Average velocity calculus questions

Question

An object moves on a line so that its position in meters at time t seconds is s(t) = sin(t).

a) Find the average velocity over the time interval [1, 1+h]

b) Estimate the instantaneous velocity when t = 1

Answer 1

Average velocity is given as [f(1+h) - f(1)]/[(1+h)-1] or, plugging in the function and simplifying, [sin(1+h) - sin(1)]/h

Note that since you can pick any value for h, you can't get an actual number here.

The instantaneous velocity is the limit of this as h approaches 0. To estimate it, plug this into your calculator and make h get smaller and smaller (closer to zero) to see what it converges on.

Here are values I got:

for h = 0.1: vel = 0.49736 m/s

for h = 0.05: vel = 0.51904 m/s

for h = 0.01: vel = 0.53609 m/s

for h = 0.005: vel = 0.5382 m/s

for h = 0.001: vel = 0.5399 m/s

for h = 0.0005: vel = 0.54009 m/s

for h = 0.0001: vel = 0.54026 m/s

for h = 0.00005: vel = 0.54028 m/s

for h = 0.00001: vel = 0.54029 m/s

for h = 0.000005: vel = 0.5403 m/s

for h = 0.000001: vel = 0.5403 m/s

So the instantaneous velocity is 0.05403 m/s