Calc 3 Flashcards
Tangent Line
A line that touches a curve at only 1 point and determines the instantaneous rate of change
A line that touches a curve at only 1 point and determines the instantaneous rate of change
Tangent Line
r(t) = c(r’(t))
two vectors are parallel
two vectors are parallel
r(t) = c(r’(t))
Unit Tangent Formula
T(t) = r’(t) / ||r’(t)|| (tangent vector divided by magnitude of tangent vector)
T(t) = r’(t) / ||r’(t)|| (tangent vector divided by magnitude of tangent vector)
Unit Tangent Formula
Principle Normal Vector
N(t) = T’(t) / ||T’(t)|| ( derivative of unit tangent divided by derivative unit tangent magnitude)
N(t) = T’(t) / ||T’(t)|| ( derivative of unit tangent divided by derivative unit tangent magnitude)
Principle Normal Vector
Equation for osculating plane
N(t) x T(t) = cross product of Principle Normal Vector and Unit Tangent
N(t) x T(t) = cross product of Principle Normal Vector and Unit Tangent
Equation for osculating plane
Arc Length Formula
Intergral of : square root ( x’(t) + y’(t) + z’(t) )
Intergral of : square root ( x’(t) + y’(t) + z’(t) )
Arc Length Formula
||v(t)|| = ||r’(t)||
speed at time t
speed at time t
||v(t)|| = ||r’(t)||
Curvature of a plane curve
k = |y’’| / ( 1 + (y’)^2 ) ^ 3/2
k = |y’’| / ( 1 + (y’)^2 ) ^ 3/2
Curvature of a plane curve
r(t) = x(t) + y(t) = given vector function curvature is
k = |x’y’’ - y’x’’| / ( (x’)^2 + (y’)^2 )^ 3/2
k = |x’y’’ - y’x’’| / ( (x’)^2 + (y’)^2 )^ 3/2
r(t) = x(t) + y(t) = vector function curvature
Curvature of a space curve
k = ||dT/dt|| / (ds/dt)
k = ||dT/dt|| / (ds/dt)
Curvature of a space curve
(ds/dt)
velocity
velocity notation derivative
(ds/dt)
aT = Tangential Acceleration
T dot a(t) = depends only on the change of speed
T dot a(t) = depends only on the change of speed
aT = Tangential Acceleration