Problem 43
Question
Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=y / 2 ; c=0,1,2\)
Step-by-Step Solution
Verified Answer
The level curves are horizontal lines at \(y = 0\), \(y = 2\), and \(y = 4\).
1Step 1: Understand the function
The function given is \( f(x, y) = \frac{y}{2} \). This is a function of two variables, \(x\) and \(y\). The equation for the level curves is obtained by setting this function equal to a constant \(c\).
2Step 2: Set up the level curve equation
To find the level curves, set \( f(x, y) = c \), which gives us the equation \( \frac{y}{2} = c \). Multiply both sides by 2 to get \( y = 2c \).
3Step 3: Determine equations for specific values of c
Substitute the given values of \(c\) (0, 1, and 2) into \(y = 2c\):- For \(c = 0\), we get the equation \(y = 0\).- For \(c = 1\), we get the equation \(y = 2\).- For \(c = 2\), we get the equation \(y = 4\).
4Step 4: Sketch the level curves
The level curves are horizontal lines on the Cartesian plane because the equations are of the form \(y = k\):- The line \(y = 0\) is the x-axis.- The line \(y = 2\) is a horizontal line parallel to the x-axis, 2 units above it.- The line \(y = 4\) is a horizontal line parallel to the x-axis, 4 units above it.
Key Concepts
Functions of Two VariablesCartesian PlaneEquation of a Line
Functions of Two Variables
Functions of two variables are mathematical expressions that involve two independent variables, typically denoted as \(x\) and \(y\). In these functions, the value of the dependent variable is determined by the combination of these two variables.
For instance, the function \(f(x, y) = \frac{y}{2}\) indicates that the output value is solely dependent on the variable \(y\), divided by 2, showing that \(x\) plays no role here. This type of function is often used to model scenarios where two inputs affect an outcome, such as temperature variations over a landscape. Common features of functions of two variables include:
For instance, the function \(f(x, y) = \frac{y}{2}\) indicates that the output value is solely dependent on the variable \(y\), divided by 2, showing that \(x\) plays no role here. This type of function is often used to model scenarios where two inputs affect an outcome, such as temperature variations over a landscape. Common features of functions of two variables include:
- Two inputs: \(x\) and \(y\).
- Output values that can vary independently based on the inputs.
- Used to describe surfaces or planes in a three-dimensional space.
Cartesian Plane
The Cartesian plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It is commonly used to represent points, lines, and curves in mathematics and other applications.
This plane allows us to easily visualize relationships between two variables in a function, which is particularly useful for plotting level curves.
In the context of the given function \(f(x, y) = \frac{y}{2}\), we can use the Cartesian plane to draw the level curves by plotting the lines where the function equals specific constants \(c\). Key aspects of the Cartesian plane include:
This plane allows us to easily visualize relationships between two variables in a function, which is particularly useful for plotting level curves.
In the context of the given function \(f(x, y) = \frac{y}{2}\), we can use the Cartesian plane to draw the level curves by plotting the lines where the function equals specific constants \(c\). Key aspects of the Cartesian plane include:
- Intersection of x-axis and y-axis at the origin \((0, 0)\).
- Grid structure that simplifies plotting of points and lines.
- Quadrants that indicate the signs of coordinates: positive or negative.
Equation of a Line
The equation of a line in a two-dimensional plane is a linear equation that represents all points through which the line passes. For level curves of a function with two variables, these equations often result in straight horizontal or vertical lines.
The general form of a line's equation is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In the exercise above, the level curves for \(f(x, y) = \frac{y}{2}\) are horizontal lines because they can be expressed as \(y = k\) for constants \(k\). This shows that the output of the function is the same across all values of \(x\), which explains why the lines are horizontal. Important concepts related to the equation of a line include:
The general form of a line's equation is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In the exercise above, the level curves for \(f(x, y) = \frac{y}{2}\) are horizontal lines because they can be expressed as \(y = k\) for constants \(k\). This shows that the output of the function is the same across all values of \(x\), which explains why the lines are horizontal. Important concepts related to the equation of a line include:
- Horizontal lines have a zero slope \(m = 0\).
- Vertical lines have an undefined slope.
- Slope indicates the steepness and direction of a line.
- Y-intercept \(b\) is where the line crosses the y-axis.
Other exercises in this chapter
Problem 42
Find a linear approximation to each func\mathrm{tion } \(f(x, y)\) at the indicated point. \(\mathbf{f}(x, y)=\left[\begin{array}{c}(x+y)^{2} \\ x y\end{array}\
View solution Problem 42
Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=x y^{2} ; x^{2}-y=0 $$
View solution Problem 43
Find a linear approximation to $$ \mathbf{f}(x, y)=\left[\begin{array}{l} x^{2}-x y \\ 3 y^{2}-1 \end{array}\right] $$ at \((1,2)\). Use your result to find an
View solution Problem 43
Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=x^{2} y ; x^{2}+3 y=1 $$
View solution