Why does derivative 0.1 not produce exactly the identity function on real numbers?

Because 0.1 cannot be represented exactly in a Double.

See: Why 0.1 Does Not Exist In Floating-Point

>>> derivative 10 f41 1
1.0
>>> derivative 1 f41 1
1.0
>>> derivative 0.1 f41 1
1.0000000000000002

(See functions f42, err42 below.)

Evaluating the derivative at different values of x (the second argument):

>>> err42 1 6
0.25

>>> err42 1 0.5
0.25

>>> err42 1 1
0.25

It appears empirically that the error does not depend on x at all, it only depends on a.

Exploring the error for different values of a:

>>> err42 1 1
0.25

>>> err42 0.1 1
2.500000000002167e-3

>>> err42 0.01 5
2.4999998515795596e-5

>>> err42 0.02 5
9.999999903698154e-5

>>> err42 0.03 5
2.249999984940132e-4

After some trial and error, it is deduced that it follows a pattern like:

err(a) = a²/4

This formula is the implementation of the errA function below.

When x = 4, Df (4) = 48. What value of a produces an error of 1 percent at x = 4?

Use the formula we came up with, and solving for a:

a²/4 = (0.01)(48)
a² = (0.01)(48)(4)
a = √ ((0.01)(48)(4))

>>> sqrt $ 0.01 * 48 * 4
1.3856406460551018

When x = 0.1, Df (0.1) = 0.03. What value of a produces an error of 1 percent at x = 0.1?

Again, solving for a:

a²/4 = (0.01)(0.03)
a² = (0.01)(0.03)(4)
a = √ ((0.01)(0.03)(4))

>>> sqrt $ 0.01 * 0.03 * 4
3.4641016151377546e-2

Using the expression errA a from Exercise 4.2, using 0.01 for the value of a, solve for x in errA 0.01 >= 0.10 * x.

0.01²/4 ≥ 0.10x
0.10x ≤ 0.01²/4
x ≤ 0.01²/4/0.10

>>> 0.01 ** 2 / 4 / 0.10
2.5e-4

Therefore: x ≤ 2.5e-4

• - TO DO

• - TO DO