Use 'x' as variable. Supported: +, -, *, /, ^, sin, cos, tan, sqrt, log.
Calculation Steps
Interval:
Value at f(a):
Value at f(b):
Change in y (Δy):
Change in x (Δx):
Average Rate =
What is the Average Rate of Change?
The Average Rate of Change (AROC) measures how much a function changes on average over a specific interval. In geometry, this is equivalent to finding the slope of the secant line connecting two points on a curve.
Formula: AROC = [ f(b) – f(a) ] / [ b – a ]
Where:
f(x) is the function you are analyzing.
a is the starting x-value of the interval.
b is the ending x-value of the interval.
f(a) and f(b) are the function outputs at those points.
How to Calculate Manually
To find the average rate of change without a calculator, follow these steps:
Evaluate the function at 'a': Plug your start value into the function to get f(a).
Evaluate the function at 'b': Plug your end value into the function to get f(b).
Find the Change in Output (Δy): Subtract f(a) from f(b).
Find the Change in Input (Δx): Subtract a from b.
Divide: Divide Δy by Δx to get the rate.
Real World Applications
While often used in calculus and algebra courses, this concept applies directly to the real world:
Physics: Average rate of change of position with respect to time is average velocity.
Economics: It calculates the average change in cost or revenue over a production interval.
Biology: Measuring the average growth rate of a population over a specific time period.
Example Calculation
Let's calculate the average rate of change for the function f(x) = x² on the interval [1, 3].
f(1) = 1² = 1
f(3) = 3² = 9
Δy = 9 – 1 = 8
Δx = 3 – 1 = 2
Rate = 8 / 2 = 4
function calculateAROC() {
// Inputs
var funcStr = document.getElementById('functionInput').value;
var valA = parseFloat(document.getElementById('intervalStart').value);
var valB = parseFloat(document.getElementById('intervalEnd').value);
var errorBox = document.getElementById('arocError');
var resultBox = document.getElementById('arocResult');
// Reset display
errorBox.style.display = 'none';
errorBox.innerText = ";
resultBox.style.display = 'none';
// Validation
if (!funcStr || funcStr.trim() === ") {
errorBox.innerText = "Please enter a valid function expression.";
errorBox.style.display = 'block';
return;
}
if (isNaN(valA) || isNaN(valB)) {
errorBox.innerText = "Please enter valid numeric values for the interval start (a) and end (b).";
errorBox.style.display = 'block';
return;
}
if (valA === valB) {
errorBox.innerText = "Interval start (a) and end (b) cannot be the same value (division by zero).";
errorBox.style.display = 'block';
return;
}
try {
// Parse function string safely
// Replace standard math text with JS Math object calls
// 1. Handle implicit multiplication (e.g., 2x -> 2*x)
var parsedFunc = funcStr.replace(/(\d)([a-zA-Z(])/g, '$1*$2');
// 2. Replace exponents ^ with **
parsedFunc = parsedFunc.replace(/\^/g, '**');
// 3. Replace common math functions with Math.func
// Note: Replace "sin" but prevent "Math.sin" from becoming "Math.Math.sin" if user typed Math.
// Simple replace approach for standard inputs
parsedFunc = parsedFunc.replace(/\b(sin|cos|tan|log|exp|sqrt|abs|pow|PI)\b/g, 'Math.$1');
// 4. Create function evaluator
var f = new Function('x', 'return ' + parsedFunc + ';');
// Evaluation
var fA = f(valA);
var fB = f(valB);
if (isNaN(fA) || isNaN(fB) || !isFinite(fA) || !isFinite(fB)) {
throw new Error("Function evaluation resulted in undefined or infinity.");
}
var deltaY = fB – fA;
var deltaX = valB – valA;
var aroc = deltaY / deltaX;
// Formatting output (rounding to 4 decimal places if necessary)
function formatNum(n) {
return (Math.round(n * 10000) / 10000);
}
// Display Results
document.getElementById('displayInterval').innerText = '[' + valA + ', ' + valB + ']';
document.getElementById('displayFA').innerText = formatNum(fA);
document.getElementById('displayFB').innerText = formatNum(fB);
document.getElementById('displayDeltaY').innerText = formatNum(fB) + " – " + formatNum(fA) + " = " + formatNum(deltaY);
document.getElementById('displayDeltaX').innerText = formatNum(deltaX);
document.getElementById('finalRate').innerText = formatNum(aroc);
resultBox.style.display = 'block';
} catch (e) {
console.error(e);
errorBox.innerText = "Error parsing function. Please use valid syntax (e.g., x^2 + 2*x).";
errorBox.style.display = 'block';
}
}