Final answer:
The difference quotient used to approximate the instantaneous rate of change of a function at a specific point involves the slope of the tangent to the function's curve at that point, calculated as (g(x+h) - g(x)) / h where h approaches zero.
Step-by-step explanation:
To approximate the instantaneous rate of change of a function g at x = 2.5, we can use the difference quotient. The difference quotient is represented by the formula:
f(x+h) - f(x) over h, where h approaches zero.
In this context, for the function g, we'd write g(2.5+h) - g(2.5) over h. To approximate the rate of change, h should be a small number. As h gets smaller, the quotient approaches the slope of the tangent to the curve at x = 2.5, which represents the instantaneous rate of change.
For example, if we have the graph of g or it’s defined by an algebraic expression, we can evaluate g(2.5+h) and g(2.5) for a small value of h, and compute the quotient.