Question

Write a difference quotient that best approximates the instantaneous rate of change of g at x=0

Please show how you got your answer

Answer 1

We can define the average rate of change of a function f(x) in an interval [a, b] as:

$r = \frac{f(b) - f(a)}{b - a}$

Particularly, if we want to find an approximation of the rate of change for a particular value x', we can choose:

a < x' < b

Such that a and b are really close to x'.

Then if we want to find the instantaneous rate of change of g at x = 0, we need to find values of a and b really close to zero.

Looking at the table, we can choose:

a = -0.001

b = 0.001

(are the two closer ones to zero)

Then the rate of change can be written as:

$f(x) = \frac{g(0.001) - g(-0.001)}{0.001 - (-0.001)} = \frac{2.001 - 1.999}{0.002} = 1$

From that, we can conclude that a good approximation of the instantaneous rate of change is 1.

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