Slopes from numbers

What you will be able to do

After this lesson you can

• find the slope of a curve from numbers, without a formula,
• tell why the central difference is better than the forward difference,
• use np.gradient on measured data, and
• find the turning points of a curve.

The slope of a curve

In your calculus class, the derivative $f'(x)$ is the slope of the tangent to the graph of $f$ at the point $x$. For $f(x) = x^3 - 3x$, the rules give $f'(x) = 3x^2 - 3$. But what if you have no formula? An experiment gives you a list of numbers: the positions of a car at each second. A computer can still find the slope, from the numbers alone.

The idea is old. A tangent touches the curve at one point, but a secant line cuts it at two points, and a secant line is easy: its slope is the change of $f$ divided by the change of $x$. When the two points come close, the secant becomes the tangent.

for a small $h$. The picture shows $f(x) = \sin x$ at $x = 1$. The tangent has the slope $\cos 1 = 0.540$. The secant with $h = 1.2$ is very different (slope $-0.027$), but with $h = 0.25$ it is already close (0.430).

The forward difference

Predict first. When $h$ is ten times smaller, how will the error change? Then run it.

The error is ten times smaller when $h$ is ten times smaller. It is proportional to $h$. This is called a first-order method. To get 6 correct digits, we would need $h$ of about $10^{-6}$.

The central difference

We can do better with the same effort. Instead of looking only to the right of $x$, look at both sides: the secant through $x - h$ and $x + h$.

Predict first. The forward error at $h = 0.001$ is 0.00042. What do you expect for the central difference?

Remember: the central difference is nearly always the better choice.

Halve h and watch the errors: the forward error about halves, the central error falls to about a quarter.

A smaller h is not always better

If a small $h$ is good, is the smallest $h$ the best? Let us try.

The lesson: a numerical derivative has two kinds of error. Choose $h$ that is small, but not tiny.

Slopes from measured data

When you have measured values, you cannot choose $h$: it is the gap between the measurements. NumPy has the function np.gradient, which uses the central difference at every inner point, and a one-sided rule at the two ends.

Predict first. The positions are $x = 2t^2$, so the exact speed is $v = 4t$. Which values will be exact?

Real data has errors, and the derivative magnifies them: a small jump in the data gives a big jump in the slope. This is another reason not to use a tiny gap, and to look at the results with care.

Finding the turning points

At the top or the bottom of a curve, the slope is zero. So the slope changes sign there. This is the same sign-change test as in chapter 3.

This is how a computer finds the highest point of a path, the cheapest design, or the best angle, when no formula is available.

Common mistakes

• Using the forward difference when a central one is possible.
• Choosing $h$ as small as possible. Rounding errors grow.
• Forgetting to give the x values to np.gradient. Without them, it assumes a gap of 1.
• Believing the two end values. They are less accurate.
• Using the slope of noisy data without looking at it. The noise is magnified.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Derivative from numbers. You write the central difference as a function.
2. Build. Open the problem Speed from positions and the problem The top of the path.

Remember these

• The slope of a curve is about $\dfrac{f(x + h) - f(x)}{h}$. It is first order: the error is proportional to $h$.
• The central difference $\dfrac{f(x + h) - f(x - h)}{2h}$ is second order: the error is proportional to $h^2$.
• A tiny $h$ brings rounding errors. The best $h$ for the central difference is about $10^{-5}$.
• np.gradient(values, x) gives the slope at every point.
• A turning point is where the slope changes sign.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The forward difference has an error of 0.02 at $h = 0.1$. What do you expect at $h = 0.01$?

2. The central difference has an error of 0.02 at $h = 0.1$. What do you expect at $h = 0.01$?

3. Why do the end values of np.gradient have larger errors?

4. How do you find the turning points of a curve from numbers?