Secant slope / Average rate of change (example)

Secant slope / Average rate of change

  • The secant slope between a and b is the average rate of change of f on [a,b].
  • Definition: Average rate of change=f(b)−f(a)b−a\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}

Given function and interval

  • Function: f(t)=15t−4.9t2f(t) = 15t - 4.9 t^2
  • Interval endpoints: a=0.5,b=2.5a = 0.5,\quad b = 2.5
  • This uses the formula with these endpoint values as described in the transcript.

Compute f at endpoints

  • f(2.5)=15(2.5)−4.9(2.5)2=37.5−30.625=6.875f(2.5) = 15(2.5) - 4.9(2.5)^2 = 37.5 - 30.625 = 6.875
  • f(0.5)=15(0.5)−4.9(0.5)2=7.5−1.225=6.275f(0.5) = 15(0.5) - 4.9(0.5)^2 = 7.5 - 1.225 = 6.275

Compute the secant slope

  • Difference in f: f(2.5)−f(0.5)=6.875−6.275=0.6f(2.5) - f(0.5) = 6.875 - 6.275 = 0.6
  • Difference in t: 2.5−0.5=22.5 - 0.5 = 2
  • Secant slope: f(2.5)−f(0.5)2.5−0.5=0.62=0.3\frac{f(2.5) - f(0.5)}{2.5 - 0.5} = \frac{0.6}{2} = 0.3
  • Interpretation: This is the average rate of change (average velocity) on the interval [0.5,2.5][0.5, 2.5].