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. ` ` `

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

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.

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.

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

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

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.

A particle travels along the path c (t) = (2t, 1 + t 3/2 ). Find: (a) The particle’s speed at t = 1 (assume units of meters and minutes).

A particle travels along the path c (t) = (2t, 1 + t 3/2 ). Find: (a) The particle’s speed at t = 1 (assume units of meters and minutes). (b) The distance traveled s and displacement d during the interval 0 ≤ t ≤ 4.

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 ω (lowercase Greek omega) as follows: c (t) = (R cos ωt, R sin ωt) The constant ω, called the angular velocity, is the rate of change with respect to time of the particle’s angle θ. A particle moving on a circle of radius R with angular velocity ω has speed |ω|R.

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