Problem 13 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=e^{\theta}\) (logarithmic spiral)

Step-by-Step Solution

Answer

Plot points for different \(\theta\) values, connect them smoothly to form a spiral, and label the graph.

Step 1: Understand the Equation

The given equation is a polar equation in the form of a logarithmic spiral, given by \(r = e^{\theta}\). In this equation, \(r\) represents the radius and \(\theta\) represents the angle.

Step 2: Calculate Key Points

To sketch the graph, calculate the values of \(r\) for different values of \(\theta\). For instance, when \(\theta = 0\), \(r = e^0 = 1\). When \(\theta = \frac{\pi}{4}\), \(r = e^{\frac{\pi}{4}}\), and so on. Make a table of \(\theta\) and corresponding values of \(r\).

Step 3: Plot Key Points

Using the values computed in the previous step, plot the key points on the polar coordinate system. Start from \(\theta = 0\) and keep plotting points for increasing \(\theta\).

Step 4: Draw the Spiral

Connect the plotted points smoothly to form the spiral. The curve starts at \(r = 1\) when \(\theta = 0\) and grows exponentially as \(\theta\) increases.

Step 5: Label the Graph

Make sure to label the graph with the equation \(r = e^{\theta}\) and mark key points for clarity.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in the plane using a radius and an angle instead of the usual x and y coordinates (Cartesian coordinates). In polar coordinates, each point is determined by:

• Radius (r): The distance from the origin to the point.
• Angle (θ): The counterclockwise angle from the positive x-axis to the point.

Think of it like describing the position of a point using 'how far' and 'which direction' instead of 'over and up'. In the equation given in the exercise, r = e^θ, the radius r changes depending on the angle θ.
As θ increases, you move around the origin in a counterclockwise direction, and as it changes, the distance from the origin changes too.

Exponential Function

The exponential function is an important mathematical function characterized by the constant e. It's written as e^x, where e is approximately 2.71828. In the context of the given exercise, the exponential function appears as e^θ.
The key feature of the exponential function is:

• It grows rapidly as the input (θ) increases.
• When the input is zero (θ = 0), the exponential function equals 1 (i.e., e^0 = 1).
• As θ increases, r = e^θ starts at 1 and increases exponentially.

This means that for small increases in θ, the radius r increases very quickly, creating the spiral effect in the polar coordinate system.

Graph Sketching

Graph sketching is the process of drawing a curve on a coordinate system based on a given function. For the equation r = e^θ in polar coordinates, follow these steps:

1. Calculate key points by substituting different values of θ into the equation to find the corresponding r. For example:
   - When θ = 0, r = 1.
   - When θ = π/4, r = e^{π/4}.
   - When θ = π/2, r = e^{π/2}.
2. Plot the key points on a polar graph. Keep in mind each point is represented by a pair (r, θ).


3. Connect these points smoothly to visualize the graph. You will notice the logarithmic spiral shape forming, starting close to the origin and spiraling outwards exponentially.


4. Label your graph clearly with the equation r = e^θ and mark some key points to help visualize its growth.

This way, you can accurately sketch the logarithmic spiral described by the polar equation.