What is the slope of a curve at one point?

1. 1. Zoom in

   This is $y = x^2$ around the point $(1, 1)$. It has no one slope: it is steeper on the right than on the left. The number under the figure measures how far the curve in view is from a straight line.

   Question: Zoom in on the point, a thousand times. What will the curve look like?

   A straight line. The gap from straight shrinks with the zoom: $14\%$ of the view at ten times, $1.4\%$ at a hundred, $0.14\%$ at a thousand. Up close a smooth curve is a line, and that line's slope is what "the slope at a point" means.

   The figure: The curve y = x² near the point (1, 1), magnified 1000 times.

2. 2. Two points, one slope

   To compute that slope without a microscope, take a second point a gap $h$ to the right. The line through both, a secant, has a slope you can work out: rise over run. Drag the hollow point.

   Question: As $h$ closes in on $0$, what does the secant's slope do?

   It settles on $2$. Here the slope is exactly $\frac{(1+h)^2 - 1}{h} = 2 + h$, so it is $2.1$ at $h = 0.1$ and $2.01$ at $h = 0.01$. The secant turns onto the line the zoom showed.

   The figure: The curve y = x² near the point (1, 1), magnified 0.9 times. A secant runs from (1, 1) to the point at x = 1.01, with slope 2.010. The tangent at (1, 1) has slope 2.

3. 3. From the left as well

   A negative $h$ puts the second point on the left, where the secant is shallower. A limit has to agree from both sides, so check this side too.

   Drag the hollow point in from the left until the secant's slope is within $0.02$ of $2$.

   From the left the slope is $2 + h$ as well, just under $2$ and closing in. At $h = 0$ the quotient is $0/0$: the derivative is the number the slopes approach, never the quotient at $h = 0$.

   $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$$f^{\prime }(a)$ $=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$

   The figure: The curve y = x² near the point (1, 1), magnified 0.9 times. A secant runs from (1, 1) to the point at x = 0.99, with slope 1.990. The tangent at (1, 1) has slope 2.

4. 4. A slope is a rate

   Read the same figure as a ball rolling: after $t$ seconds it has gone $s(t) = t^2$ metres. Then a secant's slope is distance over time, an average velocity, and the limit is the velocity at the instant $t = 1$.

   Question: What is the average velocity from $t = 1$ to $t = 1.5$?

   $\frac{1.5^2 - 1^2}{0.5} = \frac{1.25}{0.5} = 2.5$ m/s. The $1.25$ is the distance covered, not the speed. Shrink the interval and the averages close in on $s'(1) = 2$ m/s, the speedometer's reading at $t = 1$.

   The figure: The curve y = x² near the point (1, 1), magnified 0.9 times. A secant runs from (1, 1) to the point at t = 1.50 s, with slope 2.500. The tangent at (1, 1) has slope 2.

5. 5. Where there is no slope

   Now bend the curve into $y = |x - 1| + 1$. It passes through the same point and has no gap or jump there, but it has a corner.

   Question: Zoom in on the corner, a thousand times. Will it straighten out?

   It stays a corner at every zoom: the gap from straight holds at the same share of the view. A secant to the right has slope $1$ for every $h > 0$, and to the left $-1$. The one-sided limits disagree, so $f'(1)$ does not exist, although $f$ is continuous at $1$.

   The figure: The curve y = |x − 1| + 1 near the point (1, 1), magnified 1000 times.

The film, as said, 1:42

1. 0:00

   Here is $y = x^2$ near the point $(1, 1)$. It is steeper to the right of the point than to the left.

2. 0:07

   So magnify it. Ten times.

3. 0:11

   A hundred times. The curve is beginning to look straight.

4. 0:18

   A thousand times, and it is a straight line. That line’s slope is the slope of the curve at the point.

5. 0:27

   To compute it, take a second point a gap $h$ to the right, and the line through both.

6. 0:35

   That secant's slope is rise over run, $\frac{(1+h)^2 - 1}{h}$, which is $2 + h$.

7. 0:39

   Close the gap. The secant turns, and its slope closes in on $2$.

8. 0:48

   From the left, with $h$ negative, the secant is shallower.

9. 0:53

   Close it again: the slope closes in on $2$ from below.

10. 1:01

    The same number from both sides is the derivative. At $h = 0$ the quotient is $0/0$: it is the limit, never the value.

11. 1:08

    Read $x$ as seconds and $y$ as metres, and a secant's slope is an average velocity.

12. 1:14

    Its limit, $2$ m/s, is the speedometer's reading at the instant $t = 1$.

13. 1:21

    A corner is different. Bend the curve into $y = |x - 1| + 1$: the same point, no gap, no jump.

14. 1:29

    Magnify it a thousand times.

15. 1:35

    Still a corner: slope $1$ on the right, $-1$ on the left. No one slope, so no derivative, though the curve never breaks.