Calculus Workbench
Ten calculus operations on one function. Derivatives are exact and symbolic; integrals, limits, Taylor series, critical and inflection points, area, arc length, related rates and closed-interval optimization come with steps and a graph.
How to Use
- Type a function of one variable — use
^for powers and the usual functionssin, cos, tan, exp, ln, sqrt, asin, …pluspiande. Implicit multiplication works (2x,x sin(x)). - Pick an operation: Derivative, Integral, Limit, Taylor series, Critical points, Inflection points, Area under curve, Arc length, Related rates, or Optimization.
- Extra fields appear when needed — bounds
a, bfor area/arc length/optimization, a centre for the Taylor series, an order for higher derivatives, a point and side for limits. - Differentiation is symbolic and exact. Integration handles polynomials, the standard table, linear substitution and integration by parts; Area under curve falls back to Simpson’s rule when there is no closed form.
- Read the result and the readouts under the graph, and follow the working in Show Work; the graph marks critical and inflection points and shades the area. For Related rates, type an equation such as
A = pi*r^2and both sides are differentiated with respect to t.
Worked Example
The default cubic, f(x) = x³ − 6x² + 9x. The derivative is 3x² − 12x + 9 = 3(x − 1)(x − 3), so the critical points are x = 1 and x = 3. f″(x) = 6x − 12 is −6 at x = 1 (a local maximum, f(1) = 4) and +6 at x = 3 (a local minimum, f(3) = 0). f″ = 0 at x = 2, where the curve changes from concave down to concave up: the inflection point is (2, 2).
Area and arc length of y = x² from 0 to 1. The antiderivative is x³/3, so the area is 1/3, shown as 0.3333333333. The arc length is ∫₀¹ √(1 + 4x²) dx, which Simpson’s rule with 4,000 intervals gives as 1.478942858; the exact value is (2√5 + sinh⁻¹ 2)/4.
The common mistake: forgetting the endpoints. On [0, 5] the only local maximum of x³ − 6x² + 9x is f(1) = 4, but the endpoint gives f(5) = 125 − 150 + 45 = 20. The global maximum is 20 at x = 5, not 4. On [0, 4] the two tie, since f(4) = 4 as well.
Show Work
Formulas
Newton, Leibniz and the Series
Isaac Newton worked out his “method of fluxions” in 1665–66 but did not publish it for decades. Gottfried Wilhelm Leibniz published first, in 1684, with Nova Methodus pro Maximis et Minimis in the journal Acta Eruditorum; his dx notation and his ∫ sign, a long S for summa, are the ones still used. The priority dispute between the two men and their supporters ran for years.
Brook Taylor published the series named after him in Methodus Incrementorum Directa et Inversa (1715), and Colin Maclaurin made heavy use of the special case about 0 in his Treatise of Fluxions (1742). Thomas Simpson gave his integration rule in 1743, although Johannes Kepler had used an equivalent rule for the volume of wine barrels in 1615. Augustin-Louis Cauchy rebuilt calculus on limits in his Cours d’analyse of 1821.
About This Tool
The Calculus Workbench puts ten single-variable operations on one function: derivative (any order), integral, limit, Taylor series (up to order 12), critical points, inflection points, area, arc length, related rates and closed-interval optimization. Derivatives are exact, and so are rational and quadratic roots of f′, using rational arithmetic; limits, arc lengths and integrals without a closed form are numeric and say so in the steps.
Everything runs in your browser; nothing is uploaded.
Related tools: Derivative Calculator, Integral Calculator, and Differential Equation Solver.
Frequently Asked Questions
What is the derivative of x³ − 6x² + 9x?
3x² − 12x + 9, by the power rule term by term. The Derivative operation does this symbolically with the sum, product, quotient and chain rules, for polynomials, roots, exponentials, logarithms and the trig, inverse-trig and hyperbolic functions. Set the order to 2 for f″ = 6x − 12.
Which integrals can it do exactly?
Polynomials term by term, the standard table (sin, cos, eˣ, 1/x → ln|x|, 1/(1 + x²) → arctan x, ln x, √x), linear substitution f(ax + b), and integration by parts for a polynomial times eˣ, sin or cos: ∫ x² sin x dx = −x² cos x + 2x sin x + 2 cos x + C. When it finds no antiderivative, as for e^(−x²), it says so, and Area under curve gives ∫₀¹ e^(−x²) dx = 0.7468241328 by Simpson’s rule with 2,000 intervals.
How are limits found?
Numerically. The function is evaluated at the point ± 0.01, 0.0001, 10⁻⁶ and 10⁻⁸, and the two sides must agree to within 0.0001. So sin(x)/x gives 0.99998 at x = 0.01 and 1 at x = 10⁻⁸, and the limit is 1; (x² − 1)/(x − 1) → 2 at x = 1. If the sides disagree it reports that the limit does not exist. Type inf or -inf as the point for limits at infinity.
How are critical and inflection points classified?
It solves f′(x) = 0, exactly when f′ is a polynomial, and applies the second-derivative test. For x³ − 6x² + 9x, f′ = 3(x − 1)(x − 3); f″(1) = −6 makes (1, 4) a local maximum and f″(3) = 6 makes (3, 0) a local minimum. Inflection points solve f″ = 6x − 12 = 0 and are kept only where the concavity changes: (2, 2).
How does Optimization find the global maximum?
With the closed-interval method: it evaluates f at every critical point inside [a, b] and at both endpoints, and the largest value wins. For x³ − 6x² + 9x on [0, 5] the candidates are f(1) = 4, f(3) = 0, f(0) = 0 and f(5) = 20, so the global maximum is 20 at the endpoint x = 5, not the local maximum at x = 1.
How do I use the Calculus Workbench?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Homework checks
Confirm a derivative, an antiderivative or a limit and see the steps.
Curve sketching
Critical points, inflection points and the graph together: (1, 4), (3, 0) and (2, 2) for the default cubic.
Word problems
Closed-interval optimization, and related rates: dA/dt = 2πr·dr/dt means a circle of radius 3 growing at 2 per second gains area at 12π ≈ 37.7 per second.
Series approximations
The 5th-order Maclaurin series for eˣ sums to 2.71667 at x = 1, within 0.0016 of e.
Areas and lengths
Definite integrals and arc lengths: the parabola y = x² from 0 to 1 is 1.478942858 long.
Last updated: