Calculus I · Rates of change
Prerequisites: function notation, basic algebra and graph slopes. Goal: distinguish average and instantaneous rates and interpret their units.
Read and summarize
The average rate is the slope of the secant through two points. Move the second point toward the first: if the slopes approach a finite value, that limit is the derivative, the slope of the tangent at the first point.
f′(a) = limh → 0 [f(a + h) − f(a)] / h
OpenStax · Defining the Derivative
Worked example
Suppose charge is q(t) = t² coulombs, with time in seconds. From 1 s to 3 s the average current is (9 − 1) / (3 − 1) = 4 A. At 1 s, the difference quotient is [(1 + h)² − 1] / h = 2 + h. As h tends to zero, the instantaneous current is 2 A.
Estimate a rate from measurements
If you only have measured charge values, estimate current by dividing the change in charge by the change in time. Samples on both sides of the target time can give a useful estimate. Keep the units consistent: a change of 1 mC over 1 ms is 1 A.
Example: at 1.9 s and 2.1 s a sensor reports 3.61 C and 4.41 C. Estimate the current at 2 s as (4.41 − 3.61) / (2.1 − 1.9) = 4 A. This is an estimate from the samples. If you know the function is exactly q(t) = t², the derivative gives the exact current of 4 A at 2 s.
When the instantaneous rate is undefined
For q(t) = |t − 2|, the slope is −1 C/s before 2 s and +1 C/s after 2 s. The two sides disagree, so the derivative at the corner is undefined. Compare that with a flat segment, where the derivative is zero.
Practice
- For q(t) = 3t², find the average current from 0 s to 2 s.
- Find the instantaneous current at 2 s using the difference quotient.
Practice solutions
1. (12 − 0) / 2 = 6 A. 2. The quotient simplifies to 12 + 3h, so the limit is 12 A.
Mini-exam · 15 minutes
Close your notes. Show your work and units. For q(t) = t² + 2t coulombs:
- Find average current from 1 s to 4 s. (3 points)
- Use the limit definition to find current at 1 s. (5 points)
- Explain why the two values differ. (2 points)
Exam answer key
- q(4) = 24 and q(1) = 3. The average is (24 − 3) / 3 = 7 A.
- q(1 + h) = 3 + 4h + h². The difference quotient is 4 + h, which tends to 4 A.
- The rate rises over the interval. The secant averages the whole interval, while the tangent measures the rate at its start.
Virtual lab · Estimate current from charge
Use a forward difference and compare it with the exact derivative.
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Procedure
For q(t) = t² C, estimate current at t = 2 s with [q(2 + h) − q(2)]/h. Try h = 1, 0.1, and 0.01 s. Record the current and its absolute error against 4 A.
Expected results and answer check
The estimates are 5, 4.1, and 4.01 A. The errors are 1, 0.1, and 0.01 A. Expanding the numerator gives an estimate of 4 + h A.
These synthetic measurements are exact. Real sensors can make very small steps less reliable because of noise.