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Rate of change: uniform, varying and average

Calculus, grade 10 (5 units) Lesson 8 of 26Learning

Rate of change: uniform, varying and average

A uniform and a varying rate of change

A rate of change says by how much one quantity changes when another changes by one unit: kilometres per hour, degrees per hour, centimetres of water per minute.

  • A uniform rate: the same change every hour. A cyclist riding 15 kilometres every hour covers 15t\displaystyle 15t kilometres in t\displaystyle t hours, and the graph is a straight line. The rate is its slope.
  • A varying rate: a different change every hour. A ball rolling down a slope goes faster and faster; water poured at a steady rate into a vase wide at the bottom and narrow at the top rises slowly at first and fast at the end. The graph is not a straight line.
Steps

To see how the rate changes, split the x\displaystyle x-axis into equal pieces and build steps: each step is Δx\displaystyle \Delta x wide (read “delta x\displaystyle x”), and its height Δy\displaystyle \Delta y is how much y\displaystyle y changed along it.

On the graph of f(x)=x2\displaystyle {f(x)}={x^2} below the steps are 1 wide, from x=0\displaystyle {x}={0} to x=4\displaystyle {x}={4}:

f(x)=x2\displaystyle f(x)=x^{2}
(0,0)\displaystyle (0,0)
(1,1)\displaystyle (1,1)
(2,4)\displaystyle (2,4)
(3,9)\displaystyle (3,9)
(4,16)\displaystyle (4,16)
  • The heights are 1−0=1\displaystyle {1-0}={1}, 4−1=3\displaystyle {4-1}={3}, 9−4=5\displaystyle {9-4}={5}, 16−9=7\displaystyle {16-9}={7}.
  • The steps go up – the function rises.
  • Each step is higher than the one before – the function rises faster and faster.

In a linear function all the steps are equal, and the ratio ΔyΔx\displaystyle \frac{\Delta y}{\Delta x} is the slope. In any other function the ratio ΔyΔx\displaystyle \frac{\Delta y}{\Delta x} changes from step to step: it is the rate of change along the step. The narrower the step, the better the ratio describes the rate at each of its points.

The average rate of change

The average rate of change of f\displaystyle f from x1\displaystyle x_1 to x2\displaystyle x_2 is the change in y\displaystyle y divided by the change in x\displaystyle x:

f(x2)−f(x1)x2−x1\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}

It is exactly the slope of the line through the points (x1,f(x1))\displaystyle (x_1,f(x_1)) and (x2,f(x2))\displaystyle (x_2,f(x_2)) on the graph. Such a line, cutting the graph at two points, is called a secant.

The example below works out the average rate of change of f(x)=x2\displaystyle {f(x)}={x^2} from x=1\displaystyle {x}={1} to x=4\displaystyle {x}={4}:

f(4)−f(1)4−1=16−13=5\frac{f(4)-f(1)}{4-1}=\frac{16-1}{3}=5
f(x)=x2\displaystyle f(x)=x^{2}
y=5x−4\displaystyle y=5x-4
A=(1,1)\displaystyle A=(1,1)
B=(4,16)\displaystyle B=(4,16)
  • The secant is the line y=5x−4\displaystyle {y}={5x-4}, and its slope is 5.
  • On average, for each unit of x\displaystyle x from 1 to 4 the function went up by 5.
  • A positive rate – the function rose on average; a negative rate – it fell on average.
Average speed

When s(t)\displaystyle s(t) is the distance a body has covered by the time t\displaystyle t, the average rate of change of s\displaystyle s is the average speed:

s(t2)−s(t1)t2−t1\frac{s(t_2)-s(t_1)}{t_2-t_1}

Its units are the units of distance divided by the units of time. For example, if s\displaystyle s is in metres and t\displaystyle t in seconds, the speed is in metres per second.
A runner who covered 200 metres in 25 seconds ran at an average speed of 20025=8\displaystyle {\frac{200}{25}}={8} metres per second – even if she ran slower at the start and faster at the end.

A rate of change from a table

Ticket sales for a concert opened at 12:00 (t=0\displaystyle {t}={0}), and 9 hours later all the tickets were sold.
L(t)\displaystyle L(t) – the number of people queuing for tickets t\displaystyle t hours after the start.

t\displaystyle t (hours)0134789
L(t)\displaystyle L(t) (people)120156176126150800

The example below estimates the rate of change of the queue at 17:30, that is at t=5.5\displaystyle {t}={5.5}. The table has no value at 5.5, so we use the two nearest values on either side, t=4\displaystyle {t}={4} and t=7\displaystyle {t}={7}:

L(7)−L(4)7−4=150−1263=8\frac{L(7)-L(4)}{7-4}=\frac{150-126}{3}=8

At about 17:30 the queue was growing by about 8 people per hour. The units: people per hour.

What the steps tell

Even without a formula, the steps describe the graph:

  • The steps go up – the function rises; they go down – it falls.
  • The steps go up but get lower – the function rises, but more and more slowly.
  • The steps go down and then up – the function has a minimum point.
  • The steps go up and then down – the function has a maximum point.
Exercise 1

The function is f(x)=x2−4x\displaystyle {f(x)}={x^2-4x}.

a

Work out the average rate of change of f\displaystyle f from x=0\displaystyle {x}={0} to x=1\displaystyle {x}={1}.

f(x)=x2−4x\displaystyle f(x)=x^{2}-4x

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    Work out the average rate of change of f\displaystyle f from x=1\displaystyle {x}={1} to x=3\displaystyle {x}={3}.

    f(x)=x2−4x\displaystyle f(x)=x^{2}-4x

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

      Work out the average rate of change of f\displaystyle f from x=3\displaystyle {x}={3} to x=5\displaystyle {x}={5}.

      f(x)=x2−4x\displaystyle f(x)=x^{2}-4x

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

        From x=1\displaystyle {x}={1} to x=3\displaystyle {x}={3} the average rate of change came out 0.
        Is the function constant on this interval?
        Explain by the values f(1)\displaystyle f(1), f(2)\displaystyle f(2) and f(3)\displaystyle f(3).

        Exercise 2

        On a spring day the temperature was measured in the school yard.

        Time6:009:0012:0015:0018:00
        Temperature (degrees)1420262517
        a

        Work out the average rate of change of the temperature from 6:00 to 12:00, in degrees per hour.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          b

          Work out the average rate of change of the temperature from 12:00 to 18:00.

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            c

            What does the sign of the answer to part b tell?
            Between which two neighbouring measurements did the temperature change fastest?

            Exercise 3

            Go back to the table of the ticket queue.

            a

            Work out the average rate of change of the queue from t=1\displaystyle {t}={1} to t=3\displaystyle {t}={3}.

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b

              Estimate the rate of change of the queue at t=8.5\displaystyle {t}={8.5}.

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                c

                What does a negative rate mean for the ticket queue?
                Between which two neighbouring measurements did the queue get shorter fastest?

                Exercise 4

                The function is f(x)=x3\displaystyle {f(x)}={x^3}.

                a

                Work out the heights of three steps 1 wide, from x=0\displaