Project A08 - The Mysterious Moving Particle
A particle is moving along a line so that at any time t, 0 £ t £ 6, it's velocity is given by v(t). The graph of v and a table of values
are shown below.
| t | 0.0 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 |
| v(t) | -0.50 | 0.25 | 1.00 | 0.25 | -0.50 | 0.25 | 1.00 |
- Find a 3rd degree polynomial p(t) that approximates v(t). Does it
seem reasonable to approximate v(t) by a 3rd degree polynomial? Explain your
reasoning.
- On the interval (0,2), the particle's position function s is a
differentiable function of the particle's velocity v. Approximate ds/dv
when t = 1. (Hint: This is a related rate problem.)
- Find open intervals on which p is increasing, decreasing, concave
up, and concave down.
- Sketch a detailed graph of p on [0,6]. Compare your graph to that
of v. How well does your polynomial approximate v?
- At what times does the particle stop? Use your polynomial p and
Newton's Method to approximate these times.
- Using the actual values of v(t), evaluate a Riemann sum that
approximates the particle's displacement over the interval [0,6].
- Use your polynomial p and the Fundamental Theorem of Calculus to
approximate the particle's displacement over the interval [0,6].
- Use a definite integral to approximate the average velocity of the
particle over the interval [0,6].
- Use a definite integral to approximate the average speed of the
particle over the interval [0,6]. Set up the integral but use your calculator
to evaluate it.
- Use the actual values of v(t) and the Trapezoid Rule to approximate
the overall distance traveled by the particle over the interval [0,6].
File translated from TEX by TTH, version 1.98.
On 24 Sep 1999, 22:18.