Math  /  Calculus

QuestionFind the average value of the following function over the given interval. Draw a graph of the function and indicate the average value. f(x)=8x on [3,3e]f(x)=\frac{8}{x} \text { on }[3,3 e]

Studdy Solution

STEP 1

1. The function given is f(x)=8x f(x) = \frac{8}{x} .
2. The interval of interest is [3,3e][3, 3e].
3. We need to find the average value of the function over this interval.

STEP 2

1. Recall the formula for the average value of a function over an interval.
2. Set up the integral for the average value calculation.
3. Evaluate the integral.
4. Calculate the average value.
5. Graph the function and indicate the average value.

STEP 3

The average value fˉ \bar{f} of a continuous function f(x) f(x) over the interval [a,b][a, b] is given by:
fˉ=1baabf(x)dx\bar{f} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx

STEP 4

Substitute the given function and interval into the average value formula:
fˉ=13e333e8xdx\bar{f} = \frac{1}{3e - 3} \int_{3}^{3e} \frac{8}{x} \, dx

STEP 5

Evaluate the integral:
33e8xdx=8[lnx]33e=8(ln3eln3)\int_{3}^{3e} \frac{8}{x} \, dx = 8 \left[ \ln|x| \right]_{3}^{3e} = 8 (\ln|3e| - \ln|3|)
Simplify the expression:
=8(ln3+lneln3)=8lne=8= 8 (\ln 3 + \ln e - \ln 3) = 8 \ln e = 8

STEP 6

Calculate the average value:
fˉ=13e3×8=83(e1)\bar{f} = \frac{1}{3e - 3} \times 8 = \frac{8}{3(e-1)}

STEP 7

To graph the function f(x)=8x f(x) = \frac{8}{x} over the interval [3,3e][3, 3e], plot the curve and draw a horizontal line at y=83(e1) y = \frac{8}{3(e-1)} to indicate the average value.
The average value of the function over the interval [3,3e][3, 3e] is:
83(e1) \boxed{\frac{8}{3(e-1)}}

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