Exercises on Average of a function

Recall that if f is a continuous function on the interval [a,b] then the average

value of f on this interval is given by the formula


  1. Consider that the temperature in Edmonton on a given 24 hour period last summer was given by the formula:
    where $t$ is measured in hours, with $t=0$ being midnight.

    1. What is the coldest termperature occurring in that 24 hour period, and at

    2. what time does it occur?

    3. Compute the termperature every hour on the hour from 3pm ($t=15$) to 9pm ($t=21$), keeping 5 digits of accuracy. Report the average of these 7 discrete values. [Hint: It may help to use Maple to define $T$ as a function (see above) first; then you can sum up values of this function.]

    4. Use the formula for the average of a continuous function to compute the average value of the temperature function $T(t)$ over the same time interval as in part (c).

  2. Plot each of the following functions on the given intervals. Find the largest and the smallest value of the curve on the interval and average the two. Next, take five equally spaced intervals and average these. Lastly, compute the average over the interval using equation. Arguing from the shapes of the curves, can you explain any discrepencies between these different techniques?

    1. $g(x)=1/x$ on the interval $[1,5]$.

    2. $f(x)=\sin 4(x)$ on the interval MATH.

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