Calc 3 Flashcards Preview

Calculus > Calc 3 > Flashcards

Flashcards in Calc 3 Deck (36):
1

Tangent Line

A line that touches a curve at only 1 point and determines the instantaneous rate of change

2

A line that touches a curve at only 1 point and determines the instantaneous rate of change

Tangent Line

3

r(t) = c(r'(t))

two vectors are parallel

4

two vectors are parallel

r(t) = c(r'(t))

5

Unit Tangent Formula

T(t) = r'(t) / ||r'(t)|| (tangent vector divided by magnitude of tangent vector)

6

T(t) = r'(t) / ||r'(t)|| (tangent vector divided by magnitude of tangent vector)

Unit Tangent Formula

7

Principle Normal Vector

N(t) = T'(t) / ||T'(t)|| ( derivative of unit tangent divided by derivative unit tangent magnitude)

8

N(t) = T'(t) / ||T'(t)|| ( derivative of unit tangent divided by derivative unit tangent magnitude)

Principle Normal Vector

9

Equation for osculating plane

N(t) x T(t) = cross product of Principle Normal Vector and Unit Tangent

10

N(t) x T(t) = cross product of Principle Normal Vector and Unit Tangent

Equation for osculating plane

11

Arc Length Formula

Intergral of : square root ( x'(t) + y'(t) + z'(t) )

12

Intergral of : square root ( x'(t) + y'(t) + z'(t) )

Arc Length Formula

13

||v(t)|| = ||r'(t)||

speed at time t

14

speed at time t

||v(t)|| = ||r'(t)||

15

Curvature of a plane curve

k = |y''| / ( 1 + (y')^2 ) ^ 3/2

16

k = |y''| / ( 1 + (y')^2 ) ^ 3/2

Curvature of a plane curve

17

r(t) = x(t) + y(t) = given vector function curvature is

k = |x'y'' - y'x''| / ( (x')^2 + (y')^2 )^ 3/2

18

k = |x'y'' - y'x''| / ( (x')^2 + (y')^2 )^ 3/2

r(t) = x(t) + y(t) = vector function curvature

19

Curvature of a space curve

k = ||dT/dt|| / (ds/dt)

20

k = ||dT/dt|| / (ds/dt)

Curvature of a space curve

21

(ds/dt)

velocity

22

velocity notation derivative

(ds/dt)

23

aT = Tangential Acceleration

T dot a(t) = depends only on the change of speed

24

T dot a(t) = depends only on the change of speed

aT = Tangential Acceleration

25

aN = Normal component acceleration

|| T x a(t) || = depends on speed and curvature

26

|| T x a(t) || = depends on speed and curvatur

aN = Normal component acceleration

27

curvature of the path of acceleration

k = ||v(t) x a(t)|| / (ds/dt)^3 = cross product magnitude of velocity and acceleration divided by velocity ^ 3

28

k = ||v(t) x a(t)|| / (ds/dt)^3 = cross product magnitude of velocity and acceleration divided by velocity ^ 3

curvature of the path of acceleration

29

Directional Derivative

Gives the rate of change of F in the direction U.
(F) Gradient Vector * (U) unit Vector of Direction

30

Gives the rate of change of F in the direction U.
(F) Gradient Vector * (U) unit Vector of Direction

Directional Derivative

31

Gradient Vector

1st Partial Derivative in respect to (x, y, z)

32

1st Partial Derivative in respect to (x, y, z) forms what vector?

Gradient Vector

33

|| vT(x, y) || = Magnitude of the Directional Derivative implies

Rate of change at (x, y) (rate of increase at (x, y))

34

Rate of change at (x, y) (rate of increase at (x, y))

|| vT(x, y) || = Magnitude of the Directional Derivative

35

- || vT(x, y) || = Magnitude of the Directional Derivative negative implies

Rate of change (x, y) (rate of decrease at (x, y))

36

Rate of change (x, y) (rate of decrease at (x, y))

- || vT(x, y) || = Magnitude of the Directional Derivative negative