Problem 33

Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=x+y \quad c=-1,0,2,4 $$

Step-by-Step Solution

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Answer

The level curves are lines with equations $y = -x - 1$ for $c = -1$, $y = -x$ for $c = 0$, $y = -x + 2$ for $c = 2$ and $y = -x + 4$ for $c = 4$.

1Step 1: Understand the problem

We are given the function $z = x + y$ and are asked to sketch the level curves for $c = -1, 0, 2, 4$. This means we need to find all the (x, y) pairs such that the equation $x + y = c$ is satisfied. Each of these pairs will produce a curve on the x-y plane, and these curves are our level curves.

2Step 2: Find the level curves

Let's substitute the given values of c into our function to attain the equation of the level curves. For $c = -1$, the equation becomes $x + y = -1$, or equivalently, $y = -x -1$. For $c = 0$, the equation becomes $x + y = 0$, or equivalently, $y = -x$. For $c = 2$, the equation becomes $x + y = 2$, or equivalently, $y = -x + 2$. Lastly, for $c = 4$, the equation becomes $x + y = 4$, or equivalently, $y = -x + 4$.

3Step 3: Sketch the level curves

Now let's sketch these equations. You can notice the equations are all straight lines with negative slope. The equation $y = -x -1$ will be a line crossing the y-axis at -1. $y = -x$ will be a line crossing the origin. $y = -x + 2$ will be a line crossing y-axis at 2, and $y = -x + 4$ a line crossing y-axis at 4. The level curves should be labeled with their respective $c$ values.

Key Concepts

Function sketchingLinear equationsCoordinate plane

Function sketching

When tackling function sketching, the goal is to visually represent a mathematical relationship. For the function given here, $ z = x + y $, sketching involves plotting level curves for specified $ c $ values. Each level curve represents a set of solutions where the function equals a constant $ c $. To sketch effectively:

Identify the type of function: In our case, it's a linear function due to the addition of $ x $ and $ y $ without any powers or products.
Substitute different $ c $ values: These are given as -1, 0, 2, and 4 in our task.
Plot the results on the coordinate plane, interpreting each equation as a line, because the resulting curves are linear.

Sketching helps visualize how the function behaves across different sets of values, offering insights into the relationships between $ x $, $ y $, and the output $ z $. Always label each line with its corresponding $ c $ for clarity.

Linear equations

Linear equations are foundational in mathematics. They consist of a constant term and the product of constants with variables, resulting in expressions like $ x + y = c $. Here, the equation is treated as a guideline for sketching level curves. Each equation:

Represents a straight line on a coordinate plane.
Has a slope and an intercept. In $ y = -x + c $, the slope is -1 (indicating a diagonal line sloping downwards), and $ c $ is the y-intercept.
Highlights simplicity in relationships. When changes in $ x $ to match changes in $ y $ are equal in magnitude but opposite in direction, the result is a direct, readable relation seen in straight lines.

Understanding these concepts smoothens the learning process when visualizing and sketching larger functions or more complex equations.

Coordinate plane

The coordinate plane is a central tool for graphing functions like $ z = x + y $. For this function, we represent solutions on a two-dimensional plane, where:

The horizontal axis (x-axis) and the vertical axis (y-axis) intersect at the origin (0,0).
Points on the plane correspond to pairs $(x, y)$ where a given equation holds true for specific values of $ c $.
Lines like $ y = -x - 1 $, $ y = -x $, $ y = -x + 2 $, and $ y = -x + 4 $ will visibly intercept the y-axis at defined points derived from $ c $.

The layout of the coordinate plane allows one to easily interpret these intercepts and examine how the corresponding values alter the positions of level curves. It’s a powerful way to see mathematics come to life in a spatial format.

Problem 33

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