Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ f(x, y)=x^{2}+y^{2} \quad c=0,2,4,6,8 $$

Step-by-Step Solution

Answer

The level curves of the function \(f(x, y) = x^2 + y^2\) are circles centred at the origin (0,0), with the radius equal to the square root of the constant \(c\). As the constant \(c\) increases, the radius of the level curves also increases, representing the increasing values of the function as we move away from the origin in any direction.

Step 1: Set the function equal to a constant

First, we set the function equal to a constant, \(c\). This gives us the equation \(x^2 + y^2 = c\).

Step 2: Draw the level curves for c = 0, 2, 4, 6, 8

We then draw the level curves for \(c = 0, 2, 4, 6, 8\). These will be circles centred at the origin with radii of \(\sqrt{c}\). Here's what you'll have: \n- For \(c = 0\), the level curve is a single point at the origin (0,0).\n- For \(c = 2\), the level curve is a circle with radius \(\sqrt{2}\).\n- For \(c = 4\), the level curve is a circle with radius 2.\n- For \(c = 6\), the level curve is a circle with radius \(\sqrt{6}\).\n- For \(c = 8\), the level curve is a circle with radius \(\sqrt{8}\).\nAll circles are centred at the origin (0,0).

Step 3: Interpret the level curves

Finally, look at the plots and interpret. As \(c\) increases, the radius of the level curves increases as well. This means the values of the function \(x^2 + y^2\) increases as we move away from the origin in any direction.

Key Concepts

Function Graphing

To understand function graphing, especially for a function like \( f(x, y) = x^2 + y^2 \), it's essential to focus on the visual representation. The goal of graphing a function is to illustrate how input values relate to output results. In the case of level curves, which are also known as contour lines: \- Each curve represents all the points where the function has the same output value. \- For the function \( x^2 + y^2 = c \), each "level curve" is drawn at various constant values of \( c \), showing all the points that satisfy the equation. Function graphing involves plotting these level curves on a coordinate plane. The curves help to visualize how the function behaves over the entire plane, offering insights into its properties. The greater the constant \( c \), the larger the circle, illustrating increasing values of the function from the central point (the origin). This visual tool is especially handy in multivariable calculus, where interpreting the relationships between multiple variables and their combined outputs can become complex without the aid of graphical representation.

Circle Equations

Circle equations are fundamental when dealing with the level curves of the function \( f(x, y) = x^2 + y^2 \). A circle equation in its standard form is \( (x - h)^2 + (y - k)^2 = r^2 \), where \- \((h, k)\) represents the center of the circle, and \- \(r\) is the radius. For the given function, since it simplifies to \( x^2 + y^2 = c \), it describes circles centered at the origin \((0,0)\). The variable \( c \) determines the radius of these circles through \( r = \sqrt{c} \). So for each specified value of \( c \) such as 0, 2, 4, 6, and 8: \- These represent different radii \((\sqrt{0}, \sqrt{2}, 2, \sqrt{6}, \sqrt{8})\). Understanding circle equations in this context helps offer insights into how alterations in the function's constants affect its graphical depiction. The circles depict how the variable inputs relate in a multi-dimensional space and are an example of geometric representation in mathematics.

Multivariable Calculus

Multivariable calculus is a branch of mathematics that deals with functions of two or more variables. In this context, the function \( f(x, y) = x^2 + y^2 \) involves two variables, \( x \) and \( y \), producing a scalar output. Here are some key points: \- Level curves are fundamental tools in multivariable calculus as they help to visualize functions of several variables. \- They provide a way to see how a function's value changes without needing a 3D graph. In the exercise, when we took different \( c \) values, we essentially sketched cross-sections of the function’s surface at different heights or levels. This is crucial for understanding how transformations affect outputs as you alter one or more inputs while holding others constant. Overall, multivariable calculus extends the ideas of single-variable calculus to higher dimensions. It allows us to study things like gradients, directional derivatives, and integrals over areas that involve more than just a simple line.