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From secant to tangent

Change and motion · level 6, lesson 3 of 6

A question to start

In geometry, a tangent to a circle is a line that touches the circle at exactly one point, without cutting through it. Like a wheel resting on a road.

Is that the right idea for any curve? Try it on the U-shaped curve \(y = x^2\). The upright line \(x = 1\) meets the curve at exactly one point, \((1, 1)\), but it plainly is not "resting" on the curve: it slices straight through. And on the curve \(y = x^3\), the line that rests perfectly against the curve at \((1, 1)\) meets the curve again far away, at \((-2, -8)\).

So "touches at one point" is the wrong test. What is a tangent line, really? The answer comes from the last lesson, drawn as a picture, and it turns out to be the same thing as the rate of change at an instant.

The idea

Secant lines that swing

Take the curve \(y = x^2\) and the point \(P = (1, 1)\) on it. Pick a second point \(Q\) on the curve, further along, and draw the secant line through \(P\) and \(Q\). Its slope is the average rate of change between them (level 5, lesson 3).

Now slide \(Q\) along the curve towards \(P\). The secant line swings round, pivoting on \(P\).

PQQQ tangent at P(slope 2) 12 13
\(Q\) at \(x =\) Rise \(x^2 - 1\) Run \(x - 1\) Slope of secant \(PQ\)
2 3 1 3
1.5 1.25 0.5 2.5
1.2 0.44 0.2 2.2
1.1 0.21 0.1 2.1
1.01 0.0201 0.01 2.01

With \(Q\) at \(x = 1 + h\), the slope is \(\dfrac{(1 + h)^2 - 1}{h} = \dfrac{2h + h^2}{h} = 2 + h\). As \(Q\) slides to \(P\), \(h\) shrinks to 0, and the slopes head to 2. The secant lines settle down onto one line through \(P\) with slope 2.

That line is the tangent line to the curve at \(P\):

The tangent line at a point is the line through that point whose slope is the limit of the slopes of the secant lines.

Its slope is the rate of change at that point, also called the instantaneous rate of change. For the falling stone, the slope of the tangent to the distance-time curve is the speed at that instant.

(If \(Q\) comes from the other side, at \(x = 1 - h\), the slopes are \(2 - h\): 1.9, 1.99, ... The same limit, the same line. A tangent must work from both sides.)

The equation of the tangent

The tangent at \(P = (1, 1)\) has slope 2. A line with slope 2 is \(y = 2x + b\); it passes through \((1, 1)\), so \(1 = 2 + b\) and \(b = -1\). The tangent line is

\[y = 2x - 1.\]

Zoom in: why a smooth curve looks straight

Here is the curve \(y = x^2\) (dark) and its tangent at \(P\) (orange), seen through a zooming microscope centred on \(P\). Each window is narrower than the last.

x from 0 to 2x from 0.7 to 1.3x from 0.92 to 1.08

The more you zoom, the less the curve bends away from the line. Under a strong enough microscope, the curve and its tangent cannot be told apart. A curve that behaves like this at a point is called smooth there. This is the deep reason calculus works: close up, a smooth curve is a straight line, and straight lines are something we fully understand.

Why does it happen? Compare the curve and the tangent a small step \(h\) away from \(P\). The curve has height \((1 + h)^2 = 1 + 2h + h^2\); the tangent has height \(2(1 + h) - 1 = 1 + 2h\). The gap between them is exactly \(h^2\).

  • For \(h = 0.1\), the gap is 0.01: a tenth of the step.
  • For \(h = 0.01\), the gap is 0.0001: a hundredth of the step.

The gap shrinks much faster than the step. So when you zoom in (making the window, and so \(h\), small, then blowing the picture up to fill the screen), the gap becomes a smaller and smaller part of the picture, and vanishes.

Where there is no tangent: corners

Not every curve is smooth everywhere. A V shape, like the graph of \(y = |x|\) (the size of \(x\) without its sign), has a sharp corner at \((0, 0)\). Secants from the right all have slope \(+1\); secants from the left all have slope \(-1\). The two sides never agree, and however far you zoom, the corner stays a corner. There is no tangent there, and no single rate of change.

A real example: a ball bouncing on the floor. Its height-time graph has a sharp corner at each bounce, because its velocity flips from down to up almost instantly. At the exact instant of the bounce, "its velocity" has no one value.

Worked examples

1. Tangent to \(y = x^2\) at \(x = 3\).

  1. Secant slope from \((3, 9)\) to \((3 + h, (3 + h)^2)\): \(\dfrac{6h + h^2}{h} = 6 + h \to 6\).
  2. Tangent: \(y = 6x + b\) through \((3, 9)\): \(9 = 18 + b\), so \(b = -9\). \(y = 6x - 9\).

2. The stone's tangent. For \(d = 4.9t^2\), the tangent at \(t = 2\) has slope 19.6 (lesson 2) and passes through \((2, 19.6)\). Its equation: \(d = 19.6t + b\) with \(19.6 = 39.2 + b\), so \(b = -19.6\): \(d = 19.6t - 19.6\).

Meaning: if gravity were switched off at \(t = 2\), the stone would carry on at a steady 19.6 m/s, and its graph would follow this straight line. At \(t = 3\) it would be at \(58.8 - 19.6 = 39.2\) m instead of the real 44.1 m: the real stone is ahead because it keeps speeding up.

3. A tangent that meets the curve again. For \(y = x^3\), the tangent at \((1, 1)\) has slope 3 (last lesson, question 8), so it is \(y = 3x - 2\). Where else does it meet the curve? Try \(x = -2\): the curve gives \((-2)^3 = -8\), the line gives \(3 \times (-2) - 2 = -8\). They meet again at \((-2, -8)\). So a tangent can meet a curve more than once; what makes it a tangent is that it matches the curve's direction at the point, not how often it meets it.

4. Using a tangent as a shortcut. Close to \(P\), the curve and its tangent are nearly the same. Estimate \(1.03^2\) using the tangent \(y = 2x - 1\) at \((1, 1)\).

Tangent at \(x = 1.03\): \(2.06 - 1 = 1.06\). True value: \(1.0609\). The error is \(0.03^2 = 0.0009\), just as the gap rule says.

Try it

  1. For \(y = x^2\), work out the slope of the secant from \((1, 1)\) to \((1.05, 1.1025)\). Is it close to 2?
  2. Find the tangent line to \(y = x^2\) at \((2, 4)\).
  3. Find the tangent line to \(y = x^2\) at \((-1, 1)\). Which way does it slope, and why does that make sense?
  4. Find the tangent line to the stone's graph \(d = 4.9t^2\) at \(t = 1\). Where would the stone be at \(t = 2\) if it carried on at that steady speed?
  5. The tangent to \(y = x^2\) at \((0, 0)\) is \(y = 0\), the \(x\)-axis. How big is the gap between the curve and this tangent at \(x = 0.1\) and at \(x = 0.01\)?
  6. Use the tangent at \((1, 1)\) to estimate \(0.98^2\). What is the error?
  7. Explain why the graph of \(y = |x|\) has no tangent at \((0, 0)\).
  8. The line \(x = 1\) meets \(y = x^2\) only at \((1, 1)\). Explain why it is not the tangent there.
Answers
  1. \((1.1025 - 1) \div 0.05 = 0.1025 \div 0.05 = 2.05\). Yes (\(2 + h\) with \(h = 0.05\)).
  2. Slope \(\dfrac{(2 + h)^2 - 4}{h} = 4 + h \to 4\). Line: \(y = 4x + b\), \(4 = 8 + b\), \(b = -4\): \(y = 4x - 4\).
  3. Slope \(\dfrac{(-1 + h)^2 - 1}{h} = -2 + h \to -2\). Line: \(1 = 2 + b\), \(b = -1\): \(y = -2x - 1\). It slopes down: at \(x = -1\) the curve is still going down towards its lowest point at 0.
  4. Slope 9.8, through \((1, 4.9)\): \(4.9 = 9.8 + b\), \(b = -4.9\), so \(d = 9.8t - 4.9\). At \(t = 2\): \(19.6 - 4.9 = 14.7\) m (the real stone is at 19.6 m).
  5. The gap is \(x^2\): 0.01 at \(x = 0.1\) and 0.0001 at \(x = 0.01\).
  6. Tangent at \(x = 0.98\): \(1.96 - 1 = 0.96\). True: \(0.9604\). Error \(0.0004 = 0.02^2\).
  7. From the right, all secants through \((0, 0)\) have slope 1; from the left, slope \(-1\). They never agree, so there is no single limit slope, and zooming in always shows the same sharp corner.
  8. Its slope is undefined (upright), but the secant slopes near \((1, 1)\) all approach 2. The tangent must have the limit slope, 2. The line \(x = 1\) crosses the curve instead of matching its direction.

Common mistake

Defining a tangent as "touches the curve once". That works for circles, but not for most curves: upright lines can cross once without being tangents, and tangents can meet the curve again elsewhere. A tangent is the line with the limit of the secant slopes.

Drawing the tangent through two points of the curve. That is a secant. A tangent goes through one point of interest and points the way the curve is heading there.

Where it leads