Let’s review each step for mathematical correctness:

Step 1

• Incorrect. The expression provided, $f(\theta) = 2\cos(\theta) + \frac{1}{2}\sin(2\theta) - 1$, is not derived correctly. The goal is to find how many times the polar equation $r = 2 + \sin(\theta)$ intersects the vertical line $x = 1$ in Cartesian coordinates. In polar coordinates, $x = r\cos(\theta)$, so we need to set $r\cos(\theta) = 1$. The expression in this step does not reflect that approach, and the form of $f(\theta)$ seems arbitrary and unrelated to solving for $x = 1$.

Step 2

• Correct, but vague. This step suggests an alternative approach to analyze the problem geometrically by plotting the polar equation $r = 2 + \sin(\theta)$ and observing intersections with $x = 1$. While this is conceptually valid, it lacks mathematical rigor or specific details. It would be better to explicitly state that we are looking for where $r\cos(\theta) = 1$.

Step 3

• Correct. This step is accurate. The cardioid formed by $r = 2 + \sin(\theta)$ does indeed intersect the vertical line $x = 1$ twice. Geometrically, this is correct based on the cardioid’s shape and orientation, and the intersections with $x = 1$ can be observed.

Step 4

• Correct. The conclusion is accurate. After correctly analyzing the polar plot, we can conclude that the cardioid passes through $x = 1$ twice.

Summary:

• Step 1 is incorrect due to an erroneous expression for $f(\theta)$.
• Steps 2, 3, and 4 are correct in the geometric approach, though Step 2 could be improved by being more precise.

Rating:

• Step 1: Incorrect
• Step 2: Correct (but could be clearer)
• Step 3: Correct
• Step 4: Correct

Would you like to explore how to solve this problem algebraically or geometrically in more detail?

Here are five related questions:

1. How do you convert polar coordinates to Cartesian coordinates?
2. How do you derive the equation $r\cos(\theta) = 1$ for this problem?
3. Can other types of polar curves intersect the line $x = 1$ more than twice?
4. How would the result change if the polar equation were $r = 2 + \cos(\theta)$ instead of $r = 2 + \sin(\theta)$?
5. What is the general shape of the polar curve $r = 2 + \sin(\theta)$, and how does it compare to other cardioids?

Tip: In polar coordinates, the relationship between $r$, $\theta$, and $x$ is $x = r\cos(\theta)$. This is useful when analyzing intersections with vertical lines like $x = 1$.