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Polar functions, also known as circular functions, are a type of function that are graphed in a polar coordinate system, which uses a distance (r) from a fixed point, known as the pole, and an angle (θ) measured counter-clockwise from the positive x-axis, to determine the coordinates of a point. These functions are often used in physics and engineering to model phenomena such as waves, orbits, and fields.
When working with polar functions, it can be difficult to differentiate them using traditional Calculus techniques because the functions are defined in terms of r and θ, rather than x and y. To overcome this limitation, we can convert polar equations to Cartesian equations by using the relations x = r * cos(θ) and y = r * sin(θ). This allows us to differentiate the functions using the same techniques used for Cartesian functions, such as the power rule, product rule, quotient rule, and chain rule.
Converting polar equations to Cartesian equations also allows us to visualize the functions more easily, as they can be graphed on a traditional x-y coordinate plane. This can be especially useful when working with complex functions that have multiple parts, such as a combination of trigonometric and polynomial functions. 🦄
When we take derivatives of polar functions, we can take them as dr/dθ, which would give us the points that are furthest away from the origin on the polar coordinate system. We find dr/dθ in the same way we would find any normal derivative: by taking the derivative of the polar function: ⛰️
While dr/dθ can tell us relative maximum and minimum values, it doesn’t tell us the slope of the tangent line, since we can’t have linear graphs on the polar coordinate system. In order to find the slope of the tangent line, we need to find the derivative on the Cartesian system, which requires us to calculate dy/dx.
Of course, you can memorize this formula, but most students find it much easier to simply derive it using the chain rule. ⛓️
Still confused? Let's get into further detail:
When a curve is given by a polar equation, such as r = f(θ), it is represented in the polar coordinate system, where the position of a point is determined by its distance r from the origin and the angle θ that it makes with the positive x-axis. By taking the derivatives of the function r = f(θ) with respect to θ, we can learn important information about the curve, such as its curvature, concavity, and asymptotes.
The first derivative of r with respect to θ, denoted as r'(θ), is also known as the radial component of the curve, and it represents the instantaneous rate of change of the distance from the origin. It is also used to calculate the curvature of the curve, which is a measure of how sharply the curve bends at a given point.
The second derivative of r with respect to θ, denoted as r''(θ), is also known as the radial curvature of the curve, and it represents the rate of change of the curvature. It is also used to calculate the concavity of the curve, which is a measure of whether the curve is concave up or concave down at a given point.
In addition to derivatives of r, we can also find the x and y coordinates of a point on the curve in terms of θ by using the relations x = rcos(θ) and y = rsin(θ). By taking the derivatives of x and y with respect to θ, we can learn important information about the tangent vector of the curve, and how it changes as we move along the curve.
Finally, by using the chain rule, we can find the first and second derivatives of y with respect to x. The first derivative is known as the slope of the curve, and it represents the rate of change of y with respect to x. The second derivative, known as the curvature of the curve, represents the rate of change of the slope. 😁