Problem 2

Question

Sketch some of the level curves associated with the given function. $$ f(x, y)=y^{2}-x $$

Step-by-Step Solution

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Answer

The level curves are parabolas: shifted horizontally as $ c $ changes.

1Step 1: Understand the Function and Level Curves

The function given is $ f(x, y) = y^2 - x $. The level curves of a function are the curves for which the function takes a constant value, say $ c $. This means we set $ f(x, y) = c $, which gives us an equation in terms of $ x $ and $ y $.

2Step 2: Set Up the Level Curve Equation

Set $ f(x, y) = c $. We then have $ y^2 - x = c $. Rearrange this to express $ x $ in terms of $ y $:\[x = y^2 - c.\]

3Step 3: Identify Shapes of the Curves

The equation $ x = y^2 - c $ is a parabola opening horizontally (along the x-axis). The vertex of each parabola is at the point $ (c, 0) $. As $ c $ changes, the position of the parabola changes along the x-axis.

4Step 4: Choose Sample Values for $ c $

Select a few values for $ c $ to sketch the level curves. Commonly used values are $ c = -1, 0, 1 $.

5Step 5: Sketch the Curves

For $ c = -1 $, the level curve is $ x = y^2 + 1 $, a parabola shifted 1 unit to the right.For $ c = 0 $, the level curve is $ x = y^2 $, a parabola with the vertex at the origin.For $ c = 1 $, the level curve is $ x = y^2 - 1 $, a parabola shifted 1 unit to the left.Draw these curves on a coordinate plane.

Key Concepts

Understanding Parabolas

A parabola is a U-shaped curve that can open either vertically or horizontally. In the context of the function given, the parabolas open horizontally. This can be less intuitive because many are familiar with parabolas that open upwards or downwards, like the basic function $ y = x^2 $. In this exercise, the equation $ x = y^2 - c $ results in parabolas opening horizontally across the x-axis. Key Characteristics:

Direction: In the equation $ x = y^2 - c $, the parabolas open left and right because the squared term (\

Problem 2

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