Quarry School

Average rate of change from a formula

Explain it like I am five

When the function is given as a formula, you build a small table of your own. Plug the left endpoint into the formula to get its output, plug the right endpoint in to get its output, and then subtract and divide exactly as before. A formula is like a recipe: feed it an input and it cooks up the output. You only have to run the recipe twice, once for each end of the interval. Write the whole input in parentheses, especially if it is negative. Fractions show up often, so keep them as fractions until the end.

input d (cm)output F(d) (newtons)2[[1|2]]6[[1|18]]↓ evaluate: input given, read the output below it
Evaluate the formula at the two inputs to build the endpoint table.
Reminder
  • Common denominators. When the smaller bottom divides the bigger one, use the bigger bottom: 12 = 918. Otherwise the product of the bottoms works.
  • Dividing by a whole number. −49 ÷ 4 = −49 · 14. Division by a nonzero number means multiplication by its reciprocal.
  • Simplifying a fraction. Divide top and bottom by the same nonzero number: −818 = −49, using ÷ 2 in both.
  • Whole numbers as fractions. 4 = 41 = 82, so a whole number can be renamed to match a fraction's bottom.
Why it works. The formula supplies the output at each allowed input, so evaluating it at the two endpoints gives the points you would read off a perfect graph. Exact fractions avoid early rounding errors. For example, −19 is exact, while −0.11 differs by about 0.001. Evaluate first, subtract next, and divide last, so the output change and the rate remain separate quantities. The units come from the comparison: force in newtons divided by distance in centimeters gives newtons per centimeter.
RuleOn [a, b] = [left end, right end], with a < b and both outputs defined, average rate = f(b)−f(a)b−a. Evaluate the two endpoint outputs first, then subtract and divide. Preserve all nonzero-bottom and square-root restrictions.
The same idea, five ways
Say it

Say 'evaluate both ends, subtract the outputs, divide by the input span'.

Write it

A formula gives the two endpoint outputs needed for the same average-rate calculation.

In math
  • f(b)−f(a)b−a, a < b
  • f(a): output at the left input; f(b): output at the right input
Like

Run one recipe with two different ingredients, then compare the results per change in the ingredient.

See it
input d (cm)output F(d) (newtons)2[[1|2]]6[[1|18]]↓ evaluate: input given, read the output below it
Evaluate the formula at the two inputs to build the endpoint table.
The same idea, other ways
Run the recipe twice

The force recipe gives F(2) = 12 and F(6) = 118. Those are two outputs, not two rates. Put them in the endpoint table before computing the changes.

input d (cm)output F(d) (newtons)2[[1|2]]6[[1|18]]↓ evaluate: input given, read the output below it
Evaluate the formula at the two inputs to build the endpoint table.
Separate change from rate

The push changes by −49 newton over 4 centimeters. Dividing by 4 turns the whole-interval change into −19 newton per centimeter.

2461(2, [[1|2]])(6, [[1|18]])
The dashed ruler through the two curve points has slope −19; it is not part of F.
Parentheses hold an input together

For f(x) = 2 − x2, the input −3 makes f(−3) = 2 − (−3)2 = −7. Squaring the input happens before the outside subtraction. This prevents a sign mistake before the rate calculation begins.

−2−8−6−4−22(−3, −7)(1, 1)
The dashed ruler through the endpoint points has slope 2; it is not part of the curved function.
.1Negative inputs in a square formula

A negative input is one complete number. Keep it in parentheses so a square acts on the whole number. The minus outside x2 then subtracts that positive square.

  • (−3)2 = 9, while −32 means −(32) = −9.
  • For 2 − x2, the outside subtraction remains outside after substitution.
−2−8−6−4−22(−3, −7)(1, 1)
The dashed ruler through the endpoint points has slope 2; it is not part of the curved function.
Worked exampleA negative endpoint needs parentheses

This asks for the average output change per input unit, using one negative input. Find the average rate of f(x) = 2 − x2 on [−3, 1].

−2−8−6−4−22(−3, −7)(1, 1)
The dashed ruler through the endpoint points has slope 2; it is not part of the curved function.
  1. f(−3) = 2 − (−3)2 = 2 − 9 = −7.The whole input is −3. Squaring (−3) gives (−3)(−3) = 9; the outside subtraction stays outside.
  2. f(1) = 2 − 12 = 2 − 1 = 1.Evaluate the other endpoint in the same formula.
  3. Output change: 1 − (−7) = 1 + 7 = 8.Subtract the whole starting output; taking away a negative adds its opposite.
  4. Input change: 1 − (−3) = 1 + 3 = 4.The endpoint order must match the output subtraction.
  5. 84 = 2.Dividing the net change 8 by the span 4 gives the change per input unit.
Answer
2.
Check The starting output −7 plus 4 · 2 reaches −7 + 8 = 1, the ending output.
.2Negative inputs in a cube formula

A cube multiplies the same number three times. Three negative factors give a negative result. Keep the multiplication term separate and handle its sign too.

