Secants closing on the tangent to x²

• a point where the slope formula breaks down in both directions is a genuine crossing ×1
• and of slope minus one ×1
• and the difference quotient closes on u'v + uv' ×1
• and the gap as a fraction of the window falls with it ×1
• and the rough curve's do not settle on anything ×1
• arbitrarily short secants of slope one ×1
• at a regular point the partial derivative in y is away from zero ×1
• at the minimum the slope of the original curve is exactly the level ×1
• beside a vertical tangent one side meets the curve twice and the other not at all ×1
• between 1 and 4 mirrored pairs are marked ×1
• between 1 and 4 pairs are marked ×1
• between 2 and 4 windows are drawn ×1
• between 4 and 14 spacings are measured ×1
• both ends are higher than the minimum ×1
• both sides grow, so the picture is four pieces and not a signed sum ×1
• both sides of the rectangle are positive lengths ×1
• by the last halving the corner is under a twentieth of the increment ×1
• each magnification is between 2 and 20 ×1
• each secant is closer than the last ×1
• each shorter secant is closer to the slope the equation gives ×1
• each step magnifies by between 2 and 20 ×1
• every secant from the left has slope −1 ×1
• every secant from the right has slope 1 ×1
• inside the window round a regular point each vertical line meets the curve once ×1
• on every interval about nought the slope runs nearly from −1 to 1 ×1
• the bisection really inverts the function ×1
• the corner's share of the increment falls at every halving ×1
• the curve is one the figure knows ×1
• the curve, solved for y near the point, has the slope minus the ratio of the two partial derivatives ×1
• the derivative is drawn for the power 2 or 3 ×1
• the derivative's formula matches its difference quotients ×1
• the first window is at most 4 wide ×1
• the four pieces fill the grown rectangle ×1
• the function increases across the window it is drawn in ×1
• the function is increasing where it is marked, so it has an inverse there ×1
• the function is one the family knows ×1
• the function is one the figure knows ×1
• the gap to the tangent falls by the square of the magnification ×1
• the increment is the two strips and the corner ×1
• the last secant is near the tangent ×1
• the level c is reached again and again as x approaches nought ×1
• the level is strictly between −1 and 1 ×1
• the level lies between the slopes at the two ends ×1
• the marked point is inside the window ×1
• the marked point is not one where the tangent is vertical ×1
• the marked point lies on the curve ×1
• the minimum is inside the interval, not at an end ×1
• the pair is one the figure knows ×1
• the power is 1, 2 or 3 ×1
• the rectangle is drawn inside the domain of both functions ×1
• the secant slope from the origin is h^(p−1) sin(1/h) ×1
• the secant slope is squeezed by the envelope ×1
• the secants are drawn at a list of 2 to 24 positive offsets ×1
• the secants run one way along the curve or the other ×1
• the smooth curve's chords close on its tangent ×1
• the step is a positive length under 1.6 ×1
• the step the rectangle grows by is a positive length under 1.6 ×1
• the two sides disagree, so there is no limit ×1
• the two slopes at a matched pair multiply to one ×1
• the view is one the family draws ×1

The whole library · What the figures prove