3.15 Rates of Change in Polar Functions

🔄 Expanding and Contracting Polar Functions

If a polar function is positive and increasing, it means that as the angle θ increases, the distance from the origin, r, increases. This is illustrated graphically by the polar function moving away from the origin as the angle increases. This behavior is known as an "expanding polar function" 🌖

On the other hand, if a polar function is negative and decreasing, it means that as the angle θ increases, the distance from the origin, r, decreases. This is illustrated graphically by the polar function moving towards the origin as the angle increases. This behavior is also known as a "contracting polar function" 🌘

Source: Brilliant

This idea can be applied in different scenarios such as when working with polar coordinates to represent the position of an object moving in a circular path. A positive and increasing polar function would represent an object moving away from the origin, while a negative and decreasing polar function would represent an object moving towards the origin. 🤓

🔀 Relative Extrema

The function f(θ) is said to have a relative extremum when it changes from increasing to decreasing or decreasing to increasing on an interval. 🧑‍💻 This means that there is a point on the interval where the function reaches its maximum or minimum value, which corresponds to a point relatively closest to or farthest from the origin. 💡

For example, if a polar function, r = f(θ), is increasing on the interval 0 < θ < π/2 and decreasing on the interval π/2 < θ < π, this means that the function has a relative maximum at θ = π/2, which is the point on the interval that is relatively closest to the origin.

Similarly, if the function is decreasing on the interval 0 < θ < π/2 and increasing on the interval π/2 < θ < π, this means that the function has a relative minimum at θ = π/2, which is the point on the interval that is relatively farthest from the origin.

〽️ Rates of Change

In a polar coordinate system, the average rate of change of r with respect to θ over an interval of θ, is a measure of how the radius, r, of a polar function changes as the angle, θ, changes over an interval. It is calculated by taking the ratio of the change in radius values to the change in θ over an interval. #️⃣

Formally, the average rate of change of r with respect to θ over an interval of θ is represented by the expression : (Δr/Δθ) = (r(θ2) - r(θ1)) / (θ2 - θ1) where r(θ2) and r(θ1) are the radius values at the endpoints of the interval and θ2 and θ1 are the angle values at the endpoints of the interval. 📐

Graphically, the average rate of change indicates the rate at which the radius is changing per radian. It can be represented by the slope of the line connecting the two points on the polar graph of the function that correspond to the endpoints of the interval. ⛰️

• If the slope is positive, the radius is increasing ⬆️ as the angle increases. ⬇️
• If the slope is negative, the radius is decreasing ⬇️ as the angle increases. ⬆️

Source: Xatly

The average rate of change of r with respect to θ over an interval of θ can also be used to estimate values of a polar function within the interval.

For example, if the average rate of change of r with respect to θ over an interval of θ is positive, it means that the radius is increasing as the angle increases. ➕ Therefore, if we know the radius value at one point in the interval and the corresponding angle value, we can use the average rate of change to estimate the radius value at another point within the interval. ✌️

Similarly, if the average rate of change of r with respect to θ over an interval of θ is negative, it means that the radius is decreasing as the angle increases. ➖ In this case, we can also use the average rate of change to estimate the radius value at another point within the interval. 🤞