Reading Formulas and Substituting Values, foundations for calculus lesson

A taxi fare, the height of a thrown ball and the area of a circle are all given by formulas. Each one is a recipe that turns a number you know into a number you want. Calculus asks more of these recipes than any earlier course, because it feeds them inputs like $a + h$ and compares the outputs. Reading a formula carefully and substituting without slips is the habit everything else rests on.

Key terms

Formula

Definition A formula is an equation that expresses one quantity in terms of others by a fixed calculation. In this course every quantity in a formula is a real number, and a formula is used only for the inputs where its calculation makes sense in the real numbers and in the situation it describes.

In plain words A formula is a set of instructions written in symbols. It tells you exactly what to do with the numbers you put in.

Example The area of a circle with radius $r$ is

A = \pi r^2.

For a radius of $3$ centimeters the area is $9\pi \approx 28.27$ square centimeters. A negative radius makes no sense for a circle, so the formula is used only with $r > 0$ , or $r \ge 0$ if a single point counts as a circle of radius zero.

Variables and constants

Definition A variable is a symbol that stands for a quantity that can take different values, such as an input you are free to choose. A constant is a symbol or number whose value stays fixed throughout the problem. Some constants are named numbers, such as $\pi$ , and some are parameters that are fixed for one situation but could differ in another.

In plain words Variables are the parts that change from one use of the formula to the next. Constants stay the same every time.

Example A ball is thrown upward from a balcony, and its height above the ground after $t$ seconds is

h = 20 + 15t - 4.9t^2 \text{ meters.}

Here $t$ and $h$ are variables. The numbers $20$ , $15$ and $4.9$ are constants. The $20$ is the starting height in meters, the $15$ is the starting upward speed in meters per second, and the $4.9$ comes from gravity. A ball thrown from a different balcony would change the $20$ , but within this problem it never changes.

Function notation

Definition When a formula gives exactly one output for each allowed input, we name it with a letter and write, for example, $p(x)$ for the output at the input $x$ . In this notation the parentheses mark the input. They do not mean multiplication, so $p(x)$ is not $p$ times $x$ , and $p(2)$ means the output of $p$ when the input is $2$ .

In plain words The letter outside names the calculation and the thing inside the parentheses is what you feed it.

Example Write the height of the ball as

h(t) = 20 + 15t - 4.9t^2.

Then $h(2)$ is the height after $2$ seconds. It is not $h$ times $2$ . It is a single number, computed in the next section.

Substitution

Definition To evaluate a formula at a given input, replace the variable by that input at every place where the variable appears, and replace it by the same value each time. Then carry out the calculation. When the input is negative, a fraction or an expression, place it inside parentheses as you substitute.

In plain words Every copy of $x$ gets the same number, and that number goes in wrapped in parentheses so it stays together.

Example Take the polynomial formula $p(x) = 2x^2 - 3x + 5$ and the input $-2$ . Both copies of $x$ become $(-2)$ .

\begin{aligned} p(-2) &= 2(-2)^2 - 3(-2) + 5 \\[0.4em] &= 8 + 6 + 5 = 19. \end{aligned}

The same rule gives the ball’s height after $2$ seconds,

\begin{aligned} h(2) &= 20 + 15(2) - 4.9(2)^2 \\[0.4em] &= 20 + 30 - 19.6 \\[0.4em] &= 30.4 \text{ meters.} \end{aligned}

Parentheses

Definition Parentheses group an expression so that it is treated as a single number. Whatever operation stands just outside a pair of parentheses acts on the whole value inside. When a variable is replaced by an expression such as $a + h$ , the parentheses around it are required, and leaving them out changes the meaning.

In plain words Parentheses keep a package together. An exponent or a minus sign in front of the package applies to all of it.

Example For $q(x) = x^2 - 4x$ the input $a + h$ must go in as a package.

q(a + h) = (a + h)^2 - 4(a + h).

