C49 · Unit C7 Differential equations
Exponential models and interpreting growth constants
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 7.8
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
When a quantity changes at a rate proportional to itself, y minutes equals ky and y equals y sub 0 e^(kt). A positive k describes growth; negative k describes decay. The constant k has reciprocal-time units. Doubling time is ln2 over k for k>0; half-life is ln2/|k| for decay. A fixed percentage over a finite time interval determines k through a logarithm, not by simply inserting the percentage.
Worked example
An amount doubles every 5 hours. From e^(5k) equals 2, k equals ln2 over 5 per hour. With y sub 0 equals 30, y(t) equals 30e^((ln2)t over 5) equals 30 times 2^(t over 5). At t equals 10 hours it equals 120. The instantaneous fractional rate is ln2 over 5, not 0.2.
Explore the mathematics
Exponential models and interpreting growth constants
Trace input–output pairs and examine the domain of a function.
Scroll sideways to see the full diagram.
Predict before continuing
Choose an input in the displayed domain. Predict its output.
Read the complete explanation
- Choose an input in the displayed domain. Predict its output.
- Follow the input up to the curve, then read the output.
- Each unit input increase multiplies the output by 2. The growth is multiplicative.
- Change the example and distinguish a change of scale from a change of domain. A graph encodes permitted input–output pairs.
An initial amount 30 doubles at 5 hours to 60, then doubles again at 10 hours to 120. The continuous fractional rate is ln(2)/5 per hour.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
Continuous rate k differs from a discrete percentage growth rate. Always identify the model and time unit.
Investigate and explain
Compare discrete annual growth of 10% with continuous k equals 0.1. Find the continuous k that exactly matches the discrete annual multiplier 1.1.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.