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

### 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