Finding Average Velocity

Question

I'm having a hard time wrapping my head around this, so any assistance is greatly appreciated.

The problem is

"In a time of t seconds, a particle moves a distance of s meters from its starting point, where s = t² + t + 8

Find the average velocity between t = 3 and t = 3 + h if (a) h = 0.1 (b) h = 0.01 (c) h = -0.1 (d) h = -0.01

Based on the above results, estimate the instantaneous velocity of the particle at t = 3"

Thanks in advance for the assistance!

Answer 1

Average velocity is found by taking the slope of the distance function between two points. In this case, we are finding the slope between (3, s(3)) and (3 + h, s(3 + h)). Since there are 4 h's, we are finding 4 different slopes:

1. (3, s(3)) and (3.1, s(3.1))
2. Slope = 7.100
3. (3, s(3)) and (3.01, s(3.01))
4. Slope = 7.01
5. (3, s(3)) and (2.9, s(2.9))
6. Slope = 6.9
7. (3, s(3)) and (2.99, s(2.99))
8. Slope = 6.99

As (3 + h, s(3 + h)) gets closer and closer to (3, s(3)), the slope appears to be approaching 7 exactly. Therefore, the instantaneous velocity of the particle at t = 3 is 7 m/s

Hope this was helpful.