Rates in Context
A derivative is a rate: how fast a quantity changes per unit of its input. Learn to say what f′(a) means the way the exam scores it, with the input value, the quantity, the direction, the size of the rate, and the units.
What the derivative says
where the input value, not an interval
what the quantity, in context
which way increasing or decreasing, from the sign
how fast the size of the rate, always positive
units units of the output ÷ units of the input
Units, step by step
Right now
Try this
How the exam scores an interpretation
- WhereName the input value: “at time t = 3 hours”, “at a height of 2 kilometers”. Not an interval.
- WhatName the quantity in context: “the amount of water in the tank”.
- Which wayIncreasing for a positive rate, decreasing for a negative one.
- How fastGive the size of the rate as a positive number. “Decreasing at a rate of −8” does not earn credit.
- UnitsUnits of f ÷ units of the input, like gallons per hour. A second derivative divides again: gallons per hour per hour.
- Rate in − rate outAn amount increases when more comes in than goes out.
Every problem is generated fresh, and every answer key is computed in code and checked with a computer algebra system. Scores stay in this browser only. Topic 4.1 Interpreting the Meaning of the Derivative in Context and Topic 4.3 Rates of Change in Applied Contexts Other Than Motion. Estimates from tables use the difference quotient of Topic 2.1.