Without parentheses you would write $a + h^2 - 4a + h$ , which is a different expression. At $a = 1$ and $h = 2$ the correct version gives $9 - 12 = -3$ , which equals $q(3)$ , while the careless one gives $3$ . This exact kind of input appears in every difference quotient in calculus.

Squares and negative numbers

Definition The square of a real number $b$ is $b^2 = b \cdot b$ . An exponent applies only to the symbol or parenthesized group directly in front of it. So $(-3)^2$ means $(-3)(-3) = 9$ , while $-3^2$ means $-(3^2) = -9$ , the negative of the square of $3$ .

In plain words A square is never negative, but a minus sign written in front of a square is applied after squaring. Parentheses decide which one you mean.

Example With $f(x) = x^2 + 1$ and the input $-3$ , the correct substitution is

f(-3) = (-3)^2 + 1 = 9 + 1 = 10.

Writing $-3^2 + 1$ instead gives $-8$ , which is wrong for this question. The square applies to the whole input, sign included.

Order of operations

Definition Unless parentheses say otherwise, an expression is evaluated in this order. First simplify inside parentheses and other grouping symbols, including numerators, denominators and the inside of roots. Next evaluate powers and roots. Then do multiplication and division from left to right. Last, do addition and subtraction from left to right.

In plain words Groups first, then powers, then multiplying and dividing, then adding and subtracting.

Example Evaluate $8 - 2 \cdot 3^2 + (4 - 1)^2$ . The parentheses give $4 - 1 = 3$ . The powers give $3^2 = 9$ twice. The multiplication gives $2 \cdot 9 = 18$ . Then

8 - 18 + 9 = -1.

Doing the subtraction $8 - 2$ first would give $6 \cdot 9 + 9 = 63$ , which is wrong.

Units

Definition The unit of a quantity says what is being measured and on what scale, such as meters, seconds or dollars. In a formula from a real situation every term that is added or subtracted must have the same unit, and the unit of the output follows from the units of the inputs and constants. An answer to an applied question is complete only when it carries its unit.

In plain words Units tell you what the number means. You can only add things measured the same way, and the answer should say what it counts.

Example A plumber charges a fixed visit fee and an hourly rate,

C(n) = 40 + 12n \text{ dollars}

for $n$ hours of work. The $40$ is in dollars and the $12$ is in dollars per hour, so $12n$ is in dollars too and the sum makes sense. For a three-hour job

C(3) = 40 + 12(3) = 76 \text{ dollars.}

Reading the numbers again checks the meaning. The input $3$ is a time in hours and the output $76$ is a cost in dollars, not a rate.

Check yourself

For $f(x) = 3x^2 - x + 2$ , find $f(-1)$ and $f(2)$ .

Show the answer

Substitute with parentheses at both copies of $x$ .

\begin{aligned} f(-1) &= 3(-1)^2 - (-1) + 2 \\[0.4em] &= 3 + 1 + 2 = 6, \\[0.4em] f(2) &= 3(2)^2 - 2 + 2 = 12. \end{aligned}

Evaluate $-2^2 + 5$ and $(-2)^2 + 5$ . Which one is $g(-2)$ for $g(x) = x^2 + 5$ ?

Show the answer

The first is $-4 + 5 = 1$ and the second is $4 + 5 = 9$ . The value $g(-2)$ is $9$ , because the square applies to the whole input $-2$ .

Using the plumber’s formula $C(n) = 40 + 12n$ dollars, what does a five-hour job cost, and what are the units of the number $12$ ?

Show the answer

The cost is $C(5) = 40 + 60 = 100$ dollars. The number $12$ is a rate in dollars per hour, which is why multiplying it by hours gives dollars.

Reading Formulas and Substituting Values

Theory, part 1

Practice, part 2

Key terms

Formula

Variables and constants

Function notation

Substitution

Parentheses

Squares and negative numbers

Order of operations

Units

Check yourself

How this lesson connects

Used later in

Where this leads in Calculus