The instantaneous rate of change at a specific point on a curve is a fundamental concept in calculus. It represents the slope of the tangent line to the curve at that exact point. This is crucial for understanding velocity, acceleration, and the behavior of functions at their most granular level. Mathematically, it's found by taking the derivative of the function and evaluating it at the specific point of interest.
function calculateInstantaneousRate() {
var funcStr = document.getElementById("function").value;
var xValue = parseFloat(document.getElementById("point").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
if (isNaN(xValue)) {
resultDiv.innerHTML = "Please enter a valid number for the point.";
return;
}
try {
// A simplified approach for common functions. For a truly robust solution,
// a symbolic math library would be needed. This example handles basic polynomials and common functions.
var derivativeStr;
if (funcStr.includes('^')) {
var baseVar = funcStr.split('^')[0].trim();
var exponent = parseInt(funcStr.split('^')[1].trim());
if (baseVar === 'x' && !isNaN(exponent)) {
if (exponent === 1) {
derivativeStr = "1";
} else if (exponent === 0) {
derivativeStr = "0";
} else {
derivativeStr = (exponent * Math.pow(xValue, exponent – 1)).toString();
}
} else {
throw new Error("Unsupported polynomial format. Only 'x^n' is supported.");
}
} else if (funcStr === 'sin(x)') {
derivativeStr = Math.cos(xValue).toString();
} else if (funcStr === 'cos(x)') {
derivativeStr = (-Math.sin(xValue)).toString();
} else if (funcStr === 'tan(x)') {
derivativeStr = (1 / Math.pow(Math.cos(xValue), 2)).toString();
} else if (funcStr === 'x') {
derivativeStr = "1";
} else if (funcStr === 'constant') { // Handle simple constants
derivativeStr = "0";
}
else {
// Attempt to evaluate the function itself to check for validity,
// and try a numerical approximation if symbolic derivative fails.
// This is a fallback and less precise than symbolic differentiation.
var h = 0.00001; // A small value for numerical approximation
var f_x = eval(funcStr.replace(/x/g, xValue.toString()));
var f_x_plus_h = eval(funcStr.replace(/x/g, (xValue + h).toString()));
if (isNaN(f_x) || isNaN(f_x_plus_h)) {
throw new Error("Invalid function or evaluation error.");
}
var numericalDerivative = (f_x_plus_h – f_x) / h;
derivativeStr = numericalDerivative.toString();
resultDiv.innerHTML = "Numerical approximation used as symbolic derivative is not supported for '" + funcStr + "'.";
}
// Evaluate the derivative at the given point
var instantaneousRate = parseFloat(derivativeStr);
if (isNaN(instantaneousRate)) {
throw new Error("Could not evaluate derivative.");
}
resultDiv.innerHTML = "The instantaneous rate of change of " + funcStr + " at x = " + xValue + " is approximately " + instantaneousRate.toFixed(4) + ".";
} catch (error) {
resultDiv.innerHTML = "Error: " + error.message + ". Please check your function input.";
}
}
.calculator-wrapper {
font-family: Arial, sans-serif;
border: 1px solid #ccc;
padding: 20px;
border-radius: 8px;
max-width: 500px;
margin: 20px auto;
background-color: #f9f9f9;
}
.calculator-wrapper h2 {
text-align: center;
color: #333;
margin-bottom: 15px;
}
.calculator-wrapper p {
line-height: 1.6;
color: #555;
margin-bottom: 20px;
}
.input-section {
margin-bottom: 15px;
}
.input-section label {
display: block;
margin-bottom: 8px;
font-weight: bold;
color: #444;
}
.input-section input[type="text"],
.input-section input[type="number"] {
width: calc(100% – 20px);
padding: 10px;
margin-bottom: 10px;
border: 1px solid #ccc;
border-radius: 4px;
}
.calculator-wrapper button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #4CAF50;
color: white;
border: none;
border-radius: 5px;
cursor: pointer;
font-size: 16px;
transition: background-color 0.3s ease;
}
.calculator-wrapper button:hover {
background-color: #45a049;
}
#result {
margin-top: 20px;
padding: 15px;
border: 1px solid #ddd;
border-radius: 5px;
background-color: #fff;
min-height: 50px;
text-align: center;
}
#result p {
margin: 0;
font-size: 1.1em;
color: #333;
}
#result strong {
color: #007bff;
}