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What are the key differences between polar and Cartesian coordinate systems?
Polar: Uses distance (r) and angle (\theta). Cartesian: Uses horizontal (x) and vertical (y) distances.
Compare the average rate of change with the instantaneous rate of change in polar functions.
Average: Change over an interval. Instantaneous: Derivative at a point.
What is the difference between relative maximum and absolute maximum in polar functions?
Relative: Local maximum in an interval. Absolute: Overall maximum of the function.
Compare increasing and decreasing behavior of a polar function.
Increasing: (r) increases as (\theta) increases. Decreasing: (r) decreases as (\theta) increases.
What are the differences between symmetry about the x-axis and symmetry about the y-axis in polar graphs?
x-axis: (f(\theta) = f(-\theta)). y-axis: (f(\theta) = -f(-\theta)) or (f(\theta) = f(\pi - \theta)).
Compare the formulas for area in Cartesian and Polar coordinates.
Cartesian: \(\int f(x) dx\). Polar: \(\frac{1}{2} \int r^2 d\theta\).
What are the differences between finding the slope of a tangent line in Cartesian and polar coordinates?
Cartesian: \(\frac{dy}{dx}\). Polar: \(\frac{dy/d\theta}{dx/d\theta}\).
Compare the graphs of (r = a\cos(\theta)) and (r = a\sin(\theta)).
\(r = a\cos(\theta)): Circle on x-axis. (r = a\sin(\theta)): Circle on y-axis.
What are the differences between a cardioid and a circle in polar coordinates?
Cardioid: Heart-shaped, (r = a(1 \pm \cos(\theta))). Circle: Centered at origin, (r = a).
Compare the equations for a line in Cartesian and polar coordinates.
Cartesian: (y = mx + b). Polar: More complex, often involving (\theta = constant) for lines through the origin.
How do you find the average rate of change of (r = 2\cos(\theta)) on the interval ([0, \pi/2])?
1. Calculate (r(0) = 2\cos(0) = 2). 2. Calculate (r(\pi/2) = 2\cos(\pi/2) = 0). 3. Apply the formula: (\frac{0 - 2}{\pi/2 - 0} = -\frac{4}{\pi}\).
How do you determine where (r = 3 + 2\cos(\theta)) is increasing on ([0, 2\pi])?
1. Find (r'(\theta) = -2\sin(\theta)). 2. Set (r'(\theta) = 0) and solve for (\theta): (\theta = 0, \pi, 2\pi). 3. Test intervals: ((\pi, 2\pi)) is increasing.
How do you find relative extrema for (r = 1 + \sin(\theta)) on ([0, 2\pi])?
1. Find (r'(\theta) = \cos(\theta)). 2. Set (r'(\theta) = 0) and solve for (\theta): (\theta = \pi/2, 3\pi/2). 3. Check endpoints and critical points for maximum and minimum values.
How do you convert the polar equation (r = 4\cos(\theta)) to Cartesian form?
1. Multiply both sides by (r): (r^2 = 4r\cos(\theta)). 2. Substitute (r^2 = x^2 + y^2) and (x = r\cos(\theta)): (x^2 + y^2 = 4x). 3. Rearrange: ((x-2)^2 + y^2 = 4).
How do you find the area enclosed by the polar curve (r = 2\theta) from (\theta = 0) to (\theta = \pi/2)?
1. Use the area formula: (A = \frac{1}{2} \int_{0}^{\pi/2} (2\theta)^2 d\theta). 2. Simplify: (A = 2 \int_{0}^{\pi/2} \theta^2 d\theta). 3. Integrate: (A = \frac{2}{3} \theta^3 \Big|_{0}^{\pi/2} = \frac{\pi^3}{12}\).
Given (r(\theta) = 2 + \cos(\theta)), find the values of (\theta) where the curve is farthest from the origin on the interval ([0, 2\pi]).
1. Find (r'(\theta) = -\sin(\theta)). 2. Set (r'(\theta) = 0) to find critical points: (\theta = 0, \pi, 2\pi). 3. Evaluate (r(\theta)) at these points to find the maximum value.
How do you find the slope of the tangent line to the polar curve (r = \sin(2\theta)) at (\theta = \pi/4)?
1. Find (\frac{dr}{d\theta} = 2\cos(2\theta)). 2. Use the formula (\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}). 3. Substitute (\theta = \pi/4) and simplify.
How do you determine the points of intersection of the polar curves (r = 2\cos(\theta)) and (r = 1)?
1. Set (2\cos(\theta) = 1). 2. Solve for (\theta): (\cos(\theta) = \frac{1}{2}), so (\theta = \pm \frac{\pi}{3}). 3. The points of intersection are ((1, \frac{\pi}{3})) and ((1, -\frac{\pi}{3})).
How do you find the arc length of the spiral (r = \theta) from (\theta = 0) to (\theta = 2\pi)?
1. Find (\frac{dr}{d\theta} = 1). 2. Use the arc length formula: (L = \int_{0}^{2\pi} \sqrt{\theta^2 + 1^2} d\theta). 3. Evaluate the integral (requires trigonometric substitution).
How do you find the equation of the tangent line to (r = 1 + \cos(\theta)) at (\theta = \frac{\pi}{2})?
1. Find (\frac{dr}{d\theta} = -\sin(\theta)). 2. Compute (x(\theta) = r\cos(\theta)) and (y(\theta) = r\sin(\theta)). 3. Find (\frac{dx}{d\theta}) and (\frac{dy}{d\theta}). 4. Compute (\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}). 5. Use point-slope form with the calculated slope and the point ((x(\frac{\pi}{2}), y(\frac{\pi}{2}))).
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).