Problem 18

Question

Sketch the level curve $f(x, y)=c$. $f(x, y)=2 y-\cos x ; c=0,1,2$

Step-by-Step Solution

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Answer

Level curves are wave-like: for $ c=0 $ at $ y = \frac{1}{2} \cos x $; $ c=1 $ at $ y = \frac{1}{2} (1 + \cos x) $; $ c=2 $ at $ y = 1 + \frac{1}{2} \cos x $.

1Step 1: Identify the Function and Level Curves

The function given is $ f(x, y) = 2y - \cos x $. To sketch level curves, we need to set $ f(x, y) = c $, where $ c = 0, 1, 2 $.

2Step 2: Write Equation for Level Curve with c=0

Set $ f(x, y) = 0 $: \[ 2y - \cos x = 0 \]. Solve for $ y $: \[ 2y = \cos x \Rightarrow y = \frac{1}{2} \cos x \]. This is the equation of the level curve when $ c = 0 $.

3Step 3: Write Equation for Level Curve with c=1

Set $ f(x, y) = 1 $: \[ 2y - \cos x = 1 \]. Solve for $ y $: \[ 2y = 1 + \cos x \Rightarrow y = \frac{1}{2} (1 + \cos x) \]. This equation represents the level curve for $ c = 1 $.

4Step 4: Write Equation for Level Curve with c=2

Set $ f(x, y) = 2 $: \[ 2y - \cos x = 2 \]. Solve for $ y $: \[ 2y = 2 + \cos x \Rightarrow y = 1 + \frac{1}{2} \cos x \]. This equation is the level curve for $ c = 2 $.

5Step 5: Sketch the Level Curves

For each $ c $, you have linear trigonometric functions in terms of $ x $. Sketch these curves on the $ xy $-plane: - For $ c = 0 $, sketch $ y = \frac{1}{2} \cos x $.- For $ c = 1 $, sketch $ y = \frac{1}{2} (1 + \cos x) $.- For $ c = 2 $, sketch $ y = 1 + \frac{1}{2} \cos x $. These curves will be wave-like due to the $ \cos x $ term, vertically translated as per $ c $.

Key Concepts

Trigonometric FunctionsCosine FunctionXY-PlaneLinear Equations

Trigonometric Functions

Trigonometric functions are essential tools in mathematics that help to relate the angles and sides of triangles. One of these functions, the cosine, is particularly important due to its properties and applications in various mathematical fields.
Trigonometric functions such as sine, cosine, and tangent can describe waveforms and periodic phenomena. In our exercise, we use the cosine function within the expression for level curves. This means we'll be describing periodic, wave-like motions across a plane.
These functions are also utilized to solve problems involving oscillations and repetitive patterns.

They help in calculating distances and angles in circular motion.
Trigonometric functions are also used in physics for wave equations and sound modeling.

Understanding these functions is foundational to solving problems involving periodic and oscillatory systems.

Cosine Function

The cosine function, denoted as $ \cos x $, is a trigonometric function that derives from the adjacent side over the hypotenuse in a right triangle. It exhibits a repeating wave-like pattern called a "cosine wave".
This function has a fundamental period of $2\pi$, which means it repeats every $2\pi$ units. The range is between -1 and 1, and it plays a huge role in defining our level curves since it modifies the linear equation output.

The cosine function starts at its maximum value, descending to minimum and then ascending back. This results in its distinctive wave pattern.
In real-world contexts, it can be used to model things like daylight hours or sound waves.

In our given problem, the cosine function $ \cos x $ alters the $x$-values of our curves, which are periodically influenced by this function.

XY-Plane

The XY-plane is a flat, two-dimensional surface that spans across all possible values of $x$ and $y$. It is crucial for plotting functions, graphs, and especially level curves.
On this plane, every point is denoted by a pair of numbers $ (x, y) $, which represent its position. The horizontal axis is the $x$-axis, while the vertical axis is the $y$-axis.

In our exercise, the XY-plane is used to display level curves for different constants, $c=0, 1, 2$.
Each level curve is basically a line that shows where the function output equals a constant value.

Understanding the XY-plane is necessary as it serves as the canvas where graphical representations in problems like this come to life. Sketched functions or curves can show relationships between variables.

Linear Equations

Linear equations form the backbone of mathematical modeling. A linear equation typically has the form $ ax + by = c $, where $a$, $b$, and $c$ are constants.
These equations describe straight lines in a two-dimensional space, and in the context of our exercise, they are altered by trigonometric components to become "linear trigonometric" functions. In our problem, each level curve after setting the variable $c$ becomes a linear equation in terms of $y$, affected by the cosine function.

Linear equations are used in problems requiring modeling of constant rates of change.
They are instrumental in predicting values and finding intersections of lines.

The beauty of linear equations is their simplicity and power in defining relationships between quantities. Understanding how to manipulate them is foundational to solving more complex problems.

Problem 18

Problem 19

Other exercises in this chapter

Problem 18

Find the first partial derivatives of the function. $$ w=\left(\frac{x}{y}\right)^{z} $$

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Problem 18

Determine whether $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} \sin x}{x^{2}+y^{2}}$ exists.

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Problem 19

$$ \begin{aligned} &\text { Let } x, y, \text { and } z \text { denote the angles of an arbitrary triangle. }\\\ &\text { Find the maximum value of } \sin x \si

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Problem 19

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=e^{x y}

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