AP Calc BC parametric and polar motion study guide

Define parametric equations in motion.

Parametric equations define motion by expressing coordinates as functions of a variable, often time. For example, x(t) = 5t and y(t) = 2t² describe a path in the plane.

Find the position at t = 3 for x(t) = 2t, y(t) = t².

x(3) = 2(3) = 6, y(3) = 3² = 9. Position at t = 3 is (6, 9).

True or False: Parametric equations can describe a vertical line.

True. If x(t) = k (a constant) and y(t) varies, it describes a vertical line.

What does 't' represent in parametric motion?

't' typically represents time. It allows for the description of movement through space as it changes over time.

How to find velocity from parametric equations?

Velocity is the derivative of position with respect to time. For x(t) and y(t), compute dx/dt and dy/dt for the velocity vector (dx/dt, dy/dt).

Fill in the blank: The equation for horizontal motion is _____ and vertical is _____.

x(t) for horizontal; y(t) for vertical.

Compare circular motion: parametric vs polar.

Parametric: x(t) = r cos(t), y(t) = r sin(t). Polar: r(θ) directly describes radius as a function of angle.

Calculate the speed at t = 2 for x(t) = t², y(t) = t.

dx/dt = 2t, dy/dt = 1. At t = 2, speed = √((4)² + (1)²) = √17.

Determine the acceleration vector for x(t) = 3t², y(t) = 4t.

Acceleration: a(t) = (d²x/dt², d²y/dt²) = (6, 0).

True or False: Parametric equations can only represent curves.

False. They can represent lines, curves, and complex paths, depending on the functions.

Describe the role of the parameter in motion.

The parameter (often time) changes the coordinates, allowing for tracking the path and speed of an object.

What are parametric equations for an object moving in a straight line?

x(t) = vt + x₀, y(t) = constant. Represents uniform motion.

Evaluate the path at t = 1 for x(t) = 4t, y(t) = 3t + 1.

x(1) = 4, y(1) = 4. Path at t = 1 is (4, 4).

Fill in the blank: The curve traced by a particle is called its _____.

trajectory.

Find the coordinates at t = π for x(t) = cos(t), y(t) = sin(t).

x(π) = -1, y(π) = 0. Coordinates are (-1, 0).

Cause: Changing speed in parametric motion. Effect?

The path becomes steeper or flatter, altering the trajectory shape.

Differentiate parametric equations: x(t) and y(t)

Use $\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. This calculates the slope of the curve defined by the parameters t.

True or False: Area under a parametric curve is $\displaystyle \frac{1}{2} imes (y_1 + y_2) imes (x_2 - x_1)$

False. Area is given by $\displaystyle ext{Area} = \frac{1}{2} imes ext{integral}(y \frac{dx}{dt}) dt$.

Find the arc length of a parametric curve.

Arc length $\displaystyle L = ext{integral} \bigg( \frac{dx}{dt}^2 + \frac{dy}{dt}^2 \bigg)^{1/2} dt$.

Calculate area between parametric curves.

Area $\displaystyle A = ext{integral}(y_1 - y_2) dx$ evaluated between intersection points.

True or False: The derivative of $\displaystyle x(t) = t^2$, $\displaystyle y(t) = t^3$ is $\displaystyle 2t^2$.

False. $\displaystyle \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3}{2}t$.

Integrate the parametric curve: x(t)=t, y(t)=t^2 from 0 to 2.

$\displaystyle ext{Area} = ext{integral}(t^2 \frac{dx}{dt})dt$ from 0 to 2 yields 4.

Find arc length of $\displaystyle x(t)=t^3$, $\displaystyle y(t)=t^2$ from t=0 to t=1.

$\displaystyle L = ext{integral} \bigg( (3t^2)^2 + (2t)^2 \bigg)^{1/2} dt$ from 0 to 1.

Fill in the blank: The formula for area under a parametric curve is __________.

$\displaystyle A = ext{integral} y \frac{dx}{dt} dt$.

Differentiate \displaystyle \begin{cases} x = t^2 \ y = t^3 \\ ext{Find } \frac{dy}{dx} ext{ at } t=1 \\ ext{Answer:} \\ \frac{dy}{dt} = 3t^2, \frac{dx}{dt} = 2t

$\displaystyle \frac{dy}{dx} = \frac{3(1)^2}{2(1)} = \frac{3}{2}$.

True or False: The arc length is a function of both x and y.

True. Arc length depends on both $\displaystyle \frac{dx}{dt}$ and $\displaystyle \frac{dy}{dt}$.

What is the arc length of $\displaystyle x(t) = \frac{1}{2}t^2$, $\displaystyle y(t)=t$ from t=0 to t=2?

$\displaystyle L = ext{integral} \bigg( 1 + t \bigg)^{1/2} dt = 2 ext{ units}$.

Compare: Area and arc length for parametric equations.

Area measures the space enclosed; arc length measures the distance along the curve.

Find the area under the parametric curve $\displaystyle x(t) = t$, $\displaystyle y(t) = t^2$ from t=0 to t=1.

$\displaystyle A = ext{integral} t^2 \frac{dx}{dt} dt = \frac{1}{3} ext{ square units}$.

What does $\displaystyle \frac{dy}{dx}$ represent in parametric equations?

It represents the slope of the tangent line at any point on the parametric curve.

Differentiate the parametric equations $\displaystyle x(t) = an(t)$, $\displaystyle y(t) = an^2(t)$.

