Let \[r\] be the polar function \[r(\theta)=\cos(8\theta)+4\]. Here is its graph for \[0\leq\theta\leq2\pi\]:

\[\small{1}\]\[\small{2}\]\[\small{3}\]\[\small{4}\]\[\small{5}\]\[\small{6}\]\[\small{7}\]\[\small{\llap{-}2}\]\[\small{\llap{-}3}\]\[\small{\llap{-}4}\]\[\small{\llap{-}5}\]\[\small{\llap{-}6}\]\[\small{\llap{-}7}\]\[\small{1}\]\[\small{2}\]\[\small{3}\]\[\small{4}\]\[\small{5}\]\[\small{6}\]\[\small{7}\]\[\small{\llap{-}2}\]\[\small{\llap{-}3}\]\[\small{\llap{-}4}\]\[\small{\llap{-}5}\]\[\small{\llap{-}6}\]\[\small{\llap{-}7}\]\[y\]\[x\]\[P\]

**What is the rate of change of the \[x\]-coordinate with respect to \[\theta\] at the point \[P\]?**

point p is at (5,0)

Choose 1 answer:

To find the rate of change of the \( x \)-coordinate with respect to \( \theta \) at the point \( P \), we need to calculate \( \frac{dx}{d\theta} \) for the polar function \( r(\theta) = \cos(8\theta) + 4 \).

### Step 1: Express \( x \) in terms of \( \theta \)
In polar coordinates, the \( x \)-coordinate is given by:
\[
x = r(\theta) \cos(\theta)
\]
Substituting \( r(\theta) = \cos(8\theta) + 4 \) into this expression:
\[
x(\theta) = \left(\cos(8\theta) + 4\right) \cos(\theta)
\]

### Step 2: Differentiate \( x(\theta) \) with respect to \( \theta \)
We need to find \( \frac{dx}{d\theta} \), which involves using the product rule:
\[
\frac{dx}{d\theta} = \frac{d}{d\theta} \left[ \left(\cos(8\theta) + 4\right) \cos(\theta) \right]
\]
Applying the product rule:
\[
\frac{dx}{d\theta} = \left(\frac{d}{d\theta}[\cos(8\theta) + 4]\right) \cos(\theta) + \left(\cos(8\theta) + 4\right) \frac{d}{d\theta}[\cos(\theta)]
\]
Calculate each derivative:
\[
\frac{d}{d\theta}[\cos(8\theta) + 4] = -8 \sin(8\theta)
\]
\[
\frac{d}{d\theta}[\cos(\theta)] = -\sin(\theta)
\]
Substituting these into the equation:
\[
\frac{dx}{d\theta} = -8 \sin(8\theta) \cos(\theta) - (\cos(8\theta) + 4) \sin(\theta)
\]

### Step 3: Evaluate \( \frac{dx}{d\theta} \) at the point \( P \)
The point \( P \) is given as \( (5, 0) \), which corresponds to \( r(\theta) = 5 \) and \( x = 5 \), \( y = 0 \). This occurs when \( \cos(\theta) = 1 \) (hence \( \theta = 0 \) or multiples of \( 2\pi \)).

For \( \theta = 0 \):
\[
r(0) = \cos(0) + 4 = 1 + 4 = 5
\]
So, at \( \theta = 0 \), \( x(\theta) = 5 \).

Substituting \( \theta = 0 \) into \( \frac{dx}{d\theta} \):
\[
\frac{dx}{d\theta}\bigg|_{\theta = 0} = -8 \sin(0) \cdot \cos(0) - (\cos(0) + 4) \sin(0)
\]
Since \( \sin(0) = 0 \), both terms are 0, so:
\[
\frac{dx}{d\theta}\bigg|_{\theta = 0} = 0
\]

### Final Answer
The rate of change of the \( x \)-coordinate with respect to \( \theta \) at the point \( P \) is **0**.

The correct answer is **0**.

Would you like further details, or do you have any questions? Here are five related questions:

1. How do you interpret the polar coordinates \( r(\theta) \) and \( \theta \) in the context of a graph?
2. What is the geometric significance of \( \frac{dx}{d\theta} = 0 \) at a point on a polar curve?
3. How would the rate of change \( \frac{dx}{d\theta} \) change if the function were \( r(\theta) = 2\cos(4\theta) + 3 \)?
4. What would \( \frac{dy}{d\theta} \) be at the point \( P \) on the same curve?
5. How does the product rule apply when differentiating functions involving trigonometric identities?

**Tip:** When working with polar coordinates, always remember to differentiate each component (like \( r(\theta) \)) and apply trigonometric identities carefully to simplify your results.

Learn how to calculate the rate of change of the x-coordinate with respect to θ at a specific point (5, 0) on a polar curve defined by r(θ) = cos(8θ) + 4. Detailed step-by-step solution using trigonometric identities and differentiation techniques.

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