Problem 48
Question
A CAS can be used to calculate and graph partial derivatives. Draw the graphs of each of the following: (a) \(\sin \left(x+y^{2}\right)\) (b) \(D_{x} \sin \left(x+y^{2}\right)\) (c) \(D_{y} \sin \left(x+y^{2}\right)\) (d) \(D_{x}\left(D_{y} \sin \left(x+y^{2}\right)\right)\)
Step-by-Step Solution
Verified Answer
Draw the graphs of \( \sin(x + y^2) \), \( \cos(x + y^2) \), \( 2y \cos(x + y^2) \), and \( -2y \sin(x + y^2) \) using a CAS.
1Step 1: Understand the Problem
The task involves drawing graphs of a function and its partial derivatives with respect to different variables. We need to plot the function \( f(x, y) = \sin(x + y^2) \) along with its derivatives using a Computer Algebra System (CAS).
2Step 2: Define the Function
Consider the function \( f(x, y) = \sin(x + y^2) \). This function depends on two variables, \( x \) and \( y \), where the argument of the sine function is the sum of \( x \) and \( y^2 \).
3Step 3: Compute the Partial Derivative with respect to x
To find \( D_x \sin(x + y^2) \), we apply the chain rule. The derivative is given by \( D_x [\sin(u)] = \cos(u) \cdot D_x[u] \), where \( u = x + y^2 \). Thus, \( D_x \sin(x + y^2) = \cos(x + y^2) \).
4Step 4: Compute the Partial Derivative with respect to y
Compute \( D_y \sin(x + y^2) \). Here, apply the chain rule: \( D_y [\sin(u)] = \cos(u) \cdot D_y[u] \), where \( u = x + y^2 \). Then \( D_y[x + y^2] = 2y \) so \( D_y \sin(x + y^2) = 2y \cos(x + y^2) \).
5Step 5: Compute the Second Partial Derivative \( D_{xy} \)
To find \( D_x(D_y \sin(x + y^2)) \), we start with \( D_y \sin(x + y^2) = 2y \cos(x + y^2) \). Now differentiate with respect to \( x \): \( D_x(2y \cos(x + y^2)) = 2y (-\sin(x + y^2)) \) because the derivative of \( \cos(u) \) is \( -\sin(u) \). Thus, \( D_x(D_y \sin(x + y^2)) = -2y \sin(x + y^2) \).
6Step 6: Use CAS to Plot the Functions
Use a CAS to plot \( \sin(x + y^2) \), \( \cos(x + y^2) \), \( 2y \cos(x + y^2) \), and \( -2y \sin(x + y^2) \). Each graph corresponds to parts (a), (b), (c), and (d) of the problem.
Key Concepts
Partial DerivativesChain RuleGraphing FunctionsComputer Algebra System
Partial Derivatives
Partial derivatives help us find how a function changes when we alter just one variable, keeping others constant. If we have a function of two variables, like \( f(x, y) = \sin(x + y^2) \), we can explore how it changes with respect to \( x \) or \( y \).
For example, the partial derivative of \( f \) with respect to \( x \), written as \( D_x \sin(x + y^2) \), shows how \( f \) reacts to changes in \( x \) while \( y \) stays fixed.
Similarly, finding the partial derivative with respect to \( y \), \( D_y \sin(x + y^2) \), can indicate how sensitive the function is to changes in \( y \) alone. Partial derivatives are essential in many areas like optimization, where we need to understand gradients and slopes.
For example, the partial derivative of \( f \) with respect to \( x \), written as \( D_x \sin(x + y^2) \), shows how \( f \) reacts to changes in \( x \) while \( y \) stays fixed.
Similarly, finding the partial derivative with respect to \( y \), \( D_y \sin(x + y^2) \), can indicate how sensitive the function is to changes in \( y \) alone. Partial derivatives are essential in many areas like optimization, where we need to understand gradients and slopes.
Chain Rule
The chain rule in calculus is a powerful tool for finding derivatives of composite functions. It helps us tackle functions involving nested operations, like \( \sin(x + y^2) \).
When we compute partial derivatives using the chain rule, we differentiate the outer function and multiply it by the derivative of the inner function. For instance, to find \( D_x \sin(x + y^2) \), we identify the outer function as \( \sin(u) \), where \( u = x + y^2 \), and the derivative of \( \sin(u) \) is \( \cos(u) \).
This means \( D_x \sin(x + y^2) = \cos(x + y^2) \cdot D_x[x + y^2] \), where \( D_x[x + y^2] = 1 \). The chain rule is especially useful for functions with layers of operations.
When we compute partial derivatives using the chain rule, we differentiate the outer function and multiply it by the derivative of the inner function. For instance, to find \( D_x \sin(x + y^2) \), we identify the outer function as \( \sin(u) \), where \( u = x + y^2 \), and the derivative of \( \sin(u) \) is \( \cos(u) \).
This means \( D_x \sin(x + y^2) = \cos(x + y^2) \cdot D_x[x + y^2] \), where \( D_x[x + y^2] = 1 \). The chain rule is especially useful for functions with layers of operations.
Graphing Functions
Graphing a function is a visual way to understand its behavior across different values of its variables. For multi-variable functions like \( \sin(x + y^2) \), plotting its surface can offer insights into how values of \( x \) and \( y \) affect changes in the function's output.
A graph will show hills, valleys, and changes that inform us about the function's gradient and curvature. For partial derivatives, such as \( D_x \sin(x + y^2) \) and \( D_y \sin(x + y^2) \), their graphs highlight the function’s rate of change along specific directions. By examining these plots, we see how each variable contributes separately to the function’s total variation.
A graph will show hills, valleys, and changes that inform us about the function's gradient and curvature. For partial derivatives, such as \( D_x \sin(x + y^2) \) and \( D_y \sin(x + y^2) \), their graphs highlight the function’s rate of change along specific directions. By examining these plots, we see how each variable contributes separately to the function’s total variation.
Computer Algebra System
A Computer Algebra System (CAS) is like having a mathematical assistant who never gets tired. It helps perform symbolic calculations, including derivatives and plotting functions quickly.
A CAS allows us to compute things like \( D_x \sin(x + y^2) \) without manual errors. It can handle tedious calculations and visualize complex graphs in seconds. This is useful for our example because we can easily chart \( \sin(x + y^2) \) and its derivatives. With a CAS, students and scientists can focus on higher-level mathematics without getting bogged down by arithmetic.
Overall, a CAS is an invaluable tool in education and research that enhances our ability to explore and understand functions and their behaviors.
A CAS allows us to compute things like \( D_x \sin(x + y^2) \) without manual errors. It can handle tedious calculations and visualize complex graphs in seconds. This is useful for our example because we can easily chart \( \sin(x + y^2) \) and its derivatives. With a CAS, students and scientists can focus on higher-level mathematics without getting bogged down by arithmetic.
Overall, a CAS is an invaluable tool in education and research that enhances our ability to explore and understand functions and their behaviors.
Other exercises in this chapter
Problem 48
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Give definitions in terms of limits for the following partial derivatives: (a) \(f_{y}(x, y, z)\) (b) \(f_{z}(x, y, z)\) (c) \(G_{x}(w, x, y, z)\) (d) \(\frac{\
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