$\displaystyle \frac{dy}{dx} = \frac{2 an(t) ext{sec}^2(t)}{ ext{sec}^2(t)} = 2 an(t)$.

Fill in the blank: To find the length of a curve, we need to integrate __________.

$\displaystyle L = ext{integral} \bigg( \frac{dx}{dt}^2 + \frac{dy}{dt}^2 \bigg)^{1/2} dt$.

What are polar coordinates?

A point in polar coordinates is represented as (r, θ), where r is the radius and θ is the angle in radians.

Convert (3, π/4) to rectangular coordinates.

Using the formulas: x = r cos(θ) and y = r sin(θ), we get: x = 3√2/2, y = 3√2/2.

True or False: Polar equations can represent curves like circles.

True - For example, r = 2 represents a circle of radius 2 centered at the origin.

Identify the effect of increasing θ in r = 2 + sin(θ).

Increasing θ will create oscillations as r changes, producing a cardioid shape.

Fill in the blank: The area A enclosed by a polar curve r(θ) from θ=a to θ=b is given by ___

A = \frac{1}{2} imes ext{integral from a to b} (r(θ))^2 dθ.

What is the length of a polar curve?

The arc length L of a polar curve r(θ) from θ=a to θ=b is given by: L = ext{integral from a to b} \frac{ ext{sqrt}((dr/dθ)^2 + r^2)}{dθ}.

Compare r = 3 and r = 3 cos(θ).

r = 3 is a circle centered at the origin, while r = 3 cos(θ) is a circle centered at (1.5, 0).

If r = 4 + 2 sin(θ), what is r when θ = π/2?

r = 4 + 2 sin(π/2) = 4 + 2 = 6.

True or False: Polar coordinates are always positive.

False - r can be negative, which reverses the direction of the angle θ.

What is the polar equation of a limaçon?

The general form is r = a + b sin(θ) or r = a + b cos(θ) where a and b are constants.

Describe how to find the tangent line to a polar curve.

Use the slope formula: \frac{dy}{dx} = \frac{(dr/dθ) sin(θ) + r cos(θ)}{(dr/dθ) cos(θ) - r sin(θ)}.

When is a polar graph symmetric about the polar axis?

A polar graph is symmetric about the polar axis if replacing θ with -θ in the equation yields the same r.

How to determine the number of loops in r = 1 + 2 sin(θ)?

Count the number of times r becomes zero within the interval [0, 2π]. This one has 2 loops.

Find the area of the petal for r = 1 + cos(θ).

Area of one petal: A = \frac{1}{2} imes ext{integral from 0 to π} (1 + cos(θ))^2 dθ = \frac{3π}{4}.

What shape does r = 2 + 2 cos(θ) form?

This polar equation represents a limaçon with an inner loop.

Effect of r = 2 on a graph when θ increases.

As θ increases, r remains constant, creating a circle of radius 2 centered at the origin.

A car's position is modeled parametrically. What do x(t) and y(t) represent?

x(t) is the horizontal position, and y(t) is the vertical position of the car at time t.

True or False: Parametric equations can represent motion in two dimensions.

True. They allow for modeling paths in the x-y plane based on time.

Fill in the blank: The parametric equations for a car moving in a circle are ____.

x(t) = r cos(t), y(t) = r sin(t) for radius r.

A cyclist's motion is modeled as x(t) = 5t, y(t) = 3t^2. What is their speed?

Speed = $\displaystyle \frac{ds}{dt}$, where $\displaystyle ds = \frac{dx}{dt}^2 + \frac{dy}{dt}^2$. Calculate derivatives and evaluate.

True or False: Polar coordinates can depict a spiral pattern.

True. For example, r(θ) = θ shows a spiral increasing outward.

What is the area under a parametric curve from t=a to t=b?

Area = $\displaystyle \frac{1}{2} imes ext{integral from } a ext{ to } b (x(t) \frac{dy}{dt} dt)$.

What do you need to find the total distance traveled using parametric equations?

You need to compute the integral of the speed function from the start to end time.

A store's delivery truck follows a path defined by x(t) = 4t^2, y(t) = 2t. What is its position at t=3?

At t=3, x(3)=36 and y(3)=6. Position is (36, 6).

Compare polar and parametric equations in motion representation.

Polar focuses on radius and angle, while parametric uses time-dependent x and y coordinates.

True or False: The parametric equation x(t) = t^3, y(t) = t^2 represents a straight line.

False. It represents a curve since x and y are not linearly related.

What is the tangent line's slope to a parametric curve at t?

Slope = $\displaystyle \frac{dy/dt}{dx/dt}$, evaluated at t.

A roller coaster's height modeled as h(t) = -16t^2 + 64t. When is it 48 feet high?

Set h(t) = 48 and solve: -16t^2 + 64t - 48 = 0.

Fill in the blank: The polar equation r(θ) = 3 + 2 sin(θ) represents a ____.

limacon with an inner loop.

A boat travels in a circular path with radius 10 miles. What is its parametric equation?

x(t) = 10 cos(t), y(t) = 10 sin(t), for t in [0, 2π].

Find the velocity vector for x(t) = 4t, y(t) = 3t^2.

Velocity v(t) = (4, 6t).

The path of a basketball shot can be modeled. What can be used?

A parametric model with time-dependent x and y coordinates based on initial velocity and angle.