2.2.3 Practice set: the derivative

Rates of change and tangents

Rates of change and tangents (lesson 2.2.1)

0 of 6 correct

1.
Find the average rate of change of $x^3$ from 1 to 3
2.
A tank holds 500 litres at 2:00 pm and 380 litres at 2:30 pm. What is its average rate of change, in litres per minute?
3.
Find the chord slope of $x^2$ at 5 with gap $h = -0.1$
4.
The slope of $x^2$ at 1 is 2. Write the tangent line at 1 in the form $mx + c$
5.
Find the average rate of change of $x^2 - 4x$ from 0 to 4
6.
The tangent to $x^2$ at 1 is $y = 2x - 1$ . How far below the curve is it at $x = 3$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The derivative and finite differences

The derivative and finite differences (lesson 2.2.2)

0 of 9 correct

1.
The derivative of $x^3$ is $3x^2$ . What is the slope of $x^3$ at $x = -2$ ?
2.
The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$ . What is the slope at $x = -2$ ?
3.
The chord slope of $f(x) = x^2 - 5x$ simplifies to $2x + h - 5$ . Find $f'(3)$
4.
A tank's volume $V$ in litres after $t$ minutes has $\frac{dV}{dt} = 3$ at $t = 10$ . About how many litres flow in over the next 0.5 minutes?
5.
Find the forward difference of $\frac{1}{x}$ at 2 with $h = 0.5$
6.
A central difference has error 0.0004 with $h = 0.01$ . About what error do you expect with $h = 0.001$ ?
7.
Find the chord slope of $\sqrt[3]{x}$ at 0 with gap $h = -0.001$
8.
What is the slope of $|x|$ at $x = -3$ ?
9.
The computer stores $2 + 10^{-20}$ and $2 - 10^{-20}$ as exactly 2. What does the central difference of $x^3$ at 2 give with $h = 10^{-20}$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Forward, backward and central differences

Write forward_diff(f, x, h), backward_diff(f, x, h) and central_diff(f, x, h), the three estimates from lesson 2.2.2. Each should also work when x is a NumPy array.

The tests compare them with known derivatives. They also check how fast each error shrinks: halving $h$ should about halve the forward error and quarter the central one.

See the tests

import math

import numpy as np


def _cube(x):
    return x**3


def test_forward():
    """Forward difference of x³ at 1 with h = 0.1 is 3.31"""
    expect_close(forward_diff(_cube, 1.0, 0.1), 3.31, what="forward_diff(x³, 1, 0.1)")


def test_backward():
    """Backward difference of x³ at 1 with h = 0.1 is 2.71"""
    expect_close(backward_diff(_cube, 1.0, 0.1), 2.71, what="backward_diff(x³, 1, 0.1)")


def test_central():
    """Central difference of x³ at 1 with h = 0.1 is 3.01"""
    expect_close(central_diff(_cube, 1.0, 0.1), 3.01, what="central_diff(x³, 1, 0.1)")


def test_central_more_accurate():
    """For e^x at 0 (true slope 1), the central difference is closer than the forward and backward ones"""
    fwd = abs(forward_diff(math.exp, 0.0, 0.01) - 1)
    bwd = abs(backward_diff(math.exp, 0.0, 0.01) - 1)
    cen = abs(central_diff(math.exp, 0.0, 0.01) - 1)
    assert cen < fwd and cen < bwd, f"central error {cen:.2e} should be below forward {fwd:.2e} and backward {bwd:.2e}"


def test_error_orders():
    """For x³ at 1, halving h from 0.01 about halves the forward error and quarters the central one"""
    fwd = abs(forward_diff(_cube, 1.0, 0.01) - 3) / abs(forward_diff(_cube, 1.0, 0.005) - 3)
    cen = abs(central_diff(_cube, 1.0, 0.01) - 3) / abs(central_diff(_cube, 1.0, 0.005) - 3)
    assert 1.9 < fwd < 2.1, f"the forward error should shrink about 2 times, it shrank {fwd:.3f} times"
    assert 3.9 < cen < 4.1, f"the central error should shrink about 4 times, it shrank {cen:.3f} times"


def test_arrays():
    """All three work on an array of points: x² has slope 2x"""
    x = np.array([0.0, 1.0, 2.5])
    expect_close(central_diff(lambda t: t**2, x, 1e-3), 2 * x, what="central_diff(x², [0, 1, 2.5])")
    expect_close(forward_diff(lambda t: t**2, x, 1e-3), 2 * x + 1e-3, what="forward_diff(x², [0, 1, 2.5])")
    expect_close(backward_diff(lambda t: t**2, x, 1e-3), 2 * x - 1e-3, what="backward_diff(x², [0, 1, 2.5])")

Rates of change and tangents

The derivative and finite differences

Coding exercise

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