Question

For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.Use the definition of the derivative \(\frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}\) and the product rule to derive the derivative of a polar equation.

Short answer

The slope is \( \frac{f'(\theta) \sin(\theta) + f(\theta) \cos(\theta)}{f'(\theta) \cos(\theta) - f(\theta) \sin(\theta)} \).

Step-by-step solution

Convert Polar to Parametric Form

Given that the polar curve is represented by \( r = f(\theta) \), we can express this in parametric form as \( x = f(\theta) \cos(\theta) \) and \( y = f(\theta) \sin(\theta) \).

Calculate Derivative of x with respect to θ

Using the product rule for differentiation \( \left( u \cdot v \right)' = u'v + uv' \), find \( \frac{d x}{d \theta} \).Let \( u = f(\theta) \) and \( v = \cos(\theta) \), then:\[ \frac{d x}{d \theta} = \frac{d}{d \theta}\left[f(\theta) \cos(\theta)\right] = f'(\theta) \cos(\theta) - f(\theta) \sin(\theta) \]

Calculate Derivative of y with respect to θ

Similarly, use the product rule to find \( \frac{d y}{d \theta} \).Let \( u = f(\theta) \) and \( v = \sin(\theta) \), then:\[ \frac{d y}{d \theta} = \frac{d}{d \theta}\left[f(\theta) \sin(\theta)\right] = f'(\theta) \sin(\theta) + f(\theta) \cos(\theta) \]

Find the Slope of the Tangent Line

The slope of the tangent line at any point on the polar curve is given by \( \frac{d y}{d x} = \frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}} \).Substitute the values we calculated:\[ \frac{d y}{d x} = \frac{f'(\theta) \sin(\theta) + f(\theta) \cos(\theta)}{f'(\theta) \cos(\theta) - f(\theta) \sin(\theta)} \]

Key concepts

Polar Coordinates

Polar coordinates represent points in the plane using a distance and angle, instead of traditional Cartesian coordinates. Here, a point is defined by two values:

• \(r\): The distance from the origin, known as the radial coordinate.
• \(\theta\): The angle from the positive x-axis, called the angular coordinate.

This system is particularly useful for circular and rotational aspects of geometry.
This contrasts with Cartesian coordinates, which use horizontal and vertical distances to define a location. Polar plots often simplify curves and make them easier to analyze, particularly when dealing with complex wave or circular patterns.

Parametric Equations

Parametric equations allow us to express Cartesian coordinates \((x, y)\) in terms of an independent variable \(\theta\). For polar equations, these are expressed as:
\[ x = f(\theta) \cos(\theta) \]
\[ y = f(\theta) \sin(\theta) \]
Using parametric forms can make analyzing curves simpler by effectively separating variables.
It's an excellent tool for tracing complicated paths, such as circles or ellipses, using simple functions such as sine and cosine. This representation singles out the dependency on \(\theta\), offering a clear view of how each coordinate is influenced by it.

Tangent Line Slope

The slope of a tangent line provides insights into the behavior of a curve at a particular point. In polar coordinates, our goal is to find \(\frac{d y}{d x}\), the change in \(y\) with respect to \(x\).
To find this, we first calculate the derivatives of \(x\) and \(y\) with respect to \(\theta\):

• \(\frac{d x}{d \theta} = f'(\theta) \cos(\theta) - f(\theta) \sin(\theta)\)
• \(\frac{d y}{d \theta} = f'(\theta) \sin(\theta) + f(\theta) \cos(\theta)\)

Then, combining these derivatives:
\[ \frac{d y}{d x} = \frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}} \]
This ratio depicts the local steepness of the curve, revealing how sharply it ascends or descends.

Product Rule

The product rule is a fundamental principle in calculus used to differentiate expressions where two functions are multiplied. It is expressed as:
\(( uv )' = u'v + uv'\).
In the context of polar derivatives, it applies when you need to find the derivative of products such as \(f(\theta) \cos(\theta)\) or \(f(\theta) \sin(\theta)\). These expressions arise when converting from polar to parametric forms.
Here's how it's used:

• Differentiate each function individually.
• Multiply each differentiated function by the other original function.
• Add the results.

Using the product rule correctly is crucial, as it ensures calculations involving derivatives of products are accurate and reliable.