average velocity and instantaneous velocity

Question

If the function  over the time interval [1,3] defines the movement of a particle along the x-axis, then

A) Find the average velocity of the particle on the time interval, and

B) Find the instantaneous velocity of the particle at each endpoint of the time interval.

Answer 1

distance traveled t=1 to t=3 is from -1 to -(1/9)

d=+8/9 length units for delta t=2 time units

v_ave=4/9 length units per time unit

d'(t)=v(t)=2t^-3

v(1)=2 length units/time unit

v(3)=2/27 length units/time unit

Answer 2

f(t) = -1/t^2

it helps to graph this, use a graphing calculator, handheld or on line

and you can see the slopes involved

f'(t) = 2/t^3

f'(1) = 2/1^3 = 2 = instantaneous rate of change at time t=1

f'(3) = 2/3^3 = 2/27 = instantaneous rate of charge at time t=3

(2-2/27)/(3-1) = (52/27)/2 = 26/27 = average rate of change from t=1 to 3

instantaneous rate of change = the slope of the tangent line

average rate of change = slope of the secant line

velocity = rate of change