Unit 9 calc parametric/vector/polar

What is a parametric equation?

an equation with 3 variables (usually t, x, and y)

What do you do to solve a parametric equation?

Get t to cancel by substitution with x and y

How to find the first and second derivative of a parametric equation?

dy/dx = (dy/dt)/(dx/dt) d²y/dx² = (d/dt(dy/dx))/(dx/dt)

What is the equation for parametric arc length?

integral of the sqrt (dx/dt)² + (dy/dt)² (remember chain rule and u sub)

How to solve vector valued functions?

magnitude: sqrt of x² + y² use unit circle and Pythagorean theorem to find direction and value

What is the equation for position, velocity, acceleration, speed, and distance for vector valued functions?

< x(t), y(t)> <x’(t), y’(t)> <x’’(t), y’’(t)> speed is the magnitude of the velocity, so just the square root of (x’(t)² + y’(t)²) distance: integral of the speed which is the integral of sqrt (x’(t)² + y(t)²)

What is the equation for going from rectangular to polar?

x = rcos(theta) y = rsin(theta)

What are polar points?

instead of (x,y), they are in (r, theta) distance from the origin and angle

What equations can you use to convert from polar to rectangular?

x = rcos(theta) y = rsin(theta) r² = x² + y² tan(theta) = y/x

How to do a derivative of a polar equation?

convert to parametric (dy/dx) using the x=rcos(theta) and y = rsin(theta). The actual equation is dy/dx = r’sin(theta) +rcos(theta)/r’cos(theta) +rsin(theta) - product rule

What is the equation for area of a polar curve?

A = integral of 1/2r² - limits of integration are the angles of the graph (thetas) (plug in 0 for r if it starts at origin)

How do you find area between polar curves?

Solve the equation for polar area, but either add or subtract sections to get the area in between (kinda like rectangular area = integral of f(x) - g(x))