Question

What is the average rate of change of the function -4≤x≤-3

Answer 1

Check the picture below.

`\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ f(x) \qquad \begin{cases} x_1=-4\\ x_2=-3 \end{cases}\implies \cfrac{f(-3)-f(-4)}{-3 - (-4)}\implies \cfrac{[-6]~~ - ~~[-4]}{-3+4} \\\\\\ \cfrac{-6~~ + ~~4}{1}\implies \text{\LARGE -2}`

Answer 2

-2

`\hrulefill`

Explanation:

The average rate of change of a function f(x) on the interval a ≤ x ≤ b can be calculated using the formula:

`\boxed{\textsf{Average rate of change}=(f(b)-f(a))/(b-a)}`

The given interval is -4 ≤ x ≤ -3, so:

• 
  `a=-4`

• 
  `b=-3`

From observation of the given graph, the values of f(a) and f(b) are:

`f(a)=f(-4)=-4`

`f(b)=f(-3)=-6`

Substitute the values of a, b, f(a) and f(b) into the formula to calculate the average rate of change of the function f(x) on the interval -4 ≤ x ≤ -3:

`\begin{aligned}\textsf{Average rate of change}&=(f(-3)-f(-4))/(-3-(-4))\\\\&=(-6-(-4))/(-3-(-4))\\\\&=(-6+4)/(-3+4)\\\\&=(-2)/(1)\\\\&=-2\end{aligned}`

Therefore, the average rate of change of the function f(x) on the given interval is -2.