Average Rate of Change of the Function Calculator

Average Rate of Change Calculator

This calculator helps you determine the average rate of change of a function between two points. The average rate of change represents the slope of the secant line connecting two points on the function's graph. It's a fundamental concept in calculus and helps understand how a function's value changes, on average, over an interval.

Result:

How it Works:

The formula for the average rate of change of a function f(x) between two points (x1, f(x1)) and (x2, f(x2)) is:

Average Rate of Change = [f(x2) – f(x1)] / (x2 – x1)

This formula calculates the total change in the function's output (y-values) divided by the total change in the input (x-values) over the specified interval.

Example:

Let's find the average rate of change of a function where:

  • First x-value (x1) = 2
  • Function value at x1 (f(x1)) = 5
  • Second x-value (x2) = 5
  • Function value at x2 (f(x2)) = 11

Using the formula:

Average Rate of Change = (11 – 5) / (5 – 2) = 6 / 3 = 2

Therefore, the average rate of change of the function between x=2 and x=5 is 2.

function calculateAverageRateOfChange() { var x1 = parseFloat(document.getElementById("x1").value); var fx1 = parseFloat(document.getElementById("fx1").value); var x2 = parseFloat(document.getElementById("x2").value); var fx2 = parseFloat(document.getElementById("fx2").value); var resultElement = document.getElementById("result"); if (isNaN(x1) || isNaN(fx1) || isNaN(x2) || isNaN(fx2)) { resultElement.innerHTML = "Please enter valid numbers for all fields."; return; } if (x1 === x2) { resultElement.innerHTML = "The two x-values cannot be the same (x2 – x1 would be zero)."; return; } var deltaY = fx2 – fx1; var deltaX = x2 – x1; var averageRateOfChange = deltaY / deltaX; resultElement.innerHTML = "The average rate of change is: " + averageRateOfChange.toFixed(4); } .calculator-widget { font-family: sans-serif; border: 1px solid #ccc; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; background-color: #f9f9f9; } .calculator-widget h2 { text-align: center; margin-bottom: 20px; color: #333; } .calculator-inputs { display: grid; grid-template-columns: repeat(2, 1fr); gap: 15px; margin-bottom: 20px; } .input-group { display: flex; flex-direction: column; } .input-group label { margin-bottom: 5px; font-weight: bold; color: #555; } .input-group input { padding: 10px; border: 1px solid #ddd; border-radius: 4px; box-sizing: border-box; font-size: 1rem; } .calculator-widget button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 1.1rem; cursor: pointer; transition: background-color 0.3s ease; margin-bottom: 20px; } .calculator-widget button:hover { background-color: #0056b3; } .calculator-result { background-color: #e9ecef; padding: 15px; border-radius: 4px; border: 1px solid #ced4da; } .calculator-result h3 { margin-top: 0; color: #333; } #result { font-size: 1.2rem; font-weight: bold; color: #28a745; } .calculator-explanation { margin-top: 25px; border-top: 1px solid #eee; padding-top: 20px; color: #444; line-height: 1.6; } .calculator-explanation h3, .calculator-explanation h4 { color: #333; margin-bottom: 10px; } .calculator-explanation p, .calculator-explanation ul { margin-bottom: 15px; } .calculator-explanation ul { list-style-type: disc; margin-left: 20px; } .calculator-explanation li { margin-bottom: 5px; } .calculator-explanation strong { font-weight: bold; }

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