Polar Slope Calculator Graph

Polar Slope Formula:

\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta} \]

Slope (dy/dx) at θ = :

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1. What is Polar Slope?

The polar slope represents the derivative dy/dx of a curve defined in polar coordinates. Unlike Cartesian coordinates where we have y = f(x), polar coordinates use r = f(θ) where r is the distance from the origin and θ is the angle from the positive x-axis.

2. How Does the Calculator Work?

The calculator uses the polar slope formula:

\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta} \]

Where:

\( r \) — The polar function value at θ
\( \frac{dr}{d\theta} \) — Derivative of the polar function with respect to θ
\( \theta \) — The angle at which to evaluate the slope

Explanation: This formula converts the polar derivative to the Cartesian slope by applying the chain rule to the relationships x = r·cosθ and y = r·sinθ.

3. Importance of Polar Slope Calculation

Details: Calculating slopes of polar curves is essential for understanding curve behavior, finding tangent lines, locating points of horizontal and vertical tangency, and analyzing curve properties in various applications including physics and engineering.

4. Using the Calculator

Tips: Enter a polar function r(θ) using standard mathematical notation. Use 'theta' for the angle variable. Provide the specific θ value (in radians) at which you want to calculate the slope.

5. Frequently Asked Questions (FAQ)

Q1: Why is the polar slope formula different from Cartesian slope?
A: Polar coordinates use a different coordinate system (r,θ) rather than (x,y), requiring conversion using the relationships x = r·cosθ and y = r·sinθ.

Q2: What does an undefined slope indicate?
A: An undefined slope (when denominator = 0) indicates a vertical tangent line at that point on the polar curve.

Q3: Can I use degrees instead of radians?
A: The calculator requires θ in radians. Convert degrees to radians by multiplying by π/180.

Q4: What functions can I input?
A: The calculator supports basic mathematical operations and functions like sin, cos, tan, exp, log, etc.

Q5: How is this different from parametric slope?
A: Polar coordinates are a special case of parametric equations where the parameter is θ and x = r(θ)cosθ, y = r(θ)sinθ.