Average Rate of Change Calculator
The average rate of change of a function describes how much the function's output value changes, on average, for a given change in its input value over an interval. For a function \(f(x)\), the average rate of change between two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) is given by the formula:
$$ \text{Average Rate of Change} = \frac{f(x_2) – f(x_1)}{x_2 – x_1} $$
This calculator helps you find the average rate of change of a function represented by an equation, given two x-values.
function calculateAverageRateOfChange() {
var functionEquationStr = document.getElementById("functionEquation").value;
var x1Str = document.getElementById("x1").value;
var x2Str = document.getElementById("x2").value;
var resultDisplay = document.getElementById("averageRateOfChangeOutput");
resultDisplay.innerHTML = ""; // Clear previous results
if (!functionEquationStr || !x1Str || !x2Str) {
resultDisplay.innerHTML = "
Please fill in all fields.";
return;
}
var x1 = parseFloat(x1Str);
var x2 = parseFloat(x2Str);
if (isNaN(x1) || isNaN(x2)) {
resultDisplay.innerHTML = "
'First x-value' and 'Second x-value' must be valid numbers.";
return;
}
if (x1 === x2) {
resultDisplay.innerHTML = "
The two x-values cannot be the same.";
return;
}
try {
// Prepare the function for evaluation
// Replace 'x' with the value for calculation.
// Using eval can be risky with untrusted input.
// For a production environment, a safer math parsing library would be recommended.
// However, adhering to the 'var' and inline JS requirement, we proceed with eval.
var f_x1 = evaluateFunction(functionEquationStr, x1);
var f_x2 = evaluateFunction(functionEquationStr, x2);
if (isNaN(f_x1) || isNaN(f_x2)) {
resultDisplay.innerHTML = "
Could not evaluate the function at the given x-values. Please check your function equation.";
return;
}
var averageRateOfChange = (f_x2 – f_x1) / (x2 – x1);
resultDisplay.innerHTML = "The average rate of change is:
" + averageRateOfChange.toFixed(4) + "";
} catch (e) {
resultDisplay.innerHTML = "
Error evaluating function. Please ensure the equation is valid (e.g., use 'x' as the variable, standard operators like +, -, *, /, ^ for power, parentheses). Details: " + e.message + "";
}
}
function evaluateFunction(equation, xValue) {
// Basic sanitization and replacement for evaluation
// This is a simplified approach. More robust parsing would handle more cases.
var sanitizedEquation = equation.replace(/[^a-zA-Z0-9+\-*/().^ ]/g, "); // Allow basic math chars and 'x'
sanitizedEquation = sanitizedEquation.replace(/x/g, '(' + xValue + ')');
sanitizedEquation = sanitizedEquation.replace(/\^/g, '**'); // Replace caret with JS exponentiation operator
// Use a more contained eval by creating a Function constructor
// This helps slightly by scoping the eval.
var func = new Function('return ' + sanitizedEquation);
return func();
}
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