Interpreting Derivatives in Context, with Units
- State the units of a derivative from the units of the two quantities
- Write a sentence interpreting f′(a) that would earn an interpretation point
- Distinguish a rate, a change, and an amount in an applied problem
Units come free, and they carry a point
The units of dy/dx are always units of y per unit of x — that is what the notation says, and it is why the notation is written as a fraction. If W(t) is the weight of a puppy in pounds and t is measured in weeks, then W′(t) is in pounds per week. If C(x) is the cost in dollars of producing x widgets, C′(x) is in dollars per widget. This is free credit on the free-response section, and it is also a diagnostic: if the units of your answer come out wrong, you have set the problem up incorrectly, and checking them takes five seconds.
The three-part interpretation sentence
An interpretation point requires a sentence containing three things, and readers check for all three. (1) When — the specific time or input value. (2) What is changing and in which direction — increasing or decreasing, stated in the problem's own language rather than as "f is increasing." (3) The rate with units. So for W′(6) = 1.4 with W in pounds and t in weeks: "At 6 weeks, the puppy's weight is increasing at a rate of 1.4 pounds per week." Compare that with "W′(6) = 1.4," which restates the given and earns nothing, or "the puppy is growing," which has no number and no units. Write the full sentence every time; it is the cheapest point on the exam.
Rate, change, and amount are three different things
Applied problems distinguish these carefully and so must your answer. If R(t) is a rate in gallons per minute, then R(5) is a rate at an instant, ∫R(t)dt over an interval is a change in gallons, and an amount requires an initial value plus that change. The tell in the wording: "how fast" or "at what rate" wants a derivative; "how much" or "by how much did it change" wants an integral of a rate; "how much is there" wants an initial amount plus an integral. A related trap is the phrase "the rate is decreasing," which is a statement about the second derivative — the quantity may still be increasing while its rate of increase slows, and that combination appears on the exam constantly.
The temperature of coffee is C(t) degrees Fahrenheit t minutes after pouring. Given C(10) = 145 and C′(10) = −3.2, write an interpretation of C′(10) and estimate C(12).
- 1.Identify the units: C is in degrees Fahrenheit and t in minutes, so C′ is in degrees Fahrenheit per minute.
- 2.The sign is negative, so the temperature is decreasing — the coffee is cooling.
- 3.Interpretation sentence: at 10 minutes after pouring, the coffee's temperature is decreasing at a rate of 3.2 degrees Fahrenheit per minute.
- 4.For the estimate, use the tangent line: C(12) ≈ C(10) + C′(10)·(12 − 10) = 145 + (−3.2)(2).
- 5.145 − 6.4 = 138.6 degrees Fahrenheit.
When you interpret a negative derivative, say "decreasing at a rate of 3.2 units per minute" rather than "increasing at −3.2." Both are mathematically true; only the first reads as an interpretation. And never write "decreasing at a rate of −3.2," which says the opposite of what you mean.
Water flows into a tank at a rate of R(t) gallons per hour. What are the units of ∫R(t)dt from t = 0 to t = 3?
A population P(t) satisfies P′(t) > 0 and P″(t) < 0. Which statement is correct?
Answer the 2 checkpoints as you read.
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