Problem 24 - Calculus an Applied Approach

Question

Evaluate \(f_{x}\) and \(f_{y}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=e^{x} y^{2} \quad(0,2) $$

Step-by-Step Solution

Answer

The evaluated partial derivatives \(f_{x}(0,2)\) and \(f_{y}(0,2)\) at the point (0,2) are both equal to 4.

Step 1: Compute the partial derivative \(f_{x}\)

The partial derivative of a function with respect to a variable treats all other variables as constants. In this case, when differentiating \(f(x, y)=e^{x} y^{2}\) with respect to x, the term \(y^{2}\) is treated as a constant. Applying this rule, the derivative of \(e^{x}\) with respect to x, is \(e^{x}\). So, the partial derivative \(f_{x}\) is \(f_{x} = e^{x} y^{2}\).

Step 2: Compute the partial derivative \(f_{y}\)

The partial derivative of a function with respect to a variable y treats all other variables as constants. So, when we differentiate \(f(x, y)=e^{x} y^{2}\) with respect to y, the term \(e^{x}\) is treated as a constant. This gives \(f_{y} = 2ye^{x}\).

Step 3: Evaluate the partial derivatives at the point (0,2)

Now, evaluate \(f_{x}\) and \(f_{y}\) at the point (0,2). Substituting x=0 and y=2 into \(f_{x}\), we find \(f_{x}(0,2) = e^{0} 2^{2} = 4\). Similarly, substituting into \(f_{y}\), we find \(f_{y}(0,2) = 2*2*e^{0} = 4\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It involves the study of derivatives, integrals, limits, and infinite series. The fundamental concepts of calculus are differentiation and integration, with differentiation focusing on the rate at which something changes, while integration deals with the total size or value, such as area under a curve. Calculus is a powerful tool in science and engineering, used to model and analyze dynamic systems.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of several variables. It involves the study of partial derivatives, multiple integrals, and the various applications in higher dimensions. The idea of a partial derivative, an essential aspect of multivariable calculus, represents the rate of change of a function with respect to one variable, while holding other variables constant.

Understanding and visualizing functions in multivariable calculus can be challenging as it requires thinking in higher dimensions. However, the applications of multivariable calculus are vast, ranging from predicting weather patterns to optimizing complex systems in economics and engineering.

Derivative Evaluation

Derivative evaluation is the process of finding the rate at which a function changes at any given point. In single-variable calculus, this involves taking the derivative of a function and plugging in a value for the variable. In multivariable calculus, we find partial derivatives, evaluating them at a specified point, much like in the given example where we computed and evaluated the partial derivatives of the function f(x, y) = e^{x}y^{2} at the point (0,2).

Being able to accurately evaluate derivatives is crucial to understanding the behavior of functions. It allows us to determine the slope of tangent lines, calculate instantaneous rates of change, and solve problems involving optimization and motion.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^{x}, where a is a positive constant called the base, and x is the exponent. These functions are characterized by their rapid growth or decay and have important applications in many areas such as compound interest, population growth, and radioactive decay.

The natural exponential function, where a is e (Euler's number, approximately 2.71828), is particularly important due to its unique properties in calculus, such as the derivative of e^{x} being equal to itself. Exponential functions appear frequently in both mathematical theory and real-world applications, making their understanding essential for students in calculus.