NAME THE TOOL

One super-thin slice

Coach

Remember the derivative? We shrank the time to a blink. Now keep that blink and look at it: one super-thin slice.

Picture a cake. Cut one very thin slice. That slice is dx: a tiny width of x. For the piggy bank, a thin slice of time is dt, and the coins that drop in during that slice are rate × dt.

Quick check: Is dx a height or a width?

THE SYMBOLS

height × dx

dx is one symbol. It does not mean d times x. Read it "a tiny bit of x."

A strip of height f(x) and width dx has area:

f(x) · dx = height × width

For the bank: a rate of 3 coins per day, for a slice of 0.1 day, drops in 3 × 0.1 = 0.3 coins.

Quick check: A strip is 4 tall and 0.1 wide. Is its area 0.4 or 4.1?

WORKED EXAMPLE

A strip is 5 tall and 0.2 wide. What is its area?

Strategy: Height times width

Name the rate or the pileTry a small caseDraw itAsk whyAnswer the exact thing asked

1

Understand

Height 5, width dx = 0.2. We want the area.

Why? A strip is a thin rectangle.

2

Plan

area = height × width.

Why? Rectangle rule.

3

Solve

5 × 0.2 = 1.

Why? 0.2 is 2 tenths; 5 × 2 tenths = 10 tenths = 1.

4

Look back

5 strips this wide make a width of 1 and an area of 5. One strip is 1/5 of that: 1. ✓

Why? A different count gives the same answer.

Tenths way

5 × 2 tenths = 10 tenths = 1.

Symbol way

f(x) · dx = 5 × 0.2 = 1

Quick check: True or false: the area is 5, because the strip is 5 tall.

SEE IT

Counting slices

A width of 2 cut into slices of dx = 0.5 gives 2 ÷ 0.5 = 4 slices. Thinner slices give more of them, and the total stays the same.

Trap: Forgetting the width

A strip 4 tall and 0.1 wide is not 4. It is 4 × 0.1 = 0.4.

Trap: Adding height and width

4 + 0.1 = 4.1 is not an area.

Trap: Multiplying when you should divide

Slices in a width of 2 at dx = 0.5: 2 ÷ 0.5 = 4, not 2 × 0.5 = 1.

Quick check: A width of 2 is cut into slices of dx = 0.1. How many slices?

WHAT TO LOOK FOR

Clue words and easy mix-ups

dx, dt, a thin slice, a stripA width. Multiply it by the height or rate.
tall, high, rateThe height of the strip.
how many sliceswidth ÷ dx.
area of the strip, coins in that sliceheight × dx, or rate × dt.

Quick check: Rate 3 coins per day, slice dt = 0.1 day. Coins in that slice?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

A strip is 10 tall and 0.5 wide. Its area is 10. It is a tall strip.

Coach

I gave only the height. The strip is half of 1 wide, so its area is half of 10: 5.

Check habit: Say both numbers out loud, height and width, before you multiply.

Quick check: 10 tall, 0.5 wide. Area?

TRAP CHECK

Run this before every answer

Which number is the height? Which is dx?
Multiply them. Do not add.
Check the decimal: × 0.1 is one tenth.
For a count of slices, divide the width by dx.