Problem 1 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r \cos \theta=4\)

Step-by-Step Solution

Answer

The graph is a vertical line at \(x = 4\).

Step 1: Understand the Equation

The given equation is in polar coordinates: \(r \cos \theta = 4\). This needs to be converted to Cartesian coordinates to sketch the graph more easily.

Step 2: Convert Polar to Cartesian Coordinates

Recall the conversion formulas: \(r \cos \theta = x\) and \(r \sin \theta = y\). Substitute \(r \cos \theta\) with \(x\) in the given equation, which results in \(x = 4\).

Step 3: Identify the Graph Type

Identify that the equation \(x = 4\) represents a vertical line in the Cartesian coordinate system passing through \(x = 4\) on the x-axis.

Step 4: Sketch the Graph

Draw the Cartesian coordinate system. Sketch a vertical line passing through the point \(x = 4\) on the x-axis. This line extends infinitely in the positive and negative y-direction.

Step 5: Label the Graph

Label the vertical line with the equation \(x = 4\), indicating that it represents the line where all points have their x-coordinate equal to 4.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of using horizontal and vertical distances like Cartesian coordinates, polar coordinates describe a point by how far it is from the origin (radius, denoted as \( r \)) and the angle (\( \theta \)) between the positive x-axis and the line from the origin to the point.

• \( r \) (radius): The distance from the origin to the point.
• \( \theta \) (theta): The counterclockwise angle from the positive x-axis to the line connecting the origin to the point.

For example, if \( r = 5 \) and \( \theta = \frac{\pi}{4} \), you have a point that is 5 units away from the origin at a 45-degree angle from the positive x-axis. This system is particularly useful when dealing with periodic functions and circular or spiral structures.

Cartesian Coordinates

Cartesian coordinates describe a point in a plane by its horizontal (\( x \)) and vertical (\( y \)) distances from the origin. Each point is represented as an ordered pair (\( x, y \)).

• \( x \): The horizontal distance from the origin.
• \( y \): The vertical distance from the origin.

The Cartesian coordinate system is formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). This system is excellent for representing linear functions and rectangular shapes. For example, the point (4, 3) means you move 4 units to the right (positive x direction) and 3 units up (positive y direction) from the origin. Converting from polar to Cartesian coordinates can be done using these formulas:

\[ x = r \cos\theta \] \[ y = r \sin\theta \] In the given exercise, we used this conversion to change the polar equation \( r \cos \theta = 4 \) to the Cartesian form \( x = 4 \).

Graphing Equations

Graphing equations is the process of drawing the curve or line that represents all the points satisfying an equation. When we graph an equation:

• Identify the type of equation in Cartesian or polar form.
• If necessary, convert the equation to the form that makes graphing easier. For example, convert polar to Cartesian coordinates if needed.
• Sketch the coordinate system, making sure to label the axes.

For the given exercise (\( r \cos \theta = 4 \)), we identified the equation as a vertical line \( x = 4 \) in Cartesian coordinates. To graph this:

• Draw the x-axis and y-axis.
• Locate the point where \( x = 4 \) on the x-axis.
• Draw a vertical line passing through this point. This line extends both upwards and downwards indefinitely through all y-values.

Graphing helps visualize how equations behave and their relationships between variables.