Fluxion′ a visual course in calculus

Derivatives · Integrals · Limits — drawn live

Calculus you can see.

Every figure on this page is a working instrument: scrub the tangents, shrink the rectangles, chase ε with δ. The ideas of Newton and Leibniz were born from pictures — here, the pictures move. Scroll to zoom any plot, drag to pan it.

↑ hover the dotted terms, everywhere on this page
Fig. 1 — a tangent sweeps y = sin x
x 0.00 sin x 0.00 slope = cos x 0.00 hover to steer · drag to scrub

Fig. 1. The tangent to sweeps the curve; its slope (orange dot, faint curve) traces — the derivative, one instant at a time.

§ 1

The Derivative

Slope, instant by instant.

The secant line through two points of a curve is honest but coarse: it reports the average slope over a whole interval. Slide the second point toward the first and the secant settles into the tangent line — the derivative , the slope at a single instant.

hover the dotted terms for their meaning

Drag the point below and watch the orange slope on the left become the orange curve on the right. Keep an eye on the turning points: where levels off, crosses zero; where climbs steepest, peaks.

Fig. 2 — f and f′, side by side
f(x)
f′(x)
drag the point on the left · scroll to zoom
a—
f(a)—
f′(a) — slope—
tangent line  —

Fig. 2. Left: the curve with its tangent (orange) and a unit-slope triangle. Right: the value of that slope, plotted — the derivative draws itself as you drag.

§ 2

The Integral

Area, slice by slice.

To measure the area under a curve, cheat: pretend the region is made of rectangles. A Riemann sum adds up their areas — wrong, but wrong in a way that fades as the slices thin. Raise below and watch the error collapse toward the exact value .

hover the dotted terms — Δx is the rectangle width you are about to drag

Drag the a and b flags on the axis to move the bounds, switch the sampling rule, and notice how the midpoint rule beats the left-endpoint rule at equal . Area below the axis is signed: it counts against the total.

Fig. 3 — Riemann sums refine the area
drag the a / b flags on the axis
a—
b—
Δx—
Riemann sum Sₙ—
exact ∫ f dx—
error |Sₙ − ∫|—

Fig. 3. Rectangles of width fill smoothly as the partition tightens; the pale green wash is the exact signed area they are chasing.

§ 3

The Limit

The ε–δ contract, made physical.

A limit is a contract. You name a tolerance ε around the output ; the function must answer with a radius δ around the input such that everything inside the δ-band lands inside the ε-band. If such a δ exists for every ε, the limit is .

hover ε, δ and both bands — the whole definition is two inequalities

Drag the green ε-edge below; when you let go, the tool finds the largest δ that honours the contract and flashes when it locks in. Then switch to manual and defend a δ yourself — if your band is too generous, the tool finds the exact that betrays you.

Fig. 4 — the ε–δ machine a = 2 · L = 5
drag the green band edge — δ snaps on release

Fig. 4. Green: the output tolerance . Orange: the input window that must earn it. The snap shows the maximal δ.

§ 4

A Proof, One Key at a Time

Seven lines from the definition to 2x.

No black boxes. Below is the entire proof that , in seven moves. Each step also shrinks a real in the picture — watch the secant line die into the tangent exactly as the algebra cancels it. Press N to advance, P to step back.

step 0 / 7
∎  Q.E.D. — the secant is gone; the tangent remains.
Fig. 5 — the shrinking secant
h1.000
secant slope = 2 + h3.000
§ 5

Practice

Exam register. Hints unfold one at a time — spend them like points.

Four applied problems, checked to two decimals. Answers are graded instantly and inline; full worked solutions are one click away when you want them.

solved 0 / 4
1.
related ratesdifficulty 2 / 3solved

A 5-metre ladder leans against a wall. Its base slides away from the wall at . At the instant the base is 3 m from the wall, how fast is the top of the ladder moving? Give the speed in m/s.

worked solution
Let be the base distance and the height of the top. Then .
Differentiate with respect to time: , so .
At the instant in question , , and .
. The top slides down at 0.3 m/s. ∎
2.
optimizationdifficulty 2 / 3solved

A rectangle has its base on the -axis and its two top corners on the parabola . What is the largest area it can have, in square units?

worked solution
Put the top corners at : the rectangle is wide and tall, so .
(taking ), and — a maximum.
square units. ∎
3.
integral · average valuedifficulty 1 / 3solved

Find the average value of on the interval .

worked solution
The average value is .
.
Average . ∎
4.
limit · techniquedifficulty 3 / 3solved

Evaluate .

worked solution
Multiply by the conjugate: .
Regroup: . As the first factor , the second .
The limit is 0.5. ∎