 | Term | Definition |
| |
parametric equations |
x = f(t) and y = g(t) |
| |
smooth curve |
a curve where for x = f(t) and y = g(t), f' and g' are continuous and are not simultaneously 0, except possibly at the end points of some spec |
| |
piecewise smooth |
smooth on some subinterval |
| |
parametric form of the derivative |
dy/dx = (dy/dt)/(dx/dt), dx.dt ≠0 |
| |
parametric form of the second derivative |
d/dx [dy/dx] = d/dt (dy/dx) = d/dt (dy/dx)/(dx/dt) |
| |
arc length in parametric form |
s = ∫√((dx/dt)² + (dy/dt)²) dt= ∫√(f'(t)² + g'(t)²) dt |
| |
area of surface of revolution in parametric form with revolution about x-axis |
S = 2π∫ g(t) √(f'(t)² + g'(t)²) dt where x=f(t) and y=g(t) |
| |
area of surface of revolution in parametric form with revolution about y-axis |
S = 2π∫ f(t) √(f'(t)² + g'(t)²) dt where x=f(t) and y=g(t) |
| |
conversion from polar to rectangular coordinates |
x = rcosθ, y = rsinθ |
| |
conversion from rectangular to polar coordinates |
tanθ = y/x, r² = x² + y² |
| |
slope in polar form |
dy/dx = (dy/dθ)/(dx/dθ) = (f(θ)cosθ + f'(θ)sinθ)/(-f(θ)sinθ + f'(θ)cosθ) where dx/dθ≠0 |
| |
tangent lines at the pole |
If f(α) = 0 and f'(α) ≠ 0, then the line θ = α is tangent at the pole to the graph r = f(θ). |
| |
area in polar coordinates |
A = ½∫[f(θ)]²dθ = ½∫r²dθ where r=f(θ) and is nonnegative and continuous |
| |
arc length of a polar curve |
∫√([f(θ)]² + [f'(θ)]²)dθ = ∫√(r² + (dr/dθ)²)dθ |
| |
area of surface of revolution in polar coordinate form with revolution about polar axis |
S = 2π∫ f(θ)sinθ √(f(θ)² + f'(θ)²) dt where r = f(θ) |
| |
area of surface of revolution in polar coordinate form with revolution about about line θ=π/2 |
S = 2π∫ f(θ)cosθ √(f(θ)² + f'(θ)²) dt where r = f(θ) |
| |
eccentricity of an ellipse |
0 < e < 1 |
| |
eccentricity of a parabola |
e = 1 |
| |
eccentricity of a hyperbola |
e > 1 |
| |
polar equations for conics |
r = ed / ( 1±ecosθ) = ed / ( 1±esinθ) where e is the eccentricity and |d| is the distance between the focus at the pole and its corresponding directrix |
| |
minor axis length of an ellipse |
b² = a² - c² = a² - (ea)² = a²(1 - e²) |
| |
minor axis length of a hyperbola |
b² = c² - a²= (ea)² - a² = a²(e² - 1) |
Share on Facebook
Share on MySpace