Here's a real quantity changing over time. Drag the dot along the curve. The tangent's steepness is the instantaneous rate — and the gauge reads it off in the right units, filling or falling. Watch how the amount (the height) and the rate (the slope) move independently. Switch scenarios: the units change with the context.
On the exam, "interpret f′(a)" wants a full sentence — never just a number. It has four slots, and dropping any one loses the point. Fill them in the same order every time.
A table or two points only give you an average rate — the slope of a secant line across an interval. The derivative is the instantaneous rate — the slope of the tangent at a single point. Shrink the interval and watch the average home in on it.
The derivative f′(a) is the slope of the curve at that instant — how fast the output changes per unit of input, right now.
A rate's units are output per input. Gallons and minutes give gal/min; dollars and widgets give $/widget. The units come straight from the context.
f(a) is how much there is; f′(a) is how fast it's changing. A tank can be nearly full yet draining fast, or nearly empty yet filling fast.
Two points give the average rate — a secant slope. One point gives the instantaneous rate — the tangent slope, the limit as the interval shrinks.
When you only have a table, you can't get the exact derivative — but you can estimate it with the average rate over the tightest interval around the point, ideally one on each side.
A balloon's volume V (liters) at time t (seconds):
| t (s) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| V (L) | 3 | 10 | 17 | 30 | 41 |
Six problems on interpreting rates in context — units, direction, amount-vs-rate, and a table estimate. Try each before revealing.
Six fast checks. Pick an answer to see whether it's right and why — then work through all six.
"f′(6) = −4" is not an interpretation. The rate is −4 gallons per minute. Units are graded, and they tell you what the number means.
"There are −4 gallons" is nonsense. f(6) is the amount; f′(6) is the rate. A negative rate means the amount is decreasing, not that the amount is negative.
A rate is output per input. Gallons per minute, not minutes per gallon. Wrong order, wrong quantity.
f′(a) > 0 means increasing; f′(a) < 0 means decreasing. Say the direction — "draining," "cooling," "shrinking" — not just the size.
An average rate over [a, b] is a secant slope, not f′(a). It only estimates the instantaneous rate — and best when the interval is small and centered.
A tall curve isn't a fast one. Steepness — not height — is the rate. The tank can be full (high) and momentarily still (rate 0) at the same instant.