Find the slope of the polar curve at the indicated point.

r equals negative 5 minus 4 cosine theta comma theta equals StartFraction pi Over 2 EndFractionr=−5−4cosθ, θ=π2

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Part 1

A.negative five fourths

negative five fourths−54

B.negative four fifths

negative four fifths−45

C.four fifths

four fifths45

D.five fourths

five fourths

To find the slope of the polar curve \( r = -5 - 4\cos\theta \) at the point where \( \theta = \frac{\pi}{2} \), we will need to follow these steps:

### 1. **Convert the Polar Equation to Cartesian Coordinates:**

The relationship between polar and Cartesian coordinates is given by:
\[
x = r\cos\theta
\]
\[
y = r\sin\theta
\]

### 2. **Find \( \frac{dy}{dx} \) in Polar Coordinates:**

The slope of the curve at a point is given by \( \frac{dy}{dx} \). In polar coordinates, this can be expressed as:
\[
\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}
\]

First, calculate \( \frac{dr}{d\theta} \):
\[
\frac{dr}{d\theta} = \frac{d}{d\theta} \left(-5 - 4\cos\theta\right) = 4\sin\theta
\]

Now, calculate \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \):
\[
\frac{dx}{d\theta} = \frac{d}{d\theta} \left(r\cos\theta\right) = \frac{dr}{d\theta}\cos\theta - r\sin\theta
\]
\[
\frac{dy}{d\theta} = \frac{d}{d\theta} \left(r\sin\theta\right) = \frac{dr}{d\theta}\sin\theta + r\cos\theta
\]

Substitute the values:
\[
\frac{dx}{d\theta} = 4\sin\theta\cos\theta - (-5 - 4\cos\theta)\sin\theta
\]
\[
\frac{dy}{d\theta} = 4\sin^2\theta + (-5 - 4\cos\theta)\cos\theta
\]

### 3. **Evaluate at \( \theta = \frac{\pi}{2} \):**

At \( \theta = \frac{\pi}{2} \), we have:
\[
\cos\left(\frac{\pi}{2}\right) = 0, \quad \sin\left(\frac{\pi}{2}\right) = 1
\]
Substitute these into the expressions for \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \):
\[
\frac{dx}{d\theta} = 4(1)(0) - (-5 - 4(0))(1) = 5
\]
\[
\frac{dy}{d\theta} = 4(1)^2 + (-5 - 4(0))(0) = 4
\]

### 4. **Compute the Slope \( \frac{dy}{dx} \):**

\[
\frac{dy}{dx} = \frac{4}{5}
\]

### 5. **Choose the Correct Answer:**

The slope of the polar curve at the point where \( \theta = \frac{\pi}{2} \) is \( \frac{4}{5} \).

Therefore, the correct answer is **C. four fifths ( \(\frac{4}{5}\) )**.

Would you like further details or have any questions?

Here are some additional related questions:
1. How do you convert a polar equation into Cartesian form?
2. What is the significance of the derivative \( \frac{dr}{d\theta} \) in polar coordinates?
3. How does the slope of a polar curve differ from that of a Cartesian curve?
4. What is the general formula for \( \frac{dy}{dx} \) in polar coordinates?
5. How do trigonometric identities simplify calculations in polar coordinates?

**Tip:** Always check the trigonometric values at the specific angles to avoid common mistakes in sign or magnitude.

Learn how to find the slope of the polar curve r = -5 - 4 cos theta at theta = pi/2 using polar coordinates. Understand the process of converting polar equations to Cartesian form and calculating derivatives in polar coordinates. Suitable for undergraduate students studying calculus and polar coordinate systems.

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