Average of a Function

Calculus Level 3

Which of the following is the best estimate for the average value of the function f ( x ) = x − 2 e 1 x f(x)= x^{-2}e^{\Large\frac{1}{x}} on the interval [ 1 , 3 ] [1,3] ?

0.675 0.661 0.683 0.644

Hana Wehbi by Hana Wehbi , 51, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

The average value of f ( x ) f(x) over the interval [ 1 , 3 ] [1,3] is given by:

μ = ∫ 1 3 f ( x ) d x 3 − 1 = 1 2 ∫ 1 3 x − 2 e 1 x d x Since d d x e 1 x = − x − 2 e 1 x = − 1 2 [ e 1 x ] 1 3 = 1 2 ( e − e 3 ) ≈ 0.661 \begin{aligned} \mu & = \frac {\int_1^3 f(x) \ dx}{3-1} \\ & = \frac 12 \int_1^3 x^{-2}e^\frac 1x \ dx \quad \quad \small \color{#3D99F6}{\text{Since }\frac d{dx} e^\frac 1x = - x^{-2}e^\frac 1x} \\ & = - \frac 12 \left[e^\frac 1x\right]_1^3 \\ & = \frac 12 \left( e - \sqrt[3] e \right) \\ & \approx \boxed{0.661} \end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes, that was a nice solution, got my vote.

Hana Wehbi - 4 years, 11 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...