what is the average rate of change from x=3 to x=5 for the function g(x)=log(x+1)

Answer 1

The average rate of change for g(x) is { g(x+h)-g(x) }/h

In our case,

{ log(5+1) - log(3+1) }/(5-3) = (.778-.602)/2 = .088

Compare it to the derivative of log(x+1) for the values: the derivative is D = (1/ln10)(1/(x+1))

at x = 3 D = .1086

at x = 4 D = .0869

at x = 5 D = .0724

Take the average of D over the interval and you get .0893 and very close to the midpoint of the x values with .0869 !