Explain how to graph a polar function.
Choose θ values, evaluate r = f(θ), plot (r, θ) points, and connect them with a smooth curve.
Describe the significance of the angle θ in polar coordinates.
θ determines the direction from the origin, measured counterclockwise from the positive x-axis.
Explain the effect of changing the coefficient in r = a cos(θ) on the graph.
Changing 'a' alters the diameter of the circle. A larger 'a' results in a larger diameter.
What is the importance of understanding symmetry when graphing polar functions?
Symmetry helps to sketch the graph more efficiently by reflecting known portions across axes or the origin.
How does periodicity affect the graph of a polar function?
Periodicity indicates how often the graph repeats itself, allowing you to focus on graphing one period.
Explain why a single point can have multiple representations in polar coordinates.
Adding multiples of 2π to θ results in the same point. Also, using a negative 'r' and adjusting θ by π yields the same point.
Describe the relationship between polar and Cartesian coordinate systems.
They are two different ways of representing points in a plane, with conversion formulas linking them.
Explain the concept of finding the area enclosed by a polar curve.
It involves integrating the square of the polar function over a specified range of angles, multiplied by 1/2.
Describe how to determine the domain and range of a polar function.
The domain is the set of possible θ values, and the range is the set of possible r values based on the function's behavior.
Explain the significance of finding the points of intersection of two polar curves.
These points represent the locations where the two curves meet, which can be found by solving their equations simultaneously.
What is the general form of a polar function?
$r = f(\theta)$
What is the polar equation of a circle passing through the origin with diameter 'a' along the x-axis?
$r = a \cos(\theta)$
What is the polar equation of a circle passing through the origin with diameter 'a' along the y-axis?
$r = a \sin(\theta)$
What is the equation for a spiral in polar coordinates?
$r = a\theta$, where 'a' is a constant.
What is the general form of a cardioid?
$r = a(1 ± \cos \theta)$ or $r = a(1 ± \sin \theta)$
What is the general form of a lemniscate?
$r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$
How do you convert from polar coordinates (r, θ) 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, θ)?
$r = \sqrt{x^2 + y^2}$, $\theta = \arctan(\frac{y}{x})$
What is the formula for finding the area enclosed by a polar curve r = f(θ) from θ = α to θ = β?
$A = \frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^2 d\theta$
What is the formula for arc length of a polar curve r = f(θ) from θ = α to θ = β?
$L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta$
What are the differences between Cartesian and Polar coordinates?
Cartesian: Uses (x, y), rectangular grid | Polar: Uses (r, θ), circular grid.
Compare the graphs of y = x and r = θ.
y = x: Straight line | r = θ: Spiral.
Compare the equations for a circle in Cartesian and Polar form.
Cartesian: (x-h)² + (y-k)² = r² | Polar: r = a (centered at origin).
Compare symmetry about the x-axis in Cartesian and Polar coordinates.
Cartesian: Replace y with -y | Polar: Replace θ with -θ.
Compare symmetry about the y-axis in Cartesian and Polar coordinates.
Cartesian: Replace x with -x | Polar: Replace θ with π - θ.
Compare the effect of changing 'a' in r = a cos(θ) and r = cos(aθ).
r = a cos(θ): Changes diameter of circle | r = cos(aθ): Changes the number of 'petals'.
Compare the concept of periodicity in Cartesian and Polar functions.
Cartesian: f(x) = f(x + p) | Polar: f(θ) = f(θ + 2π) (typically).
Compare the representation of the origin in Cartesian and Polar coordinates.
Cartesian: Unique representation (0,0) | Polar: Infinite representations (0, θ) for any θ.
Compare graphing a line in Cartesian (y = mx + b) and Polar coordinates.
Cartesian: Straight line | Polar: Requires converting to polar equation, can be more complex.
Compare the area calculation for a region in Cartesian and Polar coordinates.
Cartesian: ∫∫ dxdy | Polar: ∫∫ r dr dθ
