Math 253 PSU LP 9 - 11.2 Arc Length and Speed

1. Prove that the arc length of a continuous parametric curve given by c(t) = (f(t), g(t)) fromt = a to t = b is

s =

b∫a

√(f ′(t))2 + (g′(t))2dt.

2. Find the length of the path (t3 + 1, t2 − 3) over 0 ≤ t ≤ 1.

3. Find the length of the path given by (3 sin(2t), 3 cos(2t)) over the interval 0 ≤ t ≤ π/2 usingthe arc length formula learned in this lesson. Confirm your result using the fact that the pathgiven traces out a circle of radius 3.

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4. Suppose an object is moving along the path (ln(t2 + 1), t3), whose coordinates are given inmeters and t is given in seconds. Find its speed when t = 1.

5. Suppose an object is moving along the path (et cos(t), et sin(t)), whose coordinates are givenin meters and t is given in seconds. What is the shape of this path? What is the speed of theobject when t = π?

6. What is the minimums speed of a particle with trajectory c(t) = (t3, t−2) for t ≥ 0.5?Suppose the coordinates are given in meters and t in seconds.

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