VECTORS AND CALCULUS

Section 11-B

Vectors and Derivatives

If a smooth curve C is given by the equation

Then the slope of C at the point (x, y) is given by

Where and the second derivative is

given by :

)( and )( tgytfx

dtdxdt

dy

dx

dy

0dt

dx

dtdxdxdy

dtd

dx

dy

dx

d

dx

yd

2

2

Vectors and Derivatives

The derivative may be interpreted as the slope of the tangent line to the curve C, or as the slope of the path of a particle traveling along the curve C, or as the rate of change of y with respect to x.

The second derivative is the rate of change of the slope of the curve C with respect to x.

dx

dy

2

2

dx

yd

Velocity

Is the rate at which the x-coordinate is

changing with respect to t or the velocity of the particle in the horizontal direction.

Is the rate at which the y-coordinate is

changing with respect to t or the velocity of the particle in the vertical direction.

dt

dxtx '

dt

dyty '

Position Vector

Is the position vector at any time t.

Is the velocity vector at any time t.

Is the acceleration vector at any time t.

tytx ,

tytx ','

tytx '',''

Speed of the particle

Is the speed of the particle or the magnitude (length) of the velocity vector

Is the length of the arc (or arc length) of the curve from t = a to t = b or the distance traveled by the particle from t = a to t = b.

22

dt

dy

dt

dx

dtdt

dy

dt

dxb

a

22

1) A particle moves in the xy-plane so that at any time t, the position of the particle is given by

a) Find the velocity vector when t = 1

3423 )( ,4)( tttytttx

1) A particle moves in the xy-plane so that at any time t, the position of the particle is given by

b) Find the acceleration vector when t = 2

3423 )( ,4)( tttytttx

2) A particle moves in the xy-plane so that at any time t, t ≥ 0, the position of the particle is given by Find the magnitude of the velocity vector when t = 1

232 3)( ,3)( tttytttx

22

dt

dy

dt

dxm

dt

dy

dt

dxNote: The formula for the magnitude of the velocity vector is the same as the formula for the speed of the vector.

3) A particle moves in the xy-plane so that

The path of the particle intersects the x-axis twice. Write an expression that represents the distance traveled by the particle between the two x-intercepts. Do not evaluate

20 wheresin21 and ,cos43 ttytx

HOME WORKWorksheet: 11-B