

Rate of change is a fundamental concept involving how one quantity varies in relation to another. In the context of multivariable functions, the rate of change is illustrated by partial derivatives. Partial derivatives measure the sensitivity of a function regarding changes in one variable alone. For example:

• \( F_1 = \frac{\partial F}{\partial x} \): measures how \( F \) changes as \( x \) changes, keeping \( y \) constant.
• \( F_2 = \frac{\partial F}{\partial y} \): measures how \( F \) changes as \( y \) changes, keeping \( x \) constant.

This concept is pivotal when analyzing systems where variables interact in complex ways, such as thermodynamics, economics, and dynamic systems where the understanding of how changes in inputs affect outcomes is crucial.

Continuously Differentiable Functions

Continuously differentiable functions are those functions for which the first and second (or higher) derivatives are continuous. This continuity is crucial in calculus because it ensures that the function behaves predictably and smoothly. Important features include:

• Smoothness, meaning no abrupt changes in function direction.
• The ability to apply theorems like Clairaut's, which rely on this property.
• Ensures validity for computations involving second-order derivatives since continuity guarantees equality of mixed partials.

Continuously differentiable functions are typically easy to work with, especially in applications involving optimization and modeling since they offer mathematical tractability along with practical applicability in expressing real-world systems simply.