Question

Determine the average rate of change of the function g(x) = 1/x

from x = 1 to x = h,
with h > 1. Simplify your answer as much as you can.

Answer 1

The average rate of change of the function g(x) = 1/x from x = 1 to x = h, where h > 1, is (1/h) - 1.

Step-by-step explanation:

To determine the average rate of change of the function g(x) = 1/x from x = 1 to x = h, where h > 1, we use the formula for the average rate of change of a function over the interval [a,b]:

Average rate of change = ∆g / ∆x = (g(b) - g(a)) / (b - a)

In this case, g(a) = g(1) = 1/1 = 1 and g(b) = g(h) = 1/h. Thus, the average rate of change from x = 1 to x = h is:

(1/h - 1) / (h - 1) = (1 - h) / h(h - 1)

We can simplify this expression by dividing top and bottom by h:

(1/h - 1) / (h - 1) = (1/h - h/h) / h(h - 1) = (1 - h) / (h^2 - h)

Further simplify by factoring out -1:

(h - 1) / (h(h - 1)) = 1 / h - 1, after canceling out the (h - 1) term.

Answer 2

`-(1)/(h)`

Step-by-step explanation:

g(x) = 1/x

The average rate of change of function f(x) from a to b is

`(f(b) - f(a))/(b - a)`

Here, we have function g(x) = 1/x.

a = 1

b = h

average rate of change from 1 to x is:

`(g(h) - g(1))/(h - 1) =`

`= ((1)/(h) - (1)/(1))/(h - 1)`

`= (h((1)/(h) - (1)/(1)))/(h(h - 1))`

`= (1 - h)/(h(h - 1))`

`= (-1(h - 1))/(h(h - 1))`

`= -(1)/(h)`