Question

The average rate of change from x=2 to x=5

Answer 1

4/3 from x = 2 to x = 5.

Explanation:

What is Lagrange mean value theorem?

Lagrange mean value theorem states that, if a function f is continuous over the closed interval [a, b] and differentiable over the open interval (a, b), then there must be at least one point c in the interval (a, b) where the slope of the tangent at the point c is equal to the slope of the tangent through the curve's endpoints, resulting in the expression f'(c) = {F(b) -F(a)}/(b-a)

According to the given graph,

At point x = 2,

F(2) = 3

At point x = 5,

F(5) = 7

Since the formula for the average rate of change of the function between x = a and x = b is,

The average rate of change = {F(b) -F(a)}/(b-a)

Here a = 2, b = 5 and F(2) = 3, F(5) = 7

Substitute the values in the formula,

So the average rate of change = (7 - 3)/(5 - 2) = 4/3.

Hence, the average rate of change of the function is 4/3.

Answer 2

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\\\\ \begin{cases} x_1=2\\ x_2=5 \end{cases}\implies \cfrac{f(5)-f(2)}{5 - 2}\implies \cfrac{7-3}{5-2}\implies \cfrac{4}{3}`