Calc Midterm

Formulas to know

Curvature Formula (not effiecent)

(x’(t)y’’(t)-x’’(t)y’(t))/(x’(t)²+y’(t)²)^(3/2)

Efficient Curvature Formula

|r’(t)X r’’(t)|/|r’(t)|³

Area Under Polar Curve

A = (1/2) ∫[a, b] r^2 dθ

Cartesian to Cylinderical Conversions

x=rcos(theta)sin(phi)

y=rsin(theta)sin(phi)

z=rcos(phi)

Second Derivative of Parametric with Respect to x

(x’(t)y’’(t)-x’’(t)y’(t))/x’(t)³

Projection of u onto v

((u*v)/(v*v))v

T(t)

r’(t)/|r’(t)|

N(t)

T’(t)/|T’(t)|

Decomposition of Acceleration

a=a(t)T+a(n)N

a(t)=v’(t)

a(n)=K(t)v(t)²

Point to Line Distance

u=(r-p)-((r-p)*v)v

r= start point

p= point on line

v= directional vector of line

|u| = distance

Point to Plane Distance

u=((p-r)*n)n

r= start point

p= point on plane

n= normal vector

|u| = distance