CPM Homework Help : APCALC Problem 6-154

For each part below, calculate the average value of $g(t)$ over the given interval. You may solve analytically or use your graphing calculator to evaluate.

$g(t) = 2t^2 – 3$ over $[0, 4]$
$g(t) = 3e^t + \ln(t + 1)$ over $[0, 1]$

Hint:

Examine the Math Notes box about about Mean (average) Value. Notice that there are two strategies to compute Mean (average) Value.

Strategy 1: Compute the average area under the curve.
Strategy 2: Compute a slope of a secant line.

WARNING: These strategies can NOT be used interchangeably. The strategy you choose depends on the function you are given:
$f(x)$ or $F(x)$ .

And it depends on the question you are being asked:
Find average y-value? or Find average rate of change?

Average (Mean) Values

To calculate the mean (average) value of a finite set of items, add up the values of items and divide by the number of items.

Integrals help us add over a continuous interval. Therefore, for any continuous function $f$ :

$\frac{\int_a^bf(x)dx}{b-a}=$ mean value of $f$ over $[a, b]$

First quadrant, bell curve labeled, f of x, left end point on the y axis, labeled, a, right end point labeled, b, dashed horizontal segment, about 1 fourth up from x axis to peak, labeled average, & shaded rectangle between, A & b, segment & x axis.

Since $\int _ { a } ^ { b } f ( x ) d x$ , we can also calculate the average value of any function $f$ using its antiderivative $F$ . Its average slope gives the average rate of change of $F$ , which is the same as the average value of $f.$

$\frac{\int_a^bf(x)dx}{b-a}=\frac { F ( b ) - F ( a ) } { b - a }=$ mean rate of change of $F ′$ over $[a, b]$

First quadrant, 2 tick marks on x axis, first at the origin labeled, A, second almost to the right end, labeled b, Increasing curve labeled, capital F of x, starting at the origin, changing from concave up to concave down, in center of quadrant, ending at point corresponding to, b, almost at the top, with dashed segment labeled, m = average, from origin to end point of curve.