Intro

The Derivative as a Limit

~10 mincode completion

Implement central_difference(coeffs, x, h), where coeffs describes a polynomial and the derivative is estimated at x.

is the coefficient of , so [1, 0, 2] means . Evaluate with np.polyval(coeffs[::-1], x), which expects the highest power first.

Examples

x squared has slope 2x, which is 6 at x = 3

Input
central_difference([0, 0, 1], 3, 0.00001)
Output
6

A straight line has the same slope everywhere

Input
central_difference([4, 2], 100, 0.00001)
Output
2

A constant does not change, so its derivative is zero

Input
central_difference([7], 2, 0.00001)
Output
0

Hints

Hint 1

Convert the input with before doing elementwise work.

Hint 2

A common slip here: divides by h instead of 2h.

Requirements

  • coeffs: coefficients low power first, so [1, 0, 2] is 1 + 2x^2

  • x: the point to differentiate at

  • h: the step size

  • Return float: (f(x+h) - f(x-h)) / (2h)

Constraints

  • Allowed library: NumPy only

  • Time limit: 200 ms, Memory: 64 MB

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Where this shows up

~10 min

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Python
import numpy as np


def central_difference(coeffs, x, h):
    """
    Numerical derivative of a polynomial by central difference.

    Args:
        coeffs: coefficients low power first, so [1, 0, 2] is 1 + 2x^2
        x:      the point to differentiate at
        h:      the step size

    Returns:
        float: (f(x+h) - f(x-h)) / (2h)
    """
    # YOUR CODE HERE
    pass
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