Common Mistakes
Frequent Errors
1. Reversing the subtraction in the numerator or denominator. 2. Computing $f(b)-f(a)$ and stopping (forgetting to divide by $b-a$). 3. Using the wrong pair of points from a table or graph. 4. Dropping the negative sign when the function decreases. 5. Writing a rate without units in an applied context. 6. Comparing "steepness by eye" instead of calculating both AROCs. 7. Treating average rate on $[a,b]$ as if it were $f'(a)$ or an instantaneous rate. 8. Averaging several consecutive small rates incorrectly when the problem asks for one big-interval AROC.
Why These Happen
Rate of change looks like "subtract and divide," so students rush the order of subtraction. Tables invite grabbing nearby values that are not the requested endpoints. Graphs invite eyeballing. Applied stems invite numbers without units.
How to Fix Them
- Always write $\dfrac{f(b)-f(a)}{b-a}$ before substituting.
- Circle the two $x$-values in the stem, then find matching outputs.
- After computing, check: if $f(b)>f(a)$ and $b>a$, the rate should be positive.
- Append units: output-unit / input-unit.
- For comparisons, compute both rates side by side before concluding.
Avoidance Scripts
- "Endpoints first, formula second, units last."
- "Numerator: change in $y$. Denominator: change in $x$."
- "Sign check: up means positive rate on that interval."
Quick Self-Check
- Can I restate the rate in one sentence with units?
- Did I use exactly the interval asked?
- If I swap $a$ and $b$, does the rate correctly flip sign?