Secant to Tangent
Hold one point P still and slide a second point Q toward it.
The line through them is a secant , and its slope is an average rate of
change. Watch what that slope does as the gap closes.
Drag either point on the graph, or use the slider —
P is the dark anchored dot, Q the coloured one with a
halo.
Gap h = 1
(Q sits at x = 2 )
Slide right to left to close the gap. h never reaches 0 — at
h = 0 the difference quotient would be 0⁄0.
From the right
From the left
▶ Close the gap
gap h
endpoints a , b
PQ P P — anchoredQ — drag me
Closing the gap
Slope of the secant as h shrinks
h Q atslope of PQ
Secant to Tangent
P stays put. Q slides along the curve toward it. The line
through P and Q is a secant, and its slope is the
difference quotient
As h shrinks, the secant pivots. If it settles on one particular
line, that line is the tangent and its slope is the derivative
What to watch for
The table converges from both sides . Switch sides and check that
the two columns of numbers head for the same place — if they do not,
there is no derivative there.
Try |x| with P at 0. From the right the slopes are all
+1, from the left all −1. They never agree, so no tangent exists — a
corner, not a smooth point.
Open on the falling ball at h = 1. The secant comes out
horizontal — s (1.5) = s (0.5) = 60, so the average
velocity over that interval is zero — because P and Q sit
either side of the peak. Ask the room whether the ball was therefore not
moving, before anyone touches the slider. This is the default position
on purpose; please leave it that way.The falling ball preset is the classic: height
−16t ²+32t +48 with P at t = 0.5. Average
velocities over shorter and shorter intervals pile up at 16 ft/sec,
which is the speedometer reading at that instant.
Leave Show the tangent off until students have called out the
number they think the slopes are heading for.
Putting this in a slide or a page
Every control is URL-addressable, so a lecture slide can link a stripped-down
version. For example:
?f=-16x^2%2B32x%2B48&var=t&a=0.5&window=0,2
&embed=true&controls=h&tangent=false
gives just the figure, the h slider and the table, with no function
box and no tangent revealed. See docs/url-api.md for every
parameter.
Keyboard
← → Change h (focus the slider first)
Space Run or stop the gap closing
? This dialog
Esc Close
Close