NAME THE TOOL

Pile up, then ask how fast

Coach

Remember: the derivative asks how fast, and the integral builds the pile.

Pour water into a tank at 3 cups a minute. The tank fills up: that is the pile, the integral. Now a gauge on the side reads how fast the water level is rising. What does it read? 3 cups a minute, the same rate you poured. Piling up and asking how fast undo each other.

Quick check: Add 3 coins, then add −3 coins. Did the pile change?

THE SYMBOLS

The rate of the pile is the rate

P(x) = x∫0 f(t) dt

P(x) is the pile from 0 up to x. The opposites say:

P′(x) = f(x)

The rate of the pile at x is the rate f at x. And the other way: ∫ from a to b of a rate gives the change in the pile.

A + C is a flat start. Its rate is 0, so it never shows up in the rate.

Quick check: P(x) = ∫ from 0 to x of 3 dt. What is P′(x)?

WORKED EXAMPLE

P(x) = ∫ from 0 to x of 2t dt. How fast is P growing at x = 5?

P(x) = x∫0 2t dt

Strategy: Read the rate straight off the integral

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

1

Understand

P is a pile built from rate 2t. "How fast" asks for P′(5).

Why? Rate question, not pile question.

2

Plan

Opposites: P′(x) = 2x.

Why? The pile's rate is the rate that built it.

3

Solve

P′(5) = 2 · 5 = 10.

Why? Put 5 into the rate.

4

Look back

The pile is x² (triangle area), and the rate of x² is 2x = 10 at 5. ✓

Why? Pile first, then derivative: same answer.

Pile way

Pile = half of x × 2x = x². Rate of x² at 5: 10.

Opposites way

P′(5) = 2 · 5 = 10

Quick check: True or false: the answer is 25, the pile at 5.

SEE IT

Both directions

Trap: Giving the pile, not the rate

P(x) = ∫ from 0 to x of 4 dt at x = 7: the pile is 28; the rate is 4.

Trap: Thinking undoing leaves nothing

Undo a pile and you get the rate back, not 0.

Quick check: A pile was built from the rate 12. How fast is it growing?

WHAT TO LOOK FOR

Clue words and easy mix-ups

how fast is the pile growing, P′The rate: read it off the integral.
how much is in the pile, P(x)The pile: the area.
+ CA flat start. Rate 0.
∫ from a to b of a rateThe change in the pile.

Quick check: Discovery: you pile up a rate, then ask how fast it is piling. What do you get?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

P(x) = ∫ from 0 to x of 2t dt. P′(3)? The rate is 2t, so the answer is 2.

Coach

I read the 2 and stopped. The rate is 2t, and t is 3 here: 2 · 3 = 6.

Check habit: Put the x-value into the rate before you answer.

Quick check: P′(3) for that P?

TRAP CHECK

Run this before every answer

Rate question or pile question?
Rate of a pile = the rate that built it, at that x.
Put the x-value into the rate.
+ C does not change any rate.