  • (−2)3 = (−2)(−2)(−2) = −8.
  • −2(−2) = +4, so these two contributions have opposite signs.
−22−20−16−12−8−448121620(−2, −4)(1, −1)
The dots are samples of the smooth curve x3 − 2x. The dashed ruler through the endpoint dots has slope 1 and is not part of the function.
Worked exampleA harder negative endpoint with a cube

This asks for the average output change per input unit when the formula has a cube. Find the average rate of f(x) = x3 − 2x on [−2, 1].

−22−20−16−12−8−448121620(−2, −4)(1, −1)
The dots are samples of the smooth curve x3 − 2x. The dashed ruler through the endpoint dots has slope 1 and is not part of the function.
  1. f(−2) = (−2)3 − 2(−2) = −8 + 4 = −4.Three factors of −2 give −8; multiplying −2 by −2 gives +4.
  2. f(1) = 13 − 2(1) = 1 − 2 = −1.Do the power and multiplication before subtracting.
  3. Output change: −1 − (−4) = −1 + 4 = 3.Subtracting the negative starting output adds 4.
  4. Input change: 1 − (−2) = 1 + 2 = 3.The input subtraction uses the same endpoint order.
  5. 33 = 1.A net rise of 3 over an input span of 3 averages 1 per unit.
Answer
1.
Check −4 + 3 · 1 = −1, so the average change reaches the ending output.
Strategy: step by step
  1. Check that both inputs are allowed: no bottom may be zero, and a real square root needs a nonnegative inside.
  2. Evaluate f(a), placing the whole input in parentheses at every variable position.
  3. Evaluate f(b) the same way.
  4. Subtract f(b) − f(a). If fractions occur, make their bottoms match first.
  5. Divide by b − a. Simplify exact fractions and attach units.
Strategy
Build the endpoint table from a formula
1
Does the formula have an input in a bottom?
YesExclude inputs that make that bottom 0; verify both endpoints are allowed.
NoContinue.
↓
2
Does a real square root occur?
YesCheck its whole inside is at least 0 at each endpoint; if it is in a bottom, it must be greater than 0.
NoContinue.
↓
3
Is an endpoint negative?
YesReplace the variable by the entire negative number in parentheses.
NoUse parentheses for the input anyway.
↓
4
Are the output fractions over different bottoms?
YesChoose a common denominator, rename both fractions, and subtract only the tops.
NoSubtract the two outputs directly.
  1. Check endpoint restrictions.
  2. Evaluate the formula twice using powers before multiplication, then addition and subtraction.
  3. Compute the output difference with parentheses around a negative starting output.
  4. Divide by the matching input difference and keep the result exact.
Worked exampleAverage rate of change of the electric force as two charges move apart

Two small charged spheres repel each other. The size of the force between them is F(r) = 36r2 newtons, where r is the distance between the spheres in meters. Find the average rate of change of F on the interval [2, 4]. Give an exact answer with units.

input distance r (m)output force F (N)293442.25↓ evaluate: input given, read the output below it
Values of F(r) = 36/r2 at r = 2, 3 and 4 m. The endpoints r = 2 and r = 4 are used for the average rate of change.
  1. Check the inputs. For r = 2, r2 = 4 ≠ 0. For r = 4, r2 = 16 ≠ 0. There is no square root, and 2 < 4.Both endpoint outputs must be defined, and the bottom of a fraction may not be zero.
  2. Evaluate F(2) = 36(2)2 = 364 = 9.Putting the input in parentheses gives the left-end output.
  3. Evaluate F(4) = 36(4)2 = 3616 = 94.The right-end output is found the same way and simplified.
  4. Subtract: F(4) − F(2) = 94 − 9 = 94 − 364 = −274.Fractions can be subtracted only after their bottoms match. Here 9 is rewritten as 364.
  5. Divide by 4 − 2 = 2: −27/42 = −278 N per m.The average rate is the change in output divided by the change in input. Its units are newtons per meter.
Answer
The average rate of change is −278 N per m, which is −3.375 N/m. On average, the force drops by 278 newtons for each meter of extra separation.
Check Start at F(2) = 9. Add the rate times the change in input: 9 + (−278)(2) = 9 − 274 = 94. This equals F(4). The negative sign also makes sense, because the force weakens as the spheres move apart.

