What is the area enclosed by the cardioid $$r = 2 + 2\text{cos}(\theta)$$ ?

The area enclosed by the cardioid $$r = 2 + 2\text{cos}(\theta)$$ is $$A = 6\text{π}$$ .

What is the formula for the area between two polar curves?

The area A between two polar curves $$r_1(\theta)$$ and $$r_2(\theta)$$ from θ=a to θ=b is given by: $$A = \frac{1}{2} \times \text{integral from a to b} (r_1^2 - r_2^2) d\theta$$ .

What is a four-leaved rose in polar coordinates?

A four-leaved rose is represented in polar coordinates by the equation $$r = a \text{cos}(n\theta)$$ or $$r = a \text{sin}(n\theta)$$ , where n is an even integer.

How do you find the slope of the four-leaved rose $$r = \text{cos}(2\theta)$$ ?

The slope of the four-leaved rose $$r = \text{cos}(2\theta)$$ is given by: $$\frac{\text{cos}(2\theta)\text{cos}(\theta) - 2\text{sin}(2\theta)\text{sin}(\theta)}{-\text{cos}(2\theta)\text{sin}(\theta) - 2\text{sin}(2\theta)\text{cos}(\theta)}$$ .

What is the derivative of r with respect to θ for the cardioid $$r = 2 + 2\text{cos}(\theta)$$ ?

The derivative of r with respect to θ for the cardioid $$r = 2 + 2\text{cos}(\theta)$$ is $$\frac{dr}{d\theta} = -2\text{sin}(\theta)$$ .

What is the relationship between polar and Cartesian coordinates?

The relationship between polar and Cartesian coordinates is given by: $$x = r\text{cos}(\theta)$$ and $$y = r\text{sin}(\theta)$$ .

How do you convert a polar equation to Cartesian form?

To convert a polar equation to Cartesian form, use the relationships $$x = r\text{cos}(\theta)$$ and $$y = r\text{sin}(\theta)$$ to express r and θ in terms of x and y.

What is the importance of the cardioid shape in real-world applications?

The cardioid shape is important in acoustics and antenna design, as it describes the radiation pattern of certain types of microphones and antennas.

What is the area of the region that lies inside the circle $$r = 3\text{sin}(\theta)$$ and outside the cardioid $$r = 1 + \text{sin}(\theta)$$ ?

The area of the region that lies inside the circle $$r = 3\text{sin}(\theta)$$ and outside the cardioid $$r = 1 + \text{sin}(\theta)$$ is $$A = \text{π}$$ .

Calculus of Polar Curves - Calc BC - Calculator Allowed Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

Define a polar curve.

Back

A polar curve is a curve represented in polar coordinates, where each point on the curve is defined by a distance from a reference point (the pole) and an angle from a reference direction.

2.

FLASHCARD QUESTION

Front

What is the formula for the slope of a polar curve given by r(θ)?

Back

3.

FLASHCARD QUESTION

Front

What is a cardioid in polar coordinates?

Back

4.

FLASHCARD QUESTION

Front

How do you find the area enclosed by a polar curve?

Back

5.

FLASHCARD QUESTION

Front

Back

6.

FLASHCARD QUESTION

Front

How do you determine the points of intersection between two polar curves?

Back

To find the points of intersection between two polar curves, set their equations equal to each other and solve for θ.

7.

FLASHCARD QUESTION

Front

What is the significance of the angle θ in polar coordinates?

Back

In polar coordinates, the angle θ represents the direction of the point from the pole, measured from the positive x-axis.

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