Explain how the sign of the average rate of change relates to the behavior of (r) as (\theta) increases.
A positive average rate of change indicates that (r) is increasing as (\theta) increases. A negative average rate of change indicates that (r) is decreasing as (\theta) increases.
What does the derivative (dr/d\theta) represent in polar functions?
It represents the instantaneous rate of change of the distance (r) from the origin with respect to the angle (\theta).
How do you determine if a polar function is increasing or decreasing?
Examine the sign of the derivative (dr/d\theta). If (dr/d\theta > 0), the function is increasing; if (dr/d\theta < 0), it's decreasing.
Explain the concept of relative extrema in the context of polar functions.
Relative extrema (maxima or minima) occur where the function changes direction. A relative maximum is a point where (r) is locally largest, and a relative minimum is a point where (r) is locally smallest.
What is the significance of finding where (dr/d\theta = 0) in polar functions?
These points are critical points and potential locations of relative maxima or minima. They indicate where the function's rate of change is momentarily zero.
Describe the relationship between polar and Cartesian coordinates.
Polar coordinates use distance (r) and angle (\theta) to define a point, while Cartesian coordinates use horizontal distance (x) and vertical distance (y). They can be converted using trigonometric relationships.
Explain how the graph of a polar function is traced as (\theta) varies.
As (\theta) increases, the point ((r, \theta)) moves around the origin. The value of (r) determines how far the point is from the origin at each angle (\theta).
Describe the behavior of the polar function when (r) is negative.
When (r) is negative, the point is plotted in the opposite direction of the angle (\theta). It is a reflection through the origin.
Explain how to determine concavity of a polar curve.
Concavity can be determined by analyzing the second derivative (\frac{d^2y}{dx^2}). If (\frac{d^2y}{dx^2} > 0), the curve is concave up. If (\frac{d^2y}{dx^2} < 0), the curve is concave down.
What are some common types of symmetry found in polar graphs?
Common symmetries include symmetry about the x-axis (polar axis), symmetry about the y-axis ((\theta = \frac{\pi}{2})), and symmetry about the origin (pole).
What does a polar graph look like when (r) is constant?
It forms a circle centered at the origin with radius (r).
If the polar graph of (r = f(\theta)) is symmetric about the x-axis, what does this imply about the function?
It implies that (f(\theta) = f(-\theta)), meaning the function is even.
How can you identify relative maxima and minima on a polar graph?
Look for points where the curve reaches its farthest or closest distance from the origin locally.
What does a cardioid polar graph look like?
It has a heart-like shape, typically represented by equations of the form (r = a(1 \pm \cos(\theta))) or (r = a(1 \pm \sin(\theta))).
What does a lemniscate polar graph look like?
It has a figure-eight shape, represented by equations of the form (r^2 = a^2 \cos(2\theta)) or (r^2 = a^2 \sin(2\theta)).
How does the period of a trigonometric function in polar coordinates affect the graph?
The period determines how often the graph repeats itself as (\theta) increases. For example, (r = \sin(2\theta)) completes a full graph in (\pi) radians.
What does it mean if a polar curve passes through the origin?
It means that there exists some value of (\theta) for which (r = 0).
How can you tell if a polar graph is symmetric about the y-axis?
The graph is symmetric about the y-axis if replacing (\theta) with (-\theta) changes the sign of (r), or if replacing (\theta) with (\pi - \theta) leaves the equation unchanged.
What does the slope of the tangent line at a point on a polar graph represent?
It represents the instantaneous rate of change of (y) with respect to (x) at that point, indicating the direction of the curve.
How does the constant term in a polar equation like (r = a + b\cos(\theta)) affect the graph?
The constant term (a) affects the size and position of the graph. If (a = b), it forms a cardioid. If (a < b), it forms a looped curve.
What is the formula for average rate of change of (r) with respect to (\theta)?
\(\frac{\Delta r}{\Delta \theta} = \frac{r(\theta_2) - r(\theta_1)}{\theta_2 - \theta_1}\)
How do you convert from polar coordinates ((r, \theta)) to Cartesian coordinates ((x, y))?
x = r \cos(\theta), y = r \sin(\theta)
How do you convert from Cartesian coordinates ((x, y)) to polar coordinates ((r, \theta))?
r = \sqrt{x^2 + y^2}, \theta = \arctan(\frac{y}{x})
What is the formula for the derivative of a polar function (r = f(\theta))?
\(\frac{dr}{d\theta} = f'(\theta)\)
How do you find the slope of a tangent line to a polar curve?
\(\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}\)
What is the formula to find the area enclosed by a polar curve (r = f(\theta)) from (\theta = a) to (\theta = b)?
\(A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta\)
How to calculate the arc length of a polar curve (r = f(\theta)) from (\theta = a) to (\theta = b)?
\(L = \int_{a}^{b} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta\)
What is the general form of a polar equation for a circle centered at the origin?
r = a, where 'a' is the radius of the circle.
What is the polar equation for a line passing through the origin?
\(\theta = c\), where (c) is a constant angle.
What is the formula for finding points of intersection between two polar curves (r_1(\theta)) and (r_2(\theta))?
Solve the equation (r_1(\theta) = r_2(\theta)) for (\theta). Also, check if the pole (origin) is a point on either curve.
