A rate of change has a size and a sign. A positive value means the quantity is increasing, a negative value means it is decreasing, and zero means it is momentarily not changing. In context questions the sentence matters as much as the number.
This lesson completes the module on tangents, normals and rates. It builds on using a derivative as a rate of change and on the related-rate model.
How do you write a good context sentence?
- Name the quantity, for example “the volume of water”.
- Say the direction: increasing or decreasing.
- Give the size with units: “at 18 m³ per day”.
- Give the moment: “when t = 2 days”.
A full sentence reads: “When t = 2 days, the volume is increasing at 18 m³ per day.” Every part earns credit.
Worked example
The volume of water in a reservoir is V = 500 + 30t − 3t² m³ after t days. Find dV/dt when t = 2 and when t = 8, interpret both, and find when the rate is zero.
Step 1, differentiate: dV/dt = 30 − 6t.
Step 2, at t = 2: 30 − 12 = 18. The volume is increasing at 18 m³ per day.
Step 3, at t = 8: 30 − 48 = −18. The volume is decreasing at 18 m³ per day.
Step 4, zero rate: 30 − 6t = 0, so t = 5. At this moment the volume is momentarily not changing. Here V = 500 + 150 − 75 = 575, the greatest volume, because the rate changes from positive to negative.
Check: the two rates 18 and −18 are equal in size and opposite in direction, which fits because 2 and 8 are equally far from 5. ✓
The mistake to watch for
The common error is to misread the negative sign as “the amount is 18 less” or to state only the number.
Mistaken sentence: “The rate is −18 so the volume is 18 m³ less.”
A rate is a speed of change, in m³ per day. It is not an amount removed.
The correction is to keep the units “per day” in the sentence, and to translate the sign into “decreasing”. If your sentence has no “per” in it, the meaning is usually wrong.
Check yourself
Try these, then open each answer.
1. The number of bacteria N in a sample satisfies dN/dt = −120 when t = 3 hours. Write a sentence that explains this.
Show answer
When t = 3 hours, the number of bacteria is decreasing at 120 per hour.
2. The height of a candle above a table is h = 10 − t² cm after t minutes (t ≥ 0). Find dh/dt when t = 2 and interpret it.
Show answer
dh/dt = −2t. At t = 2: −4. The height is decreasing at 4 cm per minute.
3. For the reservoir V = 500 + 30t − 3t², the rate at t = 2 is positive. Is the rate itself increasing or decreasing at t = 2?
Show answer
dV/dt = 30 − 6t, and its derivative is −6, which is negative. So the volume is increasing, but the rate of increase is decreasing: the reservoir fills more slowly each day.
Where this leads next
Bring the whole module together in the mixed practice set, which includes interpretation questions. If you want to keep a record of which kinds of slip you make, the mistake log and retest queue helps. The quadratic structure explorer shows how a turning point links to a zero rate.
Writing the meaning in words improves fastest when someone reads your sentences. In online one-to-one Additional Mathematics tuition, a teacher can show you what an examiner looks for.