Work to write

  1. r = 2 and r = 4 both give a nonzero bottom
  2. F(2) = 36(2)2 = 9
  3. F(4) = 36(4)2 = 94
  4. F(4) − F(2) = 94 − 364 = −274
  5. Average rate = −27/44−2 = −278 N per m

The average rate of change is −278 N per m, which is −3.375 N/m. On average, the force drops by 278 newtons for each meter of extra separation.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: a line from a formula

This asks for the average output change per input unit using a formula. Find the average rate of f(x) = 6x − 7 on [1, 4].

input xoutput f(x)1−1417↓ evaluate: input given, read the output below it
The two formula evaluations become the endpoint table.
  1. f(1) = 6(1) − 7 = −1; f(4) = 6(4) − 7 = 17.Run the formula once for each endpoint input.
  2. Output change: 17 − (−1) = 18.Subtract the whole starting output; subtracting −1 adds 1.
  3. Input change: 4 − 1 = 3.Use the same endpoint order as the outputs.
  4. 183 = 6.Divide the output change by the nonzero input span.
Answer
6.
Check The starting output −1 plus 3 · 6 reaches 17, the ending output.
Rung 2Average rate of change of a ball's height on [1, 4]

A ball is thrown upward. Its height above the ground is h(t) = −5(t − 2)2 + 25 meters, where t is the time in seconds after the throw, for 0 ≤ t ≤ 4. Find the average rate of change of h on the interval [1, 4]. Give an exact value with units.

2436912151821242730(1, 20)(4, 5)
Graph of h(t) = −5(t − 2)2 + 25 with the endpoints (1, 20) and (4, 5) marked. The secant line through them has slope −5.
  1. Check the inputs t = 1 and t = 4. The formula has no fraction bottom and no square root. Both values lie in 0 ≤ t ≤ 4, so both outputs are defined.The average rate needs both endpoint outputs to exist. A square formula has no zero-bottom or root restriction, so only the stated time range matters.
  2. Evaluate h(1) = −5((1) − 2)2 + 25 = −5(−1)2 + 25 = −5(1) + 25 = 20.Putting the input in parentheses keeps the order right. We square (−1) first, then multiply by −5.
  3. Evaluate h(4) = −5((4) − 2)2 + 25 = −5(2)2 + 25 = −5(4) + 25 = 5.The right-end output is found the same way as the left-end output.
  4. Subtract: h(4) − h(1) = 5 − 20 = −15.The top of the rate is the change in output, right end minus left end. Here it is a change in height of −15 meters.
  5. Divide by 4 − 1 = 3: −153 = −5. Attach units: meters per second.Average rate = h(b)−h(a)b−a. The output is in meters and the input is in seconds.
Answer
The average rate of change is −5 meters per second. On average, the ball's height drops 5 meters each second between t = 1 and t = 4.
Check Expand: h(t) = −5(t2 − 4t + 4) + 25 = −5t2 + 20t + 5. This gives h(1) = −5 + 20 + 5 = 20 and h(4) = −80 + 80 + 5 = 5, which match. For f(t) = At2 + Bt + C, the average rate on [a, b] is A(a + b) + B, so −5(1 + 4) + 20 = −5. ✓

Work to write

  1. Both inputs t = 1 and t = 4 are allowed (0 ≤ t ≤ 4; no bottoms or roots)
  2. h(1) = −5((1) − 2)2 + 25 = 20
  3. h(4) = −5((4) − 2)2 + 25 = 5
  4. h(4) − h(1) = 5 − 20 = −15
  5. h(4)−h(1)4−1 = −153 = −5
  6. Average rate of change = −5 meters per second

The average rate of change is −5 meters per second. On average, the ball's height drops 5 meters each second between t = 1 and t = 4.

Rung 3Rung 3: parentheses for a negative endpoint

This asks for the average output change per input unit, using one negative input. Find the average rate of f(x) = 2 − x2 on [−3, 1].

−2−8−6−4−22(−3, −7)(1, 1)
The dashed ruler through the endpoint points has slope 2; it is not part of the curved function.
  1. f(−3) = 2 − (−3)2 = 2 − 9 = −7.The whole input is −3. Squaring (−3) gives (−3)(−3) = 9; the outside subtraction stays outside.
  2. f(1) = 2 − 12 = 2 − 1 = 1.Evaluate the other endpoint in the same formula.
  3. Output change: 1 − (−7) = 1 + 7 = 8.Subtract the whole starting output; taking away a negative adds its opposite.
  4. Input change: 1 − (−3) = 1 + 3 = 4.The endpoint order must match the output subtraction.
  5. 84 = 2.Dividing the net change 8 by the span 4 gives the change per input unit.
Answer
2.
Check The starting output −7 plus 4 · 2 reaches −7 + 8 = 1, the ending output.
Rung 4Average rate of change of an inverse-square force on [2, 5]

