Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=e^{x} \cos y\)

Step-by-Step Solution

Answer

\( \frac{\partial f}{\partial x} = e^{x} \cos y \), \( \frac{\partial f}{\partial y} = -e^{x} \sin y \).

Step 1: Identify the Function

We are given the function \( f(x, y) = e^{x} \cos y \). This function depends on two variables, \( x \) and \( y \). We are tasked with finding the partial derivatives with respect to each variable.

Step 2: Partial Derivative with Respect to x

To find the partial derivative of \( f \) with respect to \( x \), \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant. The derivative of \( e^{x} \) with respect to \( x \) is itself \( e^{x} \), and since \( \cos y \) is constant with respect to \( x \), it remains as a multiplier:\[ \frac{\partial f}{\partial x} = e^{x} \cos y \].

Step 3: Partial Derivative with Respect to y

To find the partial derivative of \( f \) with respect to \( y \), \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant. The derivative of \( \cos y \) with respect to \( y \) is \( -\sin y \), so:\[ \frac{\partial f}{\partial y} = e^{x} (-\sin y) = -e^{x} \sin y \].

Key Concepts

Multivariable Calculus

In multivariable calculus, we explore functions that depend on more than one variable. Unlike single-variable calculus, where each function has only one input, multivariable functions take multiple inputs. For example, the function \( f(x, y) = e^{x} \cos y \) relies on both \( x \) and \( y \). The field of multivariable calculus is vast, encompassing concepts such as partial derivatives, gradients, and multiple integrals. It plays a crucial role in various fields, including physics, engineering, and economics. By analyzing how changes in each input affect the output, we can make predictive models and solve complex problems. Understanding these functions allows us to work with surfaces, curves, and higher-dimensional spaces.

Functions of Two Variables

Functions of two variables are a special case in multivariable calculus where the input consists of two variables, typically denoted as \(x\) and \(y\). An example is \( f(x, y) = e^{x} \cos y \). In this scenario, each pair \((x, y)\) maps to a specific function value. This mapping can represent a surface in three-dimensional space.Such functions are visualized as surfaces because they have a third dimension represented by the function's value. Functions of two variables are ubiquitous in real-world applications:

• Weather patterns mapping data such as temperature and humidity.
• Economic models that use variables like price and demand.
• Engineering designs, for example, stress and strain calculations.

Working with these functions requires understanding how each variable impacts the outcome, often clarified through the calculation of partial derivatives.

Chain Rule

The chain rule is a fundamental differentiation technique in both single-variable and multivariable calculus. It is essential when functions are composed of other functions. In the context of multivariable calculus, the chain rule helps differentiate compositions of functions involving several variables. Suppose we have a multivariable function \( z = f(x, y) \) where \( x \) and \( y \) are themselves functions of another parameter, such as time \(t\) with \( x = g(t) \) and \( y = h(t) \). Applying the chain rule, the derivative of \( z \) with respect to \( t \) is:\[\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}\]This formula allows us to analyze how changes in a parameter affect a function's outcome, considering the interdependencies among the variables involved.

Differentiation Techniques

Differentiation techniques refer to a set of methods used to find the rate of change of a function. In the realm of multivariable calculus, these methods become more sophisticated due to the involvement of multiple variables.The basic differentiation tools include:

• Partial Derivatives: Derivatives of functions with respect to one variable at a time while treating other variables as constants. For instance, for \( f(x, y) = e^{x} \cos y \), partial derivatives with respect to \( x \) and \( y \) yield different expressions, highlighting the function's sensitivity to each variable.
• Gradient: A vector comprised of all partial derivatives of a function. It points in the direction of the greatest rate of increase of the function.
• Mesh Grids & Contour Plots: Used to visualize functions and their derivatives. These graphical tools help understand complex functions more intuitively.

With these techniques, we gain deeper insights into how functions behave and interact with varying inputs. Differentiation in multivariable calculus reveals the subtle interplays between variables and their influence on a function's behavior.