Consider the function #f(x) = (x)/(x+2)# on the interval [1, 4], how do you find the average or mean slope of the function on this interval?

Question

Paisley Johnson · Accepted Answer

The mean slope is #=1/9#

The average or mean slope of the function #y=f(x)# on an interval #[x_1,x_2]# is

#(Deltay)/(Deltax)=(f(x_2)-f(x_1))/(x_2-x_1)#

Here,

#y=f(x)=x/(x+2)#

And the interval is #=[1,4]#

Therefore,

#f(1)=1/(1+2)=1/3#

#f(4)=4/(4+2)=4/6=2/3#

So,

#(Deltay)/(Deltax)=(f(4)-f(1))/(4-1)=(2/3-1/3)/(4-1)=1/3*1/3=1/9#

Emily Ammerman · Answer

undefined To find the average or mean slope of the function $ f(x) = \frac{x}{x + 2} $ on the interval $[1, 4]$, you can use the formula for the average rate of change or slope.

The average slope of a function over an interval $[a, b]$ is given by:

$$ \text{Average slope} = \frac{f(b) - f(a)}{b - a} $$

Substitute $ a = 1 $ and $ b = 4 $, and compute $ f(1) $ and $ f(4) $:

$$ f(1) = \frac{1}{1 + 2} = \frac{1}{3} $$
$$ f(4) = \frac{4}{4 + 2} = \frac{4}{6} = \frac{2}{3} $$

Now, substitute these values into the formula for average slope:

$$ \text{Average slope} = \frac{\frac{2}{3} - \frac{1}{3}}{4 - 1} = \frac{\frac{1}{3}}{3} = \frac{1}{9} $$

Therefore, the average or mean slope of the function $ f(x) = \frac{x}{x + 2} $ on the interval $[1, 4]$ is $ \frac{1}{9} $.