Revise later
SpaceTo flip
If confident
All Flashcards
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}))).
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.
Define polar coordinates.
A coordinate system where a point is located by its distance (r) from the origin and an angle (\theta) from the polar axis.
What is a polar function?
A function defined in polar coordinates, typically in the form (r = f(\theta)), where (r) is the distance from the origin and (\theta) is the angle.
Define relative maximum in polar functions.
A point where the distance (r) from the origin is locally the greatest, changing from increasing to decreasing as (\theta) increases.
Define relative minimum in polar functions.
A point where the distance (r) from the origin is locally the smallest, changing from decreasing to increasing as (\theta) increases.
What is the polar axis?
The reference line ((\theta = 0)) from which the angle (\theta) is measured in polar coordinates, analogous to the positive x-axis in Cartesian coordinates.
Define average rate of change in polar functions.
The change in (r) with respect to (\theta) over an interval, calculated as (\frac{\Delta r}{\Delta \theta} = \frac{r(\theta_2) - r(\theta_1)}{\theta_2 - \theta_1}).
What does it mean for a polar function to be 'expanding'?
The distance (r) from the origin is increasing as (\theta) increases.
What does it mean for a polar function to be 'contracting'?
The distance (r) from the origin is decreasing as (\theta) increases.
Define critical points in polar functions.
Values of (\theta) where the derivative of (r) with respect to (\theta) (i.e., (r'(\theta))) is either zero or undefined. These points are candidates for relative extrema.
What is the significance of (r = 0) in polar coordinates?
It indicates that the point is at the origin, regardless of the value of (\theta).