∫ Calculus Calculator ∂

What is a Calculus Calculator?
This calculator performs advanced calculus operations including derivatives, integrals, limits, and series analysis with step-by-step explanations.

Calculation Results

Enter a function and click calculate to see results here.
Quick Examples

Frequently Asked Questions

Our calculator supports polynomial functions (x^2, x^3), trigonometric functions (sin(x), cos(x), tan(x)), exponential functions (e^x), logarithmic functions (ln(x), log(x)), and combinations of these. Use standard mathematical notation with * for multiplication and ^ for exponents.

The calculator uses numerical methods with high precision. Derivatives are calculated using finite difference methods with very small step sizes (10⁻⁸), and integrals use Simpson's rule with 1000 subdivisions. Results are typically accurate to 6-8 decimal places.

Symbolic derivatives provide the exact mathematical formula (e.g., d/dx[x²] = 2x), while numerical derivatives calculate approximate values at specific points using calculus limit definitions. Our calculator shows both the symbolic formula and numerical values when you specify a point.

L'Hôpital's Rule is used when evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. The rule states that if lim f(x)/g(x) gives an indeterminate form, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the limit of the derivatives exists.

The Ratio Test calculates L = lim |aₙ₊₁/aₙ|. If L < 1, the series converges; if L > 1, it diverges; if L = 1, the test is inconclusive and other tests are needed. Our calculator also shows partial sums to help visualize convergence behavior.

Real-Life Applications

🚗 Physics & Engineering

Derivatives: Calculate velocity (derivative of position) and acceleration (second derivative). For a car's position s(t) = 5t² + 2t, velocity is v(t) = 10t + 2.

Integrals: Find distance traveled from velocity function, or work done against friction.

📊 Economics & Business

Optimization: Find maximum profit by setting marginal cost equal to marginal revenue (both are derivatives).

Growth Models: Use exponential functions and their derivatives to model population or economic growth rates.

🧬 Biology & Medicine

Drug Concentration: Model how medication levels change over time using exponential decay functions.

Population Dynamics: Use differential equations to model predator-prey relationships or disease spread.

💰 Finance

Compound Interest: Use exponential functions and limits to understand continuous compounding.

Risk Analysis: Apply series and convergence tests to model financial risk and probability distributions.

💡 Pro Tip: Try these real examples in the calculator above! For velocity from position s(t) = 5t² + 2t, enter "5*x^2 + 2*x" and calculate the first derivative to get velocity v(t) = 10x + 2.

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