All Flashcards
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: . Polar: .
What are the differences between finding the slope of a tangent line in Cartesian and polar coordinates?
Cartesian: . Polar: .
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.
What is the formula for average rate of change of (r) with respect to (\theta)?
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))?
How do you find the slope of a tangent line to a polar curve?
What is the formula to find the area enclosed by a polar curve (r = f(\theta)) from (\theta = a) to (\theta = b)?
How to calculate the arc length of a polar curve (r = f(\theta)) from (\theta = a) to (\theta = b)?
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?
, 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.
How do you find the average rate of change of (r = 2\cos(\theta)) on the interval ([0, \pi/2])?
- 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])?
- 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])?
- 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?
- 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)?
- 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]).
- 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)?
- 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)?
- 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)?
- 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})?
- 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}))).