Slide 1

Let c (t) = (t 2 + 1, t 3 4t). Find: (a) An equation of the tangent line at t = 3 (b) The points where the tangent is horizontal. ` ` `

Slide 2

THEOREM 1 Arc Length Let c(t) = (x(t), y(t)), where x (t) and y (t) exist and are continuous. Then the arc length s of c(t) for a t b is equal to

Slide 3

The simplest parametrization of y = f (x) is c (t) = (t, f (t)). Which leads to the arc length formula derived in Section 9.1.

Slide 4

The arc length integral can be evaluated explicitly only in special cases. The circle and the cycloid are two such cases. Use THM 1 to calculate the arc length of a circle of radius R.

Slide 5

Calculate the length s of one arch of the cycloid generated by a circle of radius R = 2.

Slide 6

Speed is defined as the rate of change of distance traveled with respect to time, so by the 2 nd Fundamental Theorem of Calculus, THEOREM 2 Speed Along a Parametrized Path The speed of c (t) = (x (t), y (t)) is

Slide 7

The next example illustrates the difference between distance traveled along a path and displacement (also called net change in position). The displacement along a path is the distance between the initial point c (t 0 ) and the endpoint c (t 1 ). The distance traveled is greater than the displacement unless the particle happens to move in a straight line.

Slide 8

A particle travels along the path c (t) = (2t, 1 + t 3/2 ). Find: (a) The particles speed at t = 1 (assume units of meters and minutes).

Slide 9

A particle travels along the path c (t) = (2t, 1 + t 3/2 ). Find: (a) The particles speed at t = 1 (assume units of meters and minutes). (b) The distance traveled s and displacement d during the interval 0 t 4.

Slide 10

In physics, we often describe the path of a particle moving with constant speed along a circle of radius R in terms of a constant (lowercase Greek omega) as follows: c (t) = (R cos t, R sin t) The constant , called the angular velocity, is the rate of change with respect to time of the particles angle . A particle moving on a circle of radius R with angular velocity has speed ||R.

Slide 11

Angular Velocity Calculate the speed of the circular path of radius R and angular velocity . What is the speed if R = 3 m and = 4 rad/s? Thus, the speed is constant with value ||R. If R = 3 m and = 4 rad/s, then the speed is ||R = 3(4) = 12 m/s. c (t) = (R cos t, R sin t)

Slide 12