.calc-section { padding: 25px; background: #fff; }
.calc-header { background: #2c3e50; color: #fff; padding: 20px; text-align: center; }
.calc-header h2 { margin: 0; font-size: 24px; }
.input-group { margin-bottom: 15px; }
.input-group label { display: block; margin-bottom: 5px; font-weight: bold; font-size: 14px; }
.input-group input { width: 100%; padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; font-size: 16px; }
.calc-btn { background: #3498db; color: white; border: none; padding: 12px 20px; border-radius: 4px; cursor: pointer; width: 100%; font-size: 16px; font-weight: bold; transition: background 0.3s; }
.calc-btn:hover { background: #2980b9; }
.result-box { margin-top: 20px; padding: 15px; background: #f8f9fa; border-left: 5px solid #3498db; border-radius: 4px; }
.result-box h3 { margin: 0 0 10px 0; font-size: 18px; color: #2c3e50; }
.result-value { font-size: 20px; font-weight: bold; color: #e67e22; }
.calc-tabs { display: flex; background: #ecf0f1; }
.tab-btn { flex: 1; padding: 15px; border: none; cursor: pointer; background: #ecf0f1; font-weight: bold; border-bottom: 3px solid transparent; }
.tab-btn.active { background: #fff; border-bottom: 3px solid #3498db; }
.tab-content { display: none; }
.tab-content.active { display: block; }
.formula-display { background: #eee; padding: 10px; border-radius: 4px; font-family: "Courier New", Courier, monospace; font-size: 13px; margin-bottom: 15px; }
function openTab(evt, tabName) {
var i, tabcontent, tablinks;
tabcontent = document.getElementsByClassName("tab-content");
for (i = 0; i < tabcontent.length; i++) {
tabcontent[i].style.display = "none";
tabcontent[i].className = tabcontent[i].className.replace(" active", "");
}
tablinks = document.getElementsByClassName("tab-btn");
for (i = 0; i < tablinks.length; i++) {
tablinks[i].className = tablinks[i].className.replace(" active", "");
}
document.getElementById(tabName).style.display = "block";
document.getElementById(tabName).className += " active";
evt.currentTarget.className += " active";
}
function calculateAverageROC() {
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 resultContainer = document.getElementById('avg-result-container');
var resultDisplay = document.getElementById('avg-result');
var explanation = document.getElementById('avg-explanation');
if (isNaN(x1) || isNaN(fx1) || isNaN(x2) || isNaN(fx2)) {
alert("Please enter valid numbers for all fields.");
return;
}
if (x1 === x2) {
alert("x1 and x2 cannot be the same value for Average Rate of Change (division by zero).");
return;
}
var deltaY = fx2 – fx1;
var deltaX = x2 – x1;
var rate = deltaY / deltaX;
resultDisplay.innerHTML = rate.toFixed(4);
explanation.innerHTML = "The slope of the secant line is (" + fx2 + " – " + fx1 + ") / (" + x2 + " – " + x1 + ") = " + deltaY + " / " + deltaX;
resultContainer.style.display = "block";
}
function calculateInstantaneousROC() {
var a = parseFloat(document.getElementById('coeff').value);
var n = parseFloat(document.getElementById('exponent').value);
var x = parseFloat(document.getElementById('evalX').value);
var resultContainer = document.getElementById('inst-result-container');
var resultDisplay = document.getElementById('inst-result');
var explanation = document.getElementById('inst-explanation');
if (isNaN(a) || isNaN(n) || isNaN(x)) {
alert("Please enter valid numbers for coefficient, exponent, and x.");
return;
}
// Derivative of ax^n is (a*n)x^(n-1)
var newCoeff = a * n;
var newExp = n – 1;
var rate = newCoeff * Math.pow(x, newExp);
resultDisplay.innerHTML = rate.toFixed(4);
explanation.innerHTML = "The derivative is f'(x) = " + newCoeff + "x^" + newExp + ". Evaluating at x=" + x + " gives " + newCoeff + "(" + x + ")^" + newExp + " = " + rate.toFixed(4);
resultContainer.style.display = "block";
}
Rate of Change Calculus Calculator
Calculate the average rate of change between two points on a function.
Formula: [f(x₂) – f(x₁)] / (x₂ – x₁)
Result:
Calculate the derivative (instantaneous rate) for a polynomial term axⁿ.
Function: f(x) = axⁿ | Derivative: f'(x) = anxⁿ⁻¹
Result:
Understanding Rate of Change in Calculus
In calculus, the Rate of Change describes how one quantity changes in relation to another. This is most commonly represented as the slope of a line on a graph.
1. Average Rate of Change
This is the slope of the secant line passing through two points (x₁, y₁) and (x₂, y₂). It represents the total change divided by the total time or interval. If you are driving, your average speed for the whole trip is the average rate of change of your position.
2. Instantaneous Rate of Change
This is the core of differential calculus. It is the slope of the tangent line at a specific point. It is calculated by taking the derivative of the function. For a power function like f(x) = axⁿ, the rate of change at any point is f'(x) = n * ax^(n-1).
Practical Example
Imagine an object's position is defined by f(t) = 5t².
- Average Rate (t=1 to t=3): At t=1, f(1)=5. At t=3, f(3)=45. Average rate = (45 – 5) / (3 – 1) = 40 / 2 = 20 units/sec.
- Instantaneous Rate (at t=2): The derivative is f'(t) = 10t. At t=2, the rate is 10(2) = 20 units/sec.