0.2.3 Practice: reading formulas

Symbols, sets and functions

Symbols and functions (lesson 0.2.1)

0 of 7 correct

1.
The values $x_1, x_2, x_3$ are 7, 2, 9. Evaluate $x_1 - x_3$
2.
The values $x_1, x_2, x_3, x_4$ are 3, 1, 4, 1. Evaluate $x_3^{\,2}$
3.
How many elements does $\{n \in \mathbb{Z} : -2 \le n \le 3\}$ have?
4.
If $f(x) = x^2 - 3x$ , find $f(4)$
5.
If $f(x) = 5 - x$ , write $f(2a)$ in terms of $a$
6.
If $h(t) = 2t + 1$ , find $h(h(1))$
7.
If $f(x) = x^2$ , find $f(3) + f(-3)$

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.

Check yourself

Let

B = \{1, 4, 9, 16\}

. Which statement is false?

\forall x \in B,\ x > 0

\exists\, x \in B : x \text{ is odd}

9 \in B

\forall x \in B,\ x \text{ is even}

Sums and products

Sums and products (lesson 0.2.2)

0 of 9 correct

1.
Evaluate $\displaystyle\sum_{i=1}^{4} (i + 1)$
2.
Evaluate $\displaystyle\sum_{k=1}^{3} 10^k$
3.
Evaluate $\displaystyle\sum_{i=1}^{10} 0.5$
4.
Evaluate $\displaystyle\sum_{j=0}^{4} (-1)^j$
5.
If $\displaystyle\sum_{i=1}^{8} x_i = 40$ , what is the average $\bar{x}$ ?
6.
If $\sum a_i = 12$ and $\sum b_i = 5$ over the same indices, find $\sum (2a_i - b_i)$
7.
Evaluate $\displaystyle\prod_{i=1}^{4} \frac{i}{i+1}$
8.
Evaluate $\displaystyle\prod_{i=1}^{3} (1 + 0.1i)$
9.
Weights 0.5, 0.25, 0.25 and values 8, 4, 12. Evaluate $\sum w_i x_i$

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

Σ and Π as Python functions

Write two functions. Each takes a function term of the index, and the two limits.

sigma(term, lower, upper) returns $\sum_{i=\text{lower}}^{\text{upper}} \text{term}(i)$ .
product(term, lower, upper) returns $\prod_{i=\text{lower}}^{\text{upper}} \text{term}(i)$ .

Both limits are included. When upper < lower there are no terms: return 0 for the sum and 1 for the product, as in lesson 0.2.2. The tests pass in functions such as lambda i: i**2.

See the tests

def test_sum_of_squares():
    """The sum of i² for i from 1 to 4 is 30"""
    expect_close(sigma(lambda i: i**2, 1, 4), 30, what="sigma(lambda i: i**2, 1, 4)")


def test_upper_limit_included():
    """The upper limit is included: the sum of i for i from 1 to 100 is 5050"""
    expect_close(sigma(lambda i: i, 1, 100), 5050, what="sigma(lambda i: i, 1, 100)")


def test_other_lower_limit():
    """The sum can start anywhere: 2i − 1 for i from 2 to 5 is 24"""
    expect_close(sigma(lambda i: 2 * i - 1, 2, 5), 24, what="sigma(lambda i: 2*i - 1, 2, 5)")


def test_constant_term():
    """Adding 3 five times gives 15"""
    expect_close(sigma(lambda i: 3, 1, 5), 15, what="sigma(lambda i: 3, 1, 5)")


def test_empty_sum():
    """An empty sum (upper below lower) is 0"""
    expect_close(sigma(lambda i: i, 1, 0), 0, what="sigma(lambda i: i, 1, 0)")


def test_product_of_integers():
    """The product of i for i from 1 to 5 is 120"""
    expect_close(product(lambda i: i, 1, 5), 120, what="product(lambda i: i, 1, 5)")


def test_growth_factors():
    """Three growth factors 1.1 × 1.2 × 1.3 multiply to 1.716"""
    expect_close(product(lambda i: 1 + 0.1 * i, 1, 3), 1.1 * 1.2 * 1.3, what="product(lambda i: 1 + 0.1*i, 1, 3)")


def test_empty_product():
    """An empty product (upper below lower) is 1"""
    expect_close(product(lambda i: i, 3, 2), 1, what="product(lambda i: i, 3, 2)")

Symbols, sets and functions

Sums and products

Coding exercise

Free further reading