16 terms

408D Chapter 10 (10.1-10.5)

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how do you sketch a parametric curve
create a table of values with t, x, and y and then graph

MAKE SURE TO SHOW DIRECTION WITH ARROWS from initial point to terminal point
how to eliminate the parameter
solve for t in one equation and then sub that into the other
what is the equation for a circle centered at zero?
x^2 + y^2 = value

radius = value/2
how do you eliminate the parameter with an equation involving the square of a cos function plus the square of the sin function?
set cos^2t+ sin^2t=1 equal to x^2 + y^2 =1
what is dy/dx when there is a third variable t
dy/dx= (dy/dt)/(dx/dt) if dx/dt ≠0
formula for arc length l
L= ∫(from α to β) √(dx/dt)² + (dy/dt)² dt
how do you find when the slope of parametric equation is horizontal/vertical
horizontal: find where dy/dx=0, so where dy/dt=0 but dx/dt≠0
vertical : find where dx/dt=0 but dy/dt≠0
what do polar coordinates (r, θ) mean
theta is the angle from the polar axis, r is radius, or how far you go out from center
how do you convert from cartesian/polar and vice versa
use
x=rcosθ
y=rsinθ
x²+y²=r²
y/x=tanθ
how to sketch polar curves
can make a table of values with theta and r and then plot points
how to find points of intersection of polar equations
draw graphs
set equations equal
solve for thetas
check if pole is on both graphs
area inside a polar curve
a=∫from α to β ½(f(θ))² dθ = ∫from α to β ½(r)² dθ
find area between two polar curves
integral from alpha to beta 1/2 r1 squared - integral 1/2 r2 squared
equation for parabola
x²=4py
y²=4px
*axis of symmetry is axis of degree one variable
*focus: (0,p) or (p,0)
*directrix: y=-p or x=-p
equation for ellipse
x²/a² + y²/b² = 1
x²/b² + y²/a² = 1

*major axis is on axis of variable with larger denom (a)
*a≥b>0
*c²=a²-b²
*foci determined by c
*major verticies determined by a, minor by b
equation for hyperbola
x²/a² - y²/b² =1
y²/a² - x²/b² =1

*foci and verts on positive variable's axis
*c²=a²+b²
*verts determined by a
*foci determined by c
*slope of asymptotes: y=a/b or y=b/a