How to find the velocity vector at t=a given a position vector r(t)?

Find the derivative r'(t). 2. Evaluate r'(a).

How to find the acceleration vector at t=a given a position vector r(t)?

Find the second derivative r''(t). 2. Evaluate r''(a).

How to find the arc length of a parametric curve from t=a to t=b?

Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ . 2. Use the formula $L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$ .

How to find the area of a polar region?

Determine the limits of integration, a and b. 2. Identify the polar function r(θ). 3. Use the formula $A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$ .

How to find the area between two polar curves?

Determine the limits of integration, a and b. 2. Identify the outer curve R(θ) and the inner curve r(θ). 3. Use the formula $A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$ .

How to find the equation of a tangent line to a parametric curve at t=t₀?

Find $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ . 2. Evaluate $\frac{dy}{dx}$ at t=t₀ to find the slope. 3. Find x(t₀) and y(t₀) to find the point. 4. Use point-slope form to write the equation of the tangent line.

How to convert a point from polar coordinates (r, θ) to Cartesian coordinates (x, y)?

Use the formulas x = r cos(θ) and y = r sin(θ). 2. Substitute the values of r and θ into the formulas. 3. Calculate x and y.

How to find the maximum height of a particle moving according to parametric equations?

Find $\frac{dy}{dt}$ . 2. Set $\frac{dy}{dt} = 0$ and solve for t. 3. Find the y-coordinate at that t value.

How to determine when a particle changes direction in its x-coordinate movement?

Find $\frac{dx}{dt}$ . 2. Set $\frac{dx}{dt} = 0$ and solve for t. 3. Check if the sign of $\frac{dx}{dt}$ changes around that t value.

How to determine the speed of a particle given its velocity vector?

Find the velocity vector v(t) = <dx/dt, dy/dt>. 2. Calculate the magnitude of the velocity vector: speed = $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$

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

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>

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