Problem 19
Question
Determine \(d f\). $$ f(x, y, z)=z^{2} \sqrt{1+x^{2}+y^{2}} $$
Step-by-Step Solution
Verified Answer
The differential is \( df = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \, dx + \frac{z^2 y}{\sqrt{1 + x^2 + y^2}} \, dy + 2z \sqrt{1 + x^2 + y^2} \, dz \).
1Step 1: Identify Variables and Function
The function given is \( f(x, y, z) = z^2 \sqrt{1 + x^2 + y^2} \). The variables involved are \( x, y, \) and \( z \). We're tasked with finding the differential \( df \).
2Step 2: Calculate Partial Derivatives
To find \( df \), we need the partial derivatives of \( f \) with respect to \( x, y, \) and \( z \).- The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \).- The partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = \frac{z^2 y}{\sqrt{1 + x^2 + y^2}} \).- The partial derivative with respect to \( z \) is \( \frac{\partial f}{\partial z} = 2z \sqrt{1 + x^2 + y^2} \).
3Step 3: Write the Differential
The differential \( df \) can be expressed using the partial derivatives:\[ df = \frac{\partial f}{\partial x} \, dx + \frac{\partial f}{\partial y} \, dy + \frac{\partial f}{\partial z} \, dz \]Substituting in the partial derivatives, we have:\[ df = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \, dx + \frac{z^2 y}{\sqrt{1 + x^2 + y^2}} \, dy + 2z \sqrt{1 + x^2 + y^2} \, dz \]
Key Concepts
Partial DerivativesDifferential of a FunctionMultivariable Calculus
Partial Derivatives
In differential calculus, particularly when dealing with functions of multiple variables, partial derivatives are crucial. Imagine a function like the one given here, \( f(x, y, z) = z^2 \sqrt{1 + x^2 + y^2} \). This function depends on three variables: \( x, y, \) and \( z \). Partial derivatives allow us to understand how the function \( f(x, y, z) \) changes as each variable changes, while the others stay constant.
When you calculate the partial derivative with respect to a single variable, such as \( x \), you treat the other variables \( y \) and \( z \) as constants. For instance, the partial derivative of \( f \) with respect to \( x \) is \( \frac{\partial f}{\partial x} = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \). This tells us how \( f \) changes as \( x \) increases, ignoring changes in \( y \) and \( z \).
The same process is applied to find partial derivatives with respect to \( y \) and \( z \). These derivatives not only help in calculating the differential of the function but are also pivotal in optimization problems and analyzing critical points in multivariable calculus.
When you calculate the partial derivative with respect to a single variable, such as \( x \), you treat the other variables \( y \) and \( z \) as constants. For instance, the partial derivative of \( f \) with respect to \( x \) is \( \frac{\partial f}{\partial x} = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \). This tells us how \( f \) changes as \( x \) increases, ignoring changes in \( y \) and \( z \).
The same process is applied to find partial derivatives with respect to \( y \) and \( z \). These derivatives not only help in calculating the differential of the function but are also pivotal in optimization problems and analyzing critical points in multivariable calculus.
Differential of a Function
The differential of a function, \( df \), encapsulates the infinitesimal change of a function in response to small changes in its variables. For a multivariable function like \( f(x, y, z) \), this involves aggregating changes across each variable to give us a comprehensive view.
To find \( df \), we use the partial derivatives obtained in the first step. These partial derivatives give us the rate of change of \( f \) with respect to each individual variable. The formula for \( df \) is expressed as:
\[ df = \frac{\partial f}{\partial x} \, dx + \frac{\partial f}{\partial y} \, dy + \frac{\partial f}{\partial z} \, dz \]
Here, \( dx, dy, \) and \( dz \) represent small changes in \( x, y, \) and \( z \) respectively.
By substituting our specific partial derivatives into this equation, we construct the complete differential for the function:
\[ df = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \, dx + \frac{z^2 y}{\sqrt{1 + x^2 + y^2}} \, dy + 2z \sqrt{1 + x^2 + y^2} \, dz \]
This result allows us to estimate small changes in \( f \), enhancing our understanding of dynamic systems where multiple variables interact.
To find \( df \), we use the partial derivatives obtained in the first step. These partial derivatives give us the rate of change of \( f \) with respect to each individual variable. The formula for \( df \) is expressed as:
\[ df = \frac{\partial f}{\partial x} \, dx + \frac{\partial f}{\partial y} \, dy + \frac{\partial f}{\partial z} \, dz \]
Here, \( dx, dy, \) and \( dz \) represent small changes in \( x, y, \) and \( z \) respectively.
By substituting our specific partial derivatives into this equation, we construct the complete differential for the function:
\[ df = \frac{z^2 x}{\sqrt{1 + x^2 + y^2}} \, dx + \frac{z^2 y}{\sqrt{1 + x^2 + y^2}} \, dy + 2z \sqrt{1 + x^2 + y^2} \, dz \]
This result allows us to estimate small changes in \( f \), enhancing our understanding of dynamic systems where multiple variables interact.
Multivariable Calculus
Multivariable calculus is the extension of calculus to functions of several variables. It's used to study phenomena where change happens in multiple dimensions concurrently. The function \( f(x, y, z) = z^2 \sqrt{1 + x^2 + y^2} \) is a perfect exemplar of such scenarios.
In multivariable calculus, we often need to understand how a function behaves with respect to various variables jointly. This is where concepts like partial derivatives and differentials become extremely useful.
Multivariable calculus thus opens the doors to a deeper level of analysis, providing insights into dynamically complex systems that are more nuanced than single-variable calculus allows.
In multivariable calculus, we often need to understand how a function behaves with respect to various variables jointly. This is where concepts like partial derivatives and differentials become extremely useful.
- Partial Derivatives: These help us evaluate the influence of each individual variable on the function.
- Differential of a Function: This aids in providing a comprehensive picture by summing up influences from all variables.
Multivariable calculus thus opens the doors to a deeper level of analysis, providing insights into dynamically complex systems that are more nuanced than single-variable calculus allows.
Other exercises in this chapter
Problem 19
Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=e^{x y}
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Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y, z)=e^{x}+e^{y
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