Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the first partial derivatives of the function at the given point. \(f(x, y, z)=x^{2} y z^{3} ;(1,0,2)\)

Step-by-Step Solution

Answer

The first partial derivatives of the function \(f(x, y, z) = x^2 y z^3\) at the point \((1, 0, 2)\) are: \( \frac{\partial f}{\partial x}(1, 0, 2) = 0 \) \( \frac{\partial f}{\partial y}(1, 0, 2) = 0 \) \( \frac{\partial f}{\partial z}(1, 0, 2) = 0 \)

Step 1: Calculate the partial derivatives of the function with respect to x, y, and z.

To calculate the partial derivatives of the function, we differentiate it partially with respect to x, y, and z. For the partial derivative with respect to x, we treat y and z as constants, and for the partial derivative with respect to y, we treat x and z as constants. Similarly, for the partial derivative with respect to z, we treat x and y as constants. The partial derivatives are: \( \frac{\partial f}{\partial x} = 2xyz^3 \) \( \frac{\partial f}{\partial y} = x^2 z^3 \) \( \frac{\partial f}{\partial z} = 3x^2 y z^2 \)

Step 2: Evaluate the partial derivatives at the given point \((1, 0, 2)\)

Now, we will substitute the given point \((x, y, z) = (1, 0, 2)\) into the partial derivatives we calculated in Step 1: \( \frac{\partial f}{\partial x}(1, 0, 2) = 2(1)(0)(2^3) = 0 \) \( \frac{\partial f}{\partial y}(1, 0, 2) = (1)^2 (2^3) = 0 \) \( \frac{\partial f}{\partial z}(1, 0, 2) = 3(1)^2 (0)(2^2) = 0 \) So, the first partial derivatives of the function at the given point are: \( \frac{\partial f}{\partial x}(1, 0, 2) = 0 \) \( \frac{\partial f}{\partial y}(1, 0, 2) = 0 \) \( \frac{\partial f}{\partial z}(1, 0, 2) = 0 \)

Key Concepts

Multivariable Calculus

Exploring functions with several inputs is what multivariable calculus is all about. Unlike single-variable calculus, where functions have one input leading to one output, multivariable calculus deals with functions like the one in our exercise, where you have inputs x, y, and z, producing a single output. This results in a rich field of study because real-world phenomena often depend on multiple factors simultaneously.

In our case, functions are differentiated partially with respect to each input variable to assess how each factor individually influences the output. It's like investigating a real-life situation by changing one variable at a time while keeping others constant to see how it affects the outcome. For example, understanding how both humidity and temperature affect the condensation process together can be approached using multivariable calculus.

Derivative Evaluation

Evaluating a derivative is akin to finding a snapshot of how a function changes at a specific point. In multivariable functions, evaluating the partial derivatives is crucial for understanding the sensitivity of each variable at a specific point.

From the solution provided, the evaluation step involves plugging in the coordinates of the given point (1, 0, 2) into the partial derivative expressions. The result tells us that at the point (1, 0, 2), the function is not changing with respect to any of the variables x, y, and z, since all the partial derivatives are zero. This information is invaluable in scenarios such as optimizing processes or understanding stationary points in mathematical models.

Differentiation Techniques

To tackle multivariable functions, we extend the differentiation techniques from single-variable calculus. The partial derivative is computed similarly to the plain derivative, but with a twist—we fix all variables but one as constants. Following a systematic approach helps us understand how each variable influences the function independently.

Techniques such as the product rule and chain rule must be adapted for partial differentiation. In our exercise, the product rule would apply if we were differentiating a function where the variables were multiplied together, as in the case of the x, y, and z terms. Being proficient in these techniques ensures accurate and efficient evaluation of derivatives in complex scenarios.

Mathematical Analysis

The overarching field of mathematical analysis delves deep into the limits, continuity, and differentiability of functions. Regarding partial derivatives, analysis involves the rigorous examination of how a multivariable function behaves locally and globally.

Understanding the role of each variable within the context of a function such as our example, where the derivatives turned out to be zero at a point, resides under the umbrella of mathematical analysis. This kind of result often indicates a critical point, which may be a local maximum, minimum, or saddle point. Further analysis, such as employing the second derivative test, can help in classifying these critical points, defining a comprehensive understanding of the function's behavior throughout its domain.