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What are the differences between parametric equations and Cartesian equations?
Parametric: x and y are functions of t | Cartesian: y is a function of x
What are the differences between integrating vector-valued functions and integrating scalar functions?
Vector-Valued: Integrate each component separately, result is a vector | Scalar: Integrate a single function, result is a scalar
What are the differences between finding the area between curves in Cartesian coordinates vs. polar coordinates?
Cartesian: Integrate difference of functions wrt x, A = ∫(top - bottom) dx | Polar: Integrate difference of squared polar functions wrt θ, A = 1/2 ∫(R² - r²) dθ
What are the differences between velocity and speed?
Velocity: Vector quantity with magnitude and direction | Speed: Scalar quantity, magnitude of velocity
What are the differences between position vector and velocity vector?
Position Vector: Represents location at a given time | Velocity Vector: Represents rate of change of position at a given time
What are the differences between polar coordinates and Cartesian coordinates?
Polar: Defined by radius and angle (r, θ) | Cartesian: Defined by horizontal and vertical distance (x, y)
What are the differences between differentiating parametric equations and differentiating Cartesian equations?
Parametric: Use chain rule, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ | Cartesian: Direct differentiation, $\frac{dy}{dx}$
What are the differences between integrating parametric equations and integrating Cartesian equations?
Parametric: Integrate with respect to t | Cartesian: Integrate with respect to x
What are the differences between vector-valued functions and scalar functions?
Vector-valued: Output is a vector | Scalar: Output is a single number
What are the differences between finding arc length in Cartesian coordinates vs. parametric coordinates?
Cartesian: Integrate $\sqrt{1 + (dy/dx)^2}$ dx | Parametric: Integrate $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ dt
Explain how to find the velocity and acceleration from a position vector.
Velocity is the first derivative of the position vector with respect to time. Acceleration is the second derivative of the position vector with respect to time.
Explain how to integrate a vector-valued function.
Integrate each component of the vector-valued function separately.
Explain how to find the slope of a tangent line to a parametric curve.
Find $\frac{dy}{dx}$ by calculating $\frac{dy/dt}{dx/dt}$. Evaluate at the given t-value.
What does the derivative of a vector-valued function represent?
The velocity vector of the particle at that point.
How do you find the points where a parametric curve has a horizontal tangent?
Find where $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$.
How do you find the points where a parametric curve has a vertical tangent?
Find where $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$.
How do you find the area enclosed by a single polar curve?
Use the formula $A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$, where r is the polar function and a and b are the limits of integration.
How do you find the area enclosed by two polar curves?
Use the formula $A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$, where R is the outer curve and r is the inner curve.
Explain the relationship between position, velocity, and acceleration in vector-valued functions.
Velocity is the derivative of position, and acceleration is the derivative of velocity (or the second derivative of position).
How do you determine the direction of motion of a particle described by parametric equations?
Analyze the signs of $\frac{dx}{dt}$ and $\frac{dy}{dt}$. Positive $\frac{dx}{dt}$ means moving right, negative means moving left. Positive $\frac{dy}{dt}$ means moving up, negative means moving down.
What is the formula for the derivative of a parametric function, $\frac{dy}{dx}$?
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
What is the formula for the second derivative of a parametric function, $\frac{d^2y}{dx^2}$?
$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$
What is the formula for arc length of a parametric curve?
$L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$
What is the formula to find the area of a polar region?
$A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$
What is the formula to find the area between two polar curves?
$A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$
How do you convert from polar to Cartesian coordinates?
$x = r \cos(\theta), y = r \sin(\theta)$
How do you convert from Cartesian to polar coordinates?
$r = \sqrt{x^2 + y^2}$
What is the formula for $\frac{dy}{dx}$ in polar coordinates?
$\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}$
If r(t) = <f(t), g(t)>, what is r'(t)?
r'(t) = <f'(t), g'(t)>
If v(t) = <f(t), g(t)>, what is ∫v(t) dt?
∫v(t) dt = <∫f(t) dt, ∫g(t) dt>