The magnetic force between two magnets r centimeters apart is modeled by F(r) = 400r2 newtons, for r > 0. Find the average rate of change of F on the interval [2, 5]. Give an exact value with units.

input distance r (cm)output force F (N)2100516↓ evaluate: input given, read the output below it
Endpoint values of F(r) = 400r2: F(2) = 100 N and F(5) = 16 N. The secant slope between these two points is 16−1005−2 = −28 N per cm.
  1. Check the inputs: r = 2 gives bottom 22 = 4 and r = 5 gives bottom 52 = 25. Neither bottom is zero, and both inputs satisfy r > 0.Both endpoint outputs must be defined before the formula f(b)−f(a)b−a can be used, and F has a variable in the bottom.
  2. Evaluate the left endpoint: F(2) = 400(2)2 = 4004 = 100 newtons.Putting the whole input in parentheses at the variable position makes sure the input is squared correctly.
  3. Evaluate the right endpoint: F(5) = 400(5)2 = 40025 = 16 newtons.The right endpoint b = 5 is evaluated the same way as the left endpoint.
  4. Subtract in the order right minus left: F(5) − F(2) = 16 − 100 = −84 newtons.The numerator is f(b) − f(a). Both outputs are whole numbers, so no common bottom is needed.
  5. Divide by the change in input: b − a = 5 − 2 = 3, so the average rate = −843 = −28 newtons per centimeter.The average rate is the change in output divided by the change in input. The units are output units per input unit.
Answer
The average rate of change of F on [2, 5] is −28 newtons per centimeter. On average, the force drops 28 N for each extra centimeter of separation.
Check Start at F(2) = 100 and apply the rate over 3 cm: 100 + (−28)(3) = 100 − 84 = 16, which equals F(5). The negative sign also fits the model, because an inverse-square force weakens as r grows.

Work to write

  1. Both inputs are allowed: (2)2 = 4 ≠ 0 and (5)2 = 25 ≠ 0
  2. F(2) = 400(2)2 = 100
  3. F(5) = 400(5)2 = 16
  4. F(5) − F(2) = 16 − 100 = −84
  5. b − a = 5 − 2 = 3
  6. Average rate = −843 = −28 newtons per centimeter

The average rate of change of F on [2, 5] is −28 newtons per centimeter. On average, the force drops 28 N for each extra centimeter of separation.

Rung 5Rung 5: a negative endpoint with a cube

This asks for the average output change per input unit when the formula has a cube. Find the average rate of f(x) = x3 − 2x on [−2, 1].

−22−20−16−12−8−448121620(−2, −4)(1, −1)
The dots are samples of the smooth curve x3 − 2x. The dashed ruler through the endpoint dots has slope 1 and is not part of the function.
  1. f(−2) = (−2)3 − 2(−2) = −8 + 4 = −4.Three factors of −2 give −8; multiplying −2 by −2 gives +4.
  2. f(1) = 13 − 2(1) = 1 − 2 = −1.Do the power and multiplication before subtracting.
  3. Output change: −1 − (−4) = −1 + 4 = 3.Subtracting the negative starting output adds 4.
  4. Input change: 1 − (−2) = 1 + 2 = 3.The input subtraction uses the same endpoint order.
  5. 33 = 1.A net rise of 3 over an input span of 3 averages 1 per unit.
Answer
1.
Check −4 + 3 · 1 = −1, so the average change reaches the ending output.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: For f(x) = 2 − x2, f(−3) = 2 + 9 = 11.
The square acts on the entire input (−3), giving +9. The outside minus then subtracts 9.
✓ Instead: f(−3) = 2 − (−3)2 = 2 − 9 = −7.
✗ Not this: The force rate is −49 because that is the force change.
That change occurs over 4 centimeters. The rate still needs division by the input change.
✓ Instead: −49 ÷ 4 = −19 newton per centimeter.
✗ Not this: Round F(6) to 0.06 before subtracting: (0.06 − 0.5) ÷ 4 = −0.11 exactly.
0.06 is a rounded replacement for 118, so the rate −0.11 is approximate rather than exact.
✓ Instead: Keep 118 through the calculation; the exact rate is −19.
Tips and tricks
  • Before using the rate formula, write a tiny endpoint table so the two inputs stay paired with their outputs.
  • Use parentheses as a seat belt around every substituted input and every negative output you subtract.
  • A fraction bottom is also called a denominator. To simplify, divide both its top and bottom by the same nonzero number.
Trap. Stopping after subtracting gives only the output change. Divide by the input change to get the average rate.