Module 03
The calculus you need
Glance at your speedometer on the highway: 61. Now ask what that number means. Speed is distance covered divided by time taken, so it needs two moments, a before and an after. Sixty-one miles in one hour. A mile in 59 seconds. Some distance across some stretch of time.
But the needle is not reporting a stretch of time. It claims to know your speed now, at this instant, at a moment with no width at all. In an instant the car covers no distance and takes no time, and the recipe says zero divided by zero.
The needle works anyway. Every working speedometer is a small mechanical rebuttal to a puzzle that took two thousand years to settle. How can a rate exist at a single instant?
The takeaway
The derivative reads change at one instant by shrinking the measuring window until the ratio stops arguing. The integral adds the instants back into a total. Each undoes the other, and that pairing is the entire toolkit this course needs.
Shrink the window
Fix the puzzle by refusing to jump to the instant. Measure your speed over one second: distance covered, divided by one second. A clean number. Over a tenth of a second: another number, close to the first. A hundredth: closer still. The window keeps shrinking, and the answers stop wandering and crowd around one value. They settle. The settled value is called a limit, and the puzzle dissolves: nobody ever divides zero by zero. You divide small by small, watch where the ratios are heading, and name the destination. Speed at an instant is a destination, not a division.
Run that procedure on a whole position story and you get a machine. Feed in where you were at every time; it hands back how fast you were moving at every time. The machine is the derivative. On a graph it has a clean face: pick a point on a curve, and the derivative is the slope of the straight line the curve is impersonating right there, the tangent. Here is the zoom analogy, with its breaking point in the same breath: zoom in on any smooth curve and it straightens into a line whose slope is the derivative; the analogy breaks at corners and jumps, where no amount of zoom straightens anything and the derivative does not exist. Classical mechanics bets that motion has no corners, and the bet pays off everywhere this volume goes.
Now run the machine backward. Suppose you know your speed at every moment of a trip and want the distance covered. Chop the trip into slivers of time. In each sliver the speed barely changes, so distance is speed times sliver, a skinny rectangle on the speed graph. Add the rectangles; shrink the slivers; the sum settles the same way the speedometer ratios did. The settled total is the integral, and it is exactly the area under the speed curve. One machine reads the film one frame at a time. The other splices frames back into a trip.
A word on what you do not need, because rusty calculus mostly means remembered dread. You need no table of memorized antiderivatives, no integration by parts, no trigonometric substitutions. This volume runs on the two ideas above, the power rule, and sums and constants along for the ride. On the rare occasion a new rule is required, it gets derived on the page, in the open, the way the power rule was earned here. Notation, likewise, stays honest: dx/dt looks typographically like a fraction and behaves like a verb. Read it as the derivative of x with respect to t, one machine applied to one story, and no mystery survives.
The demon from module 01 has been waiting for precisely this instrument. Its ledger holds frame-proof snapshots, but prediction is about how snapshots flow into each other, and the derivative is the flow-reader. Hand the demon a position story and it now extracts the rate; hand it rates and the integral rebuilds the story. What it still lacks is a law connecting the two. That arrives in module 05.
The math
Let be position at time . Pick a window . Form the average rate across the window:
Shrink toward zero. The value this ratio settles on is the derivative, written .
Work one example completely. Take :
Send to zero. The leftover vanishes. The derivative of is . No zero ever divided zero: the cancelled first, then left.
The same computation on any whole-number power gives the one rule this volume leans on:
Two facts come along free. The derivative of a constant is zero: a parked car has speed zero. And the derivative of a sum is the sum of the derivatives.
The integral of a rate from time to time is written and equals the area under the graph of between those times.
The two machines undo each other. This is the fundamental theorem of calculus:
Accumulate area, then ask how fast the area grows: it grows at exactly the height of the curve. Differentiate the accumulation and the original rate comes back.
Play with it
One position story on top, its rate story below. Drag the marker through time. The slope upstairs and the height downstairs are the same number at every instant, and the shaded area downstairs recovers the distance upstairs. The fundamental theorem, witnessed rather than asserted.
Check yourself
Problem 1. Position is x(t) = 5t + 3. What is the velocity, and what does the 3 contribute?
Reveal the solution
The derivative of 5t is 5, and the derivative of the constant 3 is zero. Velocity is 5 at every instant. The 3 shifts where you started, not how fast you move.
Problem 2. Position is x(t) = t³. What is the velocity at t = 2?
Reveal the solution
The power rule gives dx/dt = 3t². At t = 2 that is 3 times 4, which is 12.
Problem 3. Speed is v(t) = 6t. How far do you travel between t = 0 and t = 3?
Reveal the solution
Distance is the area under the speed graph: a triangle with base 3 and height 18. Half of 3 times 18 is 27.
What this did to the demon
The demon's ledger was an album of stills; the derivative turns the album into a film, and the integral runs the film back into distances. Module 04 points the new instrument at a thrown ball and reads the ball's entire story off three graphs.
Motion