Calculus Studio
Zoom in on any smooth curve and it turns into a straight line: its slope is the derivative. Cut an area into thin strips and the steps flatten out: their sum is the integral. The whole subject is those two moves, and the moment they meet.
- 1Open →
Rate of change
How fast is it changing? The slope of a chord, and what it settles to.
Picture: a distance–time graph · Case: The speed camera
- 2Open →
Limits
What a rule is heading for, even where it has no value.
Picture: a hole in a graph · Case: The greedy banker
- 3Open →
The derivative
Zoom in on a curve until it is straight, and read its slope.
Picture: a zoom lens · Case: The falling phone
- 4Open →
Rules from pictures
A growing square, a growing rectangle, two gears: the rules of differentiation, seen first.
Picture: growing shapes · Case: The self-copying curve
- 5Open →
Reading the shape
Rising, falling, bending: what the slope says about the curve.
Picture: a curve and its slope graph · Case: The epidemic's turning point
- 6Open →
Optimisation
Slide to the top, where the lens is flat.
Picture: a point sliding to the top · Case: The lifeguard
- 7Open →
Related rates
Things that move together change together.
Picture: a sliding ladder · Case: The sliding ladder
- 8Open →
Straight-line guesses
The tangent line as a quick estimate, and Newton's method.
Picture: a tangent riding to the axis · Case: The Babylonian square root
- 9Open →
Area as strips
Cut the area into strips until the steps are flat.
Picture: strips · Case: Archimedes' parabola
- 10Open →
The Fundamental Theorem
The area grows as fast as the curve is high.
Picture: an area meter · Case: Exactly two
- 11Open →
Slices and averages
Solids cut into slices, and the average height of a curve.
Picture: slices · Case: Archimedes' tombstone
- 12Open →
Change that feeds itself
When the rate depends on the amount: growth, decay and slope fields.
Picture: a slope field · Case: Time of death
Final round
Ten problems from the whole studio. Nothing tells you which idea to use.