Understanding the Difference Quotient | Calculating Rate of Change in Calculus

Difference Quotient

The difference quotient is a mathematical concept used in calculus to calculate the rate of change of a function over a given interval

The difference quotient is a mathematical concept used in calculus to calculate the rate of change of a function over a given interval. It measures how a function changes as the input changes.

Mathematically, the difference quotient is expressed as:

$$\frac{{f(x+h) – f(x)}}{h}$$

where \(f(x)\) is the given function and \(h\) represents the size of the interval over which we want to calculate the rate of change. The difference quotient essentially measures the average rate of change of the function between \(x\) and \(x+h\).

To understand this better, let’s consider an example. Suppose we have a function \(f(x) = x^2\) and we want to calculate the difference quotient for a specific value of \(x\) and \(h\). Let’s choose \(x = 2\) and \(h = 0.5\).

Plugging these values into the difference quotient formula, we get:

$$\frac{{f(2+0.5) – f(2)}}{0.5} = \frac{{f(2.5) – f(2)}}{0.5}$$

Now, let’s evaluate \(f(2.5)\) and \(f(2)\):

\(f(2.5) = (2.5)^2 = 6.25\)

\(f(2) = (2)^2 = 4\)

Substituting these values into the difference quotient:

$$\frac{6.25 – 4}{0.5} = \frac{2.25}{0.5} = 4.5$$

So, in this example, the difference quotient gives us the average rate of change of the function \(f(x) = x^2\) between \(x = 2\) and \(x = 2.5\) as 4.5.

In calculus, the difference quotient is often used as an intermediate step in finding the derivative of a function. By taking the limit of the difference quotient as \(h\) approaches 0, we can obtain the instantaneous rate of change or the slope of the function at a specific point. This leads to the definition of the derivative.

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