Difference Quotient
The difference quotient is a mathematical formula used to find the average rate of change of a function over a given interval
The difference quotient is a mathematical formula used to find the average rate of change of a function over a given interval. It is represented as:
f'(x) = [f(x + h) – f(x)] / h
In this formula, f(x) represents the given function, f'(x) represents the derivative of the function, h represents the change in x-values or the size of the interval, and [f(x + h) – f(x)] represents the change in the function’s values over that interval.
The difference quotient can be thought of as measuring the slope of a secant line that passes through two points on the graph of a function. By letting h approach zero, the difference quotient becomes an instantaneous rate of change, which gives us the derivative of the function at a specific point.
For example, let’s say we have a function f(x) = x^2 and we want to find its derivative at x = 3 using the difference quotient. We can substitute these values into the formula:
f'(x) = [f(x + h) – f(x)] / h
f'(3) = [(3 + h)^2 – 3^2] / h
Expanding the squares and simplifying, we get:
f'(3) = [(9 + 6h + h^2) – 9] / h
f'(3) = (6h + h^2) / h
f'(3) = 6 + h
As h approaches zero, the derivative at x = 3 becomes f'(3) = 6.
In summary, the difference quotient is a useful tool in calculus to find the average rate of change of a function over an interval, and it can be used to calculate the derivative of a function at a specific point.
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