VisuApps
Explore sequences, limits and series visually
Apps
The derivative as a limit: three sequences hₙ → 0 (right, left, alternating) yield three sequences of secants — for x², x³, sin(x) they merge into the tangent; for |x|, √x and x·sin(1/x) they fail visibly.
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Explore a saturation limit and the shrinking gap to it. See why growth slows as the remaining distance to the limit decreases.
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Adjust the initial amount and growth rate in an exponential model. Explore proportional rates of change, doubling time and half-life.
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Drag a rectangle inside a triangle and watch its area function. Follow the objective, constraint, substitution and differentiation to find the maximum.
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Find intersections automatically and calculate the area between two curves, integrating the upper minus the lower function on each segment.
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The salami-slice argument: the area function A(x) = ∫ₐˣ f(t)dt is an antiderivative of f. A thin area strip of width dx and height f(x) is added → A′(x) = f(x). Two coupled plots show: the slope of the A curve is the height of f — even for negative values.
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Continuity via the sequential criterion: sequence xₙ → x₀ from the left, from the right, alternating — does the image sequence f(xₙ) settle into the ε-tube around f(x₀)? Six examples: parabola, jump, removable hole, oscillation sin(1/x), step, pole.
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Explore vertical, horizontal and oblique asymptotes. Compare numerator and denominator degrees and see how polynomial division reveals the slant asymptote.
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Move a tangent along a function while a second plot traces its derivative. Connect increasing values, extrema and inflection points with derivative features.
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Fly in 3D along the sequence into the ε-tube — three example sequences (harmonic, alternating without limit, rational with L = 1/2) to choose from.
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Build an arbitrarily large overhang with N equally sized bricks — the harmonic series becomes visible as a growing stack. Slider up to N = 10,000, auto-frame, centre-of-mass display.
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Both parts of the fundamental theorem intuitively animated: d/dx ∫ₐˣ f = f(x) and ∫ₐᵇ f = F(b) − F(a). The shaded area under f and the height difference of the antiderivative F grow in lockstep — the core idea made tangible.
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Enter coefficients and evaluate a polynomial step by step. Inspect the remainder and quotient, factor out roots and compare the resulting graphs.
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Two composed functions in 3D: x → u = g(x) → y = f(u). Slope triangles at P and Q share the common Δu — the teal pulsing "hinge" stroke makes visible why the derivatives multiply. Four combinations to choose from.
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Full curve discussion in eight didactic steps — domain, symmetry, zeros, monotonicity, curvature, extrema, inflection points, asymptotes — cumulatively revealing for three functions (polynomial, rational with pole, rational with parabolic asymptote).
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Find the average height of a function as an equal-area rectangle. Drag interval boundaries and locate a point where the function equals its average.
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Compare signed integrals with total geometric area. Regions above and below the axis contribute differently, and zeros divide the interval.
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Piecewise x+2 | a+bx² at the kink x=1: drag sliders a, b — continuity is a line in (a,b)-space, differentiability is a point. With a mini parameter-space plot showing the solution set live.
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Piecewise ½x²+x−4 | mx+b at the kink x=3 — sliders m, b with snap buttons onto continuity (3m+b=3.5) and differentiability (m=4, b=−8.5). Mini parameter-space plot shows line and intersection.
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The uv-rectangle argument: u·v = ∫v du + ∫u dv, hence ∫u dv = uv − ∫v du. The two partial areas and the rectangle term grow in sync. Alongside, the calculation step by step — choose u and dv (LIATE), differentiate, integrate, substitute.
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Classic polynomial long division as an animated tableau choreography: divide leading term, multiply, subtract — four examples including remainder. The factorisation appears at the end with factors glowing in sequence.
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Recover a quantity from its inflow rate using accumulated area. Explore negative rates, initial values and the link between rate and slope.
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The definite integral as a limit of Riemann sums: a 3D column landscape under the curve, five example functions, choice of sample points (lower/upper/left/midpoint/right sum), a "refine" animation (columns grow ever thinner). Two singularity special cases show: for 1/√x the integral exists despite the pole, for 1/x² the sum grows beyond all bounds — the limit does not exist.
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Rotate a curve around the x-axis. Explore the disc method, a moving cross-section and a stack of discs approximating the volume integral.
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Three classroom examples: parameter c in 2x+c (for which c is f continuous?), x/(x+2) | (x+1)/2 with pole and jump, |x+3| + |x−4| − 5 as a piecewise linear bend.
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Integration by substitution with examples: the x-area is distorted into the u-area by u = g(x) (g′(x) explains dx → du), both areas stay equal in size. In sync, the calculation step by step — find the inner function, differentiate, substitute, convert the bounds.
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Build a function from its derivatives, term by term. Drag the expansion point and explore six functions, highlighted terms and shaded convergence regions.
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Compare a function with its Taylor polynomial. Drag a measurement point to inspect the error and explore the Lagrange remainder and convergence limits.
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Reflect points across y = x to explore inverse functions, domain restrictions and reciprocal slopes using exponential, quadratic and cubic examples.
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Set polynomial roots with multiplicities one, two or three. Discover when a graph crosses or touches the axis and when it flattens.
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Three-step proof walk: (1) ln′(x) = 1/x from the difference quotient (with e as a limit), (2) inverse function rule — reflection at y=x turns slopes into reciprocals, (3) (eˣ)′ = 1 / (1/eˣ) = eˣ. Live algebra in the proof panel.
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