Question

Explain why the slope of the line tangent to the polar graph of \(r=f(\theta)\) is not \(\frac{d r}{d \theta}\).

Short answer

The slope of the line tangent to the polar graph of \(r = f(\theta)\) is not \(\frac{d r}{d \theta}\) because the correct expression for the slope takes into account the relationship between polar and Cartesian coordinates and the effect of the angle \(\theta\) on the slope, which is not considered in \(\frac{d r}{d \theta}\). The actual slope of the tangent line is given by the expression: \(m_\text{tan} = \frac{\frac{d r}{d \theta} \sin \theta + r \cos \theta}{\frac{d r}{d \theta} \cos \theta - r \sin \theta}\).

Step-by-step solution

Conversion from polar to Cartesian coordinates

To find the slope of the tangent line, we first need to convert the polar coordinates to Cartesian coordinates. The conversion formulas are given by: \(x = r \cos \theta\) \(y = r \sin \theta\)

Differentiate x and y with respect to theta

Now, we will differentiate both x and y with respect to \(\theta\): \(\frac{d x}{d \theta} = \frac{d (r \cos \theta)}{d \theta} = \frac{d r}{d \theta} \cos \theta - r \sin \theta\) \(\frac{d y}{d \theta} = \frac{d (r \sin \theta)}{d \theta} = \frac{d r}{d \theta} \sin \theta + r \cos \theta\)

Calculate the slope of the tangent line

The slope of the tangent line (\(m_\text{tan}\)) to the curve in Cartesian coordinates is given by the ratio of the change in y and the change in x. Using the derivatives calculated in Step 2: \(m_\text{tan} = \frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}\) \(m_\text{tan} = \frac{\frac{d r}{d \theta} \sin \theta + r \cos \theta}{\frac{d r}{d \theta} \cos \theta - r \sin \theta}\)

Conclusion

Comparing the expression for the slope of the tangent line to the polar graph, \(m_\text{tan}\), to \(\frac{dr}{d \theta}\), we can see they are not the same. The slope of the tangent line to the polar graph is given by the expression: \(m_\text{tan} = \frac{\frac{d r}{d \theta} \sin \theta + r \cos \theta}{\frac{d r}{d \theta} \cos \theta - r \sin \theta}\) This expression takes into account the relationship between polar and Cartesian coordinates and the effect of the angle \(\theta\) on the slope, which is not considered in \(\frac{dr}{d \theta}\). Hence, the slope of the line tangent to the polar graph of \(r = f(\theta)\) is not \(\frac{d r}{d \theta}\).

Key concepts

Polar to Cartesian Coordinate Conversion

Understanding the relationship between polar and Cartesian coordinates is essential for a variety of mathematical and engineering applications. In essence, polar coordinates describe a point in a plane using a distance and an angle, while Cartesian coordinates use a grid of horizontal and vertical lines to pinpoint a location.

To convert from polar to Cartesian coordinates, you can use the following equations:

• For the x-coordinate:

\[x = r \times \text{cos}(\theta)\]\

• For the y-coordinate:

\[y = r \times \text{sin}(\theta)\]\
Where \( r \) represents the distance from the origin to the point and \( \theta \) is the angle measured from the positive x-axis to the line segment connecting the origin to the point. This conversion is crucial when transitioning between coordinate systems for calculations, such as finding the slope of a line tangent to a curve on a graph.

Differentiating Parametric Equations

In calculus, differentiating parametric equations is a powerful technique used to find the rates of change of two related quantities. Parametric equations define a set of related variables using a common parameter. For example, in polar coordinates, the position of a point can be described by the two equations \( x = r \times \text{cos}(\theta) \) and \( y = r \times \text{sin}(\theta) \), with \( \theta \) as the parameter.

To differentiate these parametric equations with respect to \( \theta \), you apply the product rule and chain rule of differentiation:

• For \( x \):

\[ \frac{dx}{d\theta} = \frac{dr}{d\theta} \times \text{cos}(\theta) - r \times \text{sin}(\theta) \]\

• For \( y \):

\[ \frac{dy}{d\theta} = \frac{dr}{d\theta} \times \text{sin}(\theta) + r \times \text{cos}(\theta) \]\
These derivatives reflect how the x and y coordinates change as the angle \( \theta \) changes. Differentiating parametric equations is fundamental when analyzing the dynamics of systems- for instance- or finding the slope of the tangent line at a specific point.

Tangent Line in Polar Coordinates

The tangent line to a curve represents the instantaneous direction of the curve at that point and is an important concept in calculus. When dealing with polar coordinates, finding the tangent line's slope requires careful consideration because the direction of the curve is not only dependent on the rate of change of the radius but also on how the angle changes.

To calculate the slope of the tangent line in polar coordinates, one must use the derivatives of the parametric equations that represent the Cartesian coordinates of the polar graph. The formula is given by:\[ m_{\text{tan}} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]\
Using the expressions derived earlier, the slope can be found:\[ m_{\text{tan}} = \frac{\frac{dr}{d\theta} \times \text{sin}(\theta) + r \times \text{cos}(\theta)}{\frac{dr}{d\theta} \times \text{cos}(\theta) - r \times \text{sin}(\theta)} \]\
It is clear from this formula that the slope of the tangent line depends on both the rate of change of the radius with respect to \( \theta \) and the actual values of \( r \) and \( \theta \). Simply put, finding the tangent line slope in polar coordinates is more complex than just taking the derivative of the radius with respect to the angle, hence validating the assertion that the slope is not merely \( \frac{dr}{d\theta} \).