Calculus Calculator
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Calculus Operations Calculator
Perform basic calculus operations: differentiation and integration.
Result:
Enter function and operation
Understanding Calculus Operations
Calculus is a branch of mathematics that deals with rates of change and accumulation. It is broadly divided into two main branches: differential calculus and integral calculus.
Differential Calculus: The Rate of Change
Differential calculus is concerned with finding the derivative of a function. The derivative of a function at a specific point represents the instantaneous rate of change of that function at that point. Geometrically, it's the slope of the tangent line to the function's graph at that point.
Key Concepts:
- Derivative: Measures how a function changes as its input changes. Common notations include
f'(x) or dy/dx.
- Rules of Differentiation: Includes power rule (
d/dx(x^n) = nx^(n-1)), product rule, quotient rule, and chain rule, which help in differentiating complex functions.
Use Cases: Determining velocity from position, acceleration from velocity, marginal cost/revenue in economics, optimization problems (finding maximum/minimum values).
Integral Calculus: The Accumulation
Integral calculus is concerned with finding the integral of a function. It can be thought of as the reverse process of differentiation (antidifferentiation) or as finding the area under the curve of a function.
Key Concepts:
- Indefinite Integral: Represents the antiderivative of a function, including an arbitrary constant of integration
+ C. Notation: ∫f(x) dx.
- Definite Integral: Represents the accumulation of a function over a specific interval
[a, b]. It is calculated by finding the antiderivative and evaluating it at the upper and lower limits. Notation: ∫[a,b] f(x) dx.
- Fundamental Theorem of Calculus: Connects differentiation and integration, stating that
∫[a,b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
Use Cases: Calculating areas, volumes, work done, total distance traveled from velocity, probabilities.
How This Calculator Works:
This calculator uses a simplified JavaScript-based symbolic math engine to perform differentiation and indefinite integration. For integration, you can optionally provide lower and upper limits to calculate a definite integral. The calculator parses your input function, applies the relevant calculus rules, and presents the resulting expression.
Example of Use:
- Function:
3*x^2 + 2*x - 5
- Operation: Differentiate
- Result:
6*x + 2
- Operation: Integrate
- Result:
x^3 + x^2 - 5*x + C
- Operation: Integrate (Definite with limits 0 to 1)
- Result:
-3.666... (calculated as (1^3 + 1^2 – 5*1) – (0^3 + 0^2 – 5*0))
// Basic symbolic math functions (simplified for common polynomial and power functions)
function parseFunction(expression) {
expression = expression.replace(/\s+/g, "); // Remove whitespace
return expression;
}
function differentiate(expression) {
expression = parseFunction(expression);
var result = ";
var terms = expression.split(/([+-])/); // Split by + and –
var termSigns = ["];
for (var i = 1; i < terms.length; i += 2) {
termSigns.push(terms[i]);
}
terms = terms.filter(function(t) { return t !== '+' && t !== '-'; });
for (var i = 0; i < terms.length; i++) {
var term = terms[i];
var sign = termSigns[i];
var derivativeTerm = '';
if (term.includes('x')) {
var parts = term.split('x');
var coefficientStr = parts[0];
var exponentStr = parts[1];
var coefficient = 1;
if (coefficientStr === '' || coefficientStr === '+') coefficient = 1;
else if (coefficientStr === '-') coefficient = -1;
else coefficient = parseFloat(coefficientStr);
var exponent = 1;
if (exponentStr === '') exponent = 1;
else if (exponentStr.startsWith('^')) {
exponent = parseFloat(exponentStr.substring(1));
} else { // Case like '3x' where exponent is implicitly 1
exponent = 1;
}
if (exponent === 0) {
derivativeTerm = '0'; // Derivative of a constant is 0
} else {
var newCoefficient = coefficient * exponent;
var newExponent = exponent – 1;
if (newExponent === 0) {
derivativeTerm = newCoefficient.toString();
} else if (newExponent === 1) {
derivativeTerm = (newCoefficient === 1 ? '' : newCoefficient === -1 ? '-' : newCoefficient) + 'x';
} else {
derivativeTerm = (newCoefficient === 1 ? '' : newCoefficient === -1 ? '-' : newCoefficient) + 'x^' + newExponent;
}
}
} else {
derivativeTerm = '0'; // Derivative of a constant is 0
}
if (result !== '' && derivativeTerm !== '0') {
result += (sign === '-' ? '-' : '+');
}
if (derivativeTerm !== '0') {
result += derivativeTerm;
}
}
// Clean up leading/trailing signs and double signs
if (result.startsWith('+')) result = result.substring(1);
if (result === '') result = '0';
return result;
}
function integrate(expression) {
expression = parseFunction(expression);
var result = '';
var terms = expression.split(/([+-])/);
var termSigns = [''];
for (var i = 1; i < terms.length; i += 2) {
termSigns.push(terms[i]);
}
terms = terms.filter(function(t) { return t !== '+' && t !== '-'; });
for (var i = 0; i 3*x, 3( -> 3*(
safeExpression = safeExpression.replace(/([x)])(\d|[x(])/g, '$1*$2'); // x2 -> x*2, x( -> x*(, )2 -> )*2, )( -> )*(
// Handle cases like x^2 where x is a number
var regexPower = /(\d+|\(\S+\))\*\*(\d+)/g;
safeExpression = safeExpression.replace(regexPower, function(match, base, exp) {
return Math.pow(parseFloat(base.replace(/[()]/g, ")), parseFloat(exp));
});
// Handle potential function calls like sin(x), cos(x) – very limited
safeExpression = safeExpression.replace(/sin\(/g, 'Math.sin(');
safeExpression = safeExpression.replace(/cos\(/g, 'Math.cos(');
safeExpression = safeExpression.replace(/tan\(/g, 'Math.tan(');
safeExpression = safeExpression.replace(/log\(/g, 'Math.log(');
safeExpression = safeExpression.replace(/exp\(/g, 'Math.exp(');
return Function('"use strict";return (' + safeExpression + ')')();
} catch (e) {
console.error("Evaluation error:", e);
return NaN;
}
}
function calculateDefiniteIntegral(expression, lowerLimit, upperLimit) {
// Remove the '+ C' from the indefinite integral result if it exists
var indefiniteIntegralExpr = expression.replace(' + C', ");
var lowerVal = evaluateFunction(indefiniteIntegralExpr, lowerLimit);
var upperVal = evaluateFunction(indefiniteIntegralExpr, upperLimit);
if (isNaN(lowerVal) || isNaN(upperVal)) {
return "Error evaluating limits";
}
return upperVal – lowerVal;
}
function calculate() {
var expressionInput = document.getElementById('expression');
var operationSelect = document.getElementById('operation');
var resultValueElement = document.getElementById('result-value');
var expression = expressionInput.value.trim();
var operation = operationSelect.value;
var result = ";
if (!expression) {
result = "Please enter a function.";
resultValueElement.textContent = result;
return;
}
// Input validation: Check for "x" or assume constant if not present
var hasX = expression.includes('x');
if (operation === 'differentiate') {
if (!hasX) {
result = "0 (Function does not contain 'x')";
} else {
result = differentiate(expression);
}
} else if (operation === 'integrate') {
var lowerLimitInput = document.getElementById('integration_limit');
var upperLimitInput = document.getElementById('integration_upper_limit');
var lowerLimit = parseFloat(lowerLimitInput.value);
var upperLimit = parseFloat(upperLimitInput.value);
var lowerLimitIsNumber = !isNaN(lowerLimit);
var upperLimitIsNumber = !isNaN(upperLimit);
if (lowerLimitIsNumber && upperLimitIsNumber) {
if (lowerLimit > upperLimit) {
result = "Error: Lower limit cannot be greater than upper limit.";
} else {
var definiteIntegralResult = calculateDefiniteIntegral(integrate(expression), lowerLimit, upperLimit);
result = typeof definiteIntegralResult === 'number' ? definiteIntegralResult.toFixed(6) : definiteIntegralResult; // Display with precision
}
} else {
result = integrate(expression);
}
}
resultValueElement.textContent = result;
}
document.getElementById('operation').addEventListener('change', function() {
var integrationLimitGroup = document.getElementById('integration-limit-group');
var integrationUpperLimitGroup = document.getElementById('integration-upper-limit-group');
if (this.value === 'integrate') {
integrationLimitGroup.style.display = 'flex';
integrationUpperLimitGroup.style.display = 'flex';
} else {
integrationLimitGroup.style.display = 'none';
integrationUpperLimitGroup.style.display = 'none';
document.getElementById('integration_limit').value = "; // Clear limits
document.getElementById('integration_upper_limit').value = ";
}
});
// Initial call to set visibility based on default selection
document.getElementById('operation').dispatchEvent(new Event('change'));