Flashcards in Calc 3 Deck (36)

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1

## Tangent Line

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

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

### Tangent Line

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

### two vectors are parallel

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## two vectors are parallel

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

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## Unit Tangent Formula

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

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

### Unit Tangent Formula

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## Principle Normal Vector

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

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

### Principle Normal Vector

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## Equation for osculating plane

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

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

### Equation for osculating plane

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## Arc Length Formula

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

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

### Arc Length Formula

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

### speed at time t

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## speed at time t

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

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## Curvature of a plane curve

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

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

### Curvature of a plane curve

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

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

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

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

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## Curvature of a space curve

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

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

### Curvature of a space curve

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## (ds/dt)

### velocity

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## velocity notation derivative

### (ds/dt)

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## aT = Tangential Acceleration

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

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

### aT = Tangential Acceleration

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## aN = Normal component acceleration

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

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

### aN = Normal component acceleration

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## 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

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### curvature of the path of acceleration

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## Directional Derivative

###
Gives the rate of change of F in the direction U.

(F) Gradient Vector * (U) unit Vector of Direction

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##
Gives the rate of change of F in the direction U.

(F) Gradient Vector * (U) unit Vector of Direction

### Directional Derivative

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## Gradient Vector

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

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

### Gradient Vector

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

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

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

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

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

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

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