Ranking Average Values of Functions

Question

Consider the functions given above on the interval $$[0,2]$$. Their average values are computed using the formula $$\frac{1}{2}\int_0^2 f(x)\,dx$$. Rank these functions in increasing order of their average value.
 
| Function          | Interval  | Average Value (Exact or Approx.)          |
|-------------------|-----------|-------------------------------------------|
| $$f(x)=e^{-2x}$$  | $$[0,2]$$ | $$\frac{1-e^{-4}}{4}\approx0.2454$$       |
| $$f(x)=e^{-x}$$   | $$[0,2]$$ | $$\frac{1-e^{-2}}{2}\approx0.4323$$       |
| $$f(x)=e^{-0.5x}$$| $$[0,2]$$ | $$\frac{1-e^{-1}}{1}\approx0.6321$$       |
| $$f(x)=1$$        | $$[0,2]$$ | $$1$$                                     |

Accepted Answer

$$e^{-2x}$$, $$e^{-x}$$, $$e^{-0.5x}$$, $$1$$

Answer

$$1$$, $$e^{-0.5x}$$, $$e^{-x}$$, $$e^{-2x}$$

Answer

$$e^{-x}$$, $$e^{-2x}$$, $$e^{-0.5x}$$, $$1$$

Answer

$$e^{-2x}$$, $$e^{-0.5x}$$, $$e^{-x}$$, $$1$$

Function	Interval	Average Value (Exact or Approx.)
$$f(x)=e^{-2x}$$	$$[0,2]$$	$$\frac{1-e^{-4}}{4}\approx0.2454$$
$$f(x)=e^{-x}$$	$$[0,2]$$	$$\frac{1-e^{-2}}{2}\approx0.4323$$
$$f(x)=e^{-0.5x}$$	$$[0,2]$$	$$\frac{1-e^{-1}}{1}\approx0.6321$$
$$f(x)=1$$	$$[0,2]$$	$$1$$

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