Calculate the approximation

Question

Calculate the approximation 𝑀4 for the given function and interval. 𝑓(𝑥)=3^−𝑥,[1,3]

Answer 1

M4 is the midpoint approximation with 4 bins.

The interval is[1,3]

Binwidth is (3-1)/4 = .5

Midpoint of each bin is .5/2= .25 to the right of the left margin.

Thus, we calculate

.5*(f(1.25)+f(1.75)+f(2.25)+f(2.75)) =

.5*(3^-1.25 +3^-1.75+3^-2.25+3^-2.75) =

0.2663

Not asked for, but ∫₁³ 3^-x dx = -1/ln(3) 3^-x |₁³ =

-1/ln(3)(1/27 - 1/3) =

8/(27 ln(3)) =

.269701