Problem 16 - Calculus with Concepts in Calculus

Question

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=x^{2}-y ; c=-2,2\)

Step-by-Step Solution

Answer

Sketch parabolas: \(y = x^2 + 2\) and \(y = x^2 - 2\) for the level curves.

Step 1: Understand the function and level curve

The function given is a two-variable function, \(f(x, y) = x^2 - y\). A level curve occurs when the function equals a constant value \(c\). Here, we have two level curves at \(c = -2\) and \(c = 2\).

Step 2: Set equations for level curves

For \(c = -2\), set the equation \(x^2 - y = -2\). Similarly, for \(c = 2\), set \(x^2 - y = 2\). This gives us two equations: \(y = x^2 + 2\) and \(y = x^2 - 2\). These are the equations for the level curves.

Step 3: Sketch the level curve for \(c = -2\)

Rewrite \(y = x^2 + 2\), which represents a parabola opening upwards, shifted 2 units up from the standard parabola \(y = x^2\). Draw this curve on the coordinate plane.

Step 4: Sketch the level curve for \(c = 2\)

Rewrite \(y = x^2 - 2\), which is a parabola opening upwards and shifted 2 units down from \(y = x^2\). Draw this curve on the same coordinate plane.

Step 5: Compare and finalize the sketch

Ensure the parabolas are plotted accurately, keeping consistent distance from the origin according to their shifts. Verify both parabolas open upwards but are vertically shifted relative to \(y = x^2\).

Key Concepts

Multivariable Functions

Multivariable functions are an extension of simple functions that involve two or more variables instead of just one. In this context, our given function is \( f(x, y) = x^2 - y \).
This function evaluates to different outcomes depending on the values plugged in for \(x\) and \(y\).

• The level curve of a multivariable function is set by equating the function to a constant, \(c\). This means we're looking for all the combinations of \(x\) and \(y\) that satisfy the equation \(f(x, y) = c\).
• Level curves help in visualizing functions with two variables in a 2D plane.

Use these to understand how changes in variables affect the function's output.
In the exercise at hand, we explored level curves for \(f(x, y)\) at two different constant values, \(c = -2\) and \(c = 2\). This means we filled in parts of the space on the 2D graph where our function remained constant.

Parabolas

In coordinate geometry, parabolas are common shapes defined by quadratic functions. Each level curve given in the exercise, \(y = x^2 + 2\) and \(y = x^2 - 2\), represents a parabola.

• The generic form of a parabola is \( y = ax^2 + bx + c \). If \(a=0\), the curve is a straight line, but as long as \(a\) isn't zero, we have a parabola.
• The parabola \(y = x^2 + 2\) is derived from \(x^2 - y = -2\). By rearranging the terms, we observe a neat parabola opening upwards, just 2 units above the standard \(y=x^2\).
• Similarly, \(y = x^2 - 2\) is shifted 2 units downwards from the origin.

Understanding these shifts is essential to sketch a precise graph, as it depicts changes in the parabola's position about the origin without affecting its shape.

Coordinate Geometry

Coordinate geometry involves plotting functions on a coordinate plane. Here, we plot the equations of the parabolas. The plane consists of two axes:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

For accurate plotting:

• Recognize points where the curve intersects the axes.
• For \(y = x^2 + 2\), the vertex is at (0, 2), indicating an upward opening starting from y = 2.
• For \(y = x^2 - 2\), the vertex is at (0, -2), indicating a downward shift.

This understanding will guide your sketch of the level curves, allowing you to visualize them on the coordinate plane accurately.