Exercises | LLM Primer

MATH-C01-C-001Exercise: Substitute into an expression

If $x = 6$ , compute:

3x - 2

Your answer

Compute it first, then check your number.

HintSubstitute first

Replace $x$ with 6. Multiply before subtracting.

SolutionSubstitute and simplify

Given $x = 6$ :

3x - 2 = 3 \cdot 6 - 2 = 18 - 2 = 16

The important step is substitution: replace the name $x$ with the value it names, then do the arithmetic.

MATH-C01-C-002Exercise: Evaluate a function

Let

g(t) = t^2 - t

Compute $g(5)$ .

Your answer

Compute it first, then check your number.

HintUse 5 as the input

The notation $g(5)$ means use 5 as the input. Replace every $t$ with 5.

SolutionEvaluate the function

Use 5 as the input:

g(5) = 5^2 - 5 = 25 - 5 = 20

The notation $g(5)$ is a function call. It means run the same rule with 5 as the input.

MATH-C01-C-003Exercise: Read set membership

Let

Y = \{\text{cat}, \text{dog}, \text{bird}\}

Enter 1 if $\text{cat} \in Y$ means cat is one of the allowed labels.

Your answer

Compute it first, then check your number.

HintRead the symbol

The symbol $\in$ means "is in".

SolutionMembership sentence

Enter 1. The statement means cat is a member of the set $Y$ .

In this label example, it means cat is one of the allowed labels.

MATH-C01-C-004Exercise: Use mathematical indexing

Let

z = (10, 20, 30, 40)

Using mathematical indexing, what is $z_3$ ?

Your answer

Compute it first, then check your number.

HintCount mathematically

In this chapter, mathematical indexing starts at 1: $z_1$ is the first item.

SolutionThird item

Using mathematical indexing:

z_1 = 10,\quad z_2 = 20,\quad z_3 = 30,\quad z_4 = 40

So $z_3 = 30$ . The subscript names the position in mathematical indexing; it is not multiplication by 3.

MATH-C01-C-005Exercise: Expand a summation

If $a_1 = 2$ , $a_2 = 5$ , and $a_3 = 8$ , compute:

\sum_{i=1}^{3} a_i

Your answer

Compute it first, then check your number.

HintExpand first

Write the sum as $a_1 + a_2 + a_3$ .

SolutionExpand and substitute

Expand the summation:

\sum_{i=1}^{3} a_i = a_1 + a_2 + a_3

Substitute the values:

2 + 5 + 8 = 15

The sigma notation is only a compact way to write this repeated addition.

MATH-C01-C-006Exercise: Read an average-loss expression

In

\frac{1}{n}\sum_{i=1}^{n} \ell_i

enter 1 if the expression means "average the loss values."

Your answer

Compute it first, then check your number.

HintRead inside out

First read the summation. Then read the factor $\frac{1}{n}$ .

SolutionAverage loss

The summation adds the loss values:

\ell_1 + \ell_2 + \cdots + \ell_n

Multiplying by $\frac{1}{n}$ divides that total by n.

So the expression is the average loss over n examples.

MATH-C01-C-007Exercise: Separate indexing conventions

A reader says:

If $x = (4, 7, 9)$ , then $x_2$ is 9 because Python uses zero-based indexing.

Enter the correct value of $x_2$ in mathematical indexing.

Your answer

Compute it first, then check your number.

HintSeparate the conventions

In mathematical indexing, $x_1$ is the first item.

SolutionDo not mix indexing systems

In mathematical indexing, $x_1$ is the first item. So if

x = (4, 7, 9)

then:

x_1 = 4,\quad x_2 = 7,\quad x_3 = 9

In Python, the second item is x[1], not x[2].

MATH-C01-C-008Exercise: Check vocabulary membership

A model accepts one token from the set

V = \{\text{yes}, \text{no}, \text{maybe}\}

Enter 1 if $\text{yes} \in V$ is true.

Your answer

Compute it first, then check your number.

HintRead membership

Ask whether yes is listed in the set.

SolutionVocabulary membership

Enter 1. The token yes appears in $V$ , so $\text{yes} \in V$ is true.

MATH-C01-C-009Exercise: Compute average loss

Three examples have losses $4$ , $6$ , and $8$ .

What is the average loss?

Your answer

Compute it first, then check your number.

HintUse the average pattern

Average means total divided by count.

SolutionAverage

\frac{4 + 6 + 8}{3} = \frac{18}{3} = 6

Average means total divided by count. Here the total is 18 and the count is 3, so the average loss is 6.

MATH-C01-C-010Exercise: Read formulas for meaning

Enter 1 if reading a formula aloud should include what the formula means in the problem, not only the names of the symbols.

Your answer

Compute it first, then check your number.

HintThree layers

Symbol name, operation, meaning in context.

SolutionMeaning matters

Enter 1. A useful reading does not stop at "sigma" or "x sub i." It says what computation is being performed and what it means in the current problem.