All Flashcards
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 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.
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}))).