Practice · Grade 10 Calculus
Average Rate of Change Practice
Practice problems of rising difficulty on the average rate of change (f(b)−f(a))/(b−a), plus a boss question. Hint and worked solution per problem. Free.
Q1 of 6
0 correct
Find the average rate of change between the points:
(2, 5) → (6, 13)
Quick answer
What's the best way to practice the average rate of change?
Work through several problems in rising difficulty and apply the same formula every time: (f(b)−f(a))/(b−a), that is Δy divided by Δx. Watch the sign — if the function value drops, the rate is negative — and be extra careful with negative x-values in the denominator, where b − a goes wrong fast. Practice decimal results like 1.5 or −2.5 too, and after every calculation sanity-check it: if the function rises, the result must be positive.
HowTo
A 4-step solving strategy
This order works for any pair of points (a, f(a)) and (b, f(b)) with a ≠ b.
- 1Step 1 of 4
Read off the four values
Identify a, f(a), b and f(b). For the points (2, 5) and (6, 13) that means a=2, f(a)=5, b=6, f(b)=13. The x-values come first, the function values second.
- 2Step 2 of 4
Form Δy: f(b) − f(a)
Subtract the first y-value from the second: 13 − 5 = 8. If the result is negative, the function falls over the interval and the rate gets a minus sign.
- 3Step 3 of 4
Form Δx: b − a
Subtract the first x-value from the second: 6 − 2 = 4. With negative x-values take care, e.g. 2 − (−2) = 4 — minus and minus make plus.
- 4Step 4 of 4
Divide Δy by Δx and check
Divide: 8 ÷ 4 = 2. Plausibility: if the graph rises, the result is positive. With a = b this fails — then Δx = 0 and the rate is undefined.
Examples
Worked practice problems with full solutions
Four typical high-school problems. Try each one yourself first, then compare with the solution.
Easy
Compute: (1, 3) → (4, 15)
(15 − 3) ÷ (4 − 1)
= 12 ÷ 3
= 4
Plausible: graph rises → positive result ✓
Standard case. Δy = 12, Δx = 3, so the secant slope = 4.
Easy
Compute: (2, 10) → (6, 2)
(2 − 10) ÷ (6 − 2)
= −8 ÷ 4
= −2
Plausible: y-value falls → negative result ✓
On average the function falls by 2 per x-unit. Don't drop the minus sign.
Medium
Compute: (0, 2) → (5, 2)
(2 − 2) ÷ (5 − 0)
= 0 ÷ 5
= 0
Plausible: f(a) = f(b) → horizontal secant ✓
Start and end values are equal, so the average rate of change is 0.
Hard
Compute: (−1, 6) → (3, −2)
(−2 − 6) ÷ (3 − (−1))
= −8 ÷ 4
= −2
Plausible: denominator 3 − (−1) = 4, not 2 ✓
Negative x-value: b − a = 3 − (−1) = 4. Minus times minus makes plus in the denominator.
Pitfalls
Common mistakes — and how to avoid them
These five traps show up again and again with the average rate of change.
Numerator and denominator swapped
It's Δy ÷ Δx, not Δx ÷ Δy. The y-difference first, divided by the x-difference. Flip it and you get the reciprocal.
Forgetting the sign
If the function value drops, the rate is negative. (2 − 10) ÷ (6 − 2) = −2, not 2. Check: y-value got smaller → negative.
Subtracting negative x-values wrongly
In b − a with a negative a, minus times minus applies: 3 − (−1) = 3 + 1 = 4. The denominator often gets computed too small here.
Mixing up the order of the differences
Keep the order consistent: f(b) − f(a) on top, b − a on the bottom — both measured from the same endpoint. Otherwise the sign comes out wrong.
Plugging in a = b
If the two x-values are equal, Δx = 0 and the rate is undefined. You can't divide by zero — the points must have different x-values.
Study
Practice with a plan — three short tips
15 minutes at a time, not 90 in one go
Three short sessions on three days stick better than one long session the night before the test. The keyword is "spaced repetition".
Solve first, then look at the solution
Write out your working before you reveal the hint. Active recall is three to four times more effective for learning than passive reading.
On every wrong answer: ask why
Was it a sign error? A negative x-value subtracted wrongly? Note the cause — next time you'll spot the mistake instantly.
FAQ
Frequently asked questions about practicing
Glossary
Terms in one sentence
- Average rate of change
- The average change per x-unit over an interval: (f(b)−f(a))/(b−a).
- Secant
- The straight line through two points of a function's graph.
- Δy (delta y)
- The change in the y-values: f(b) − f(a).
- Δx (delta x)
- The change in the x-values: b − a.
- Slope
- The ratio Δy/Δx, the measure of how steep a line is.
- Interval [a, b]
- The range of x between the two points, over which the average is taken.
- Derivative
- The limiting case of the average rate as b approaches a — the instantaneous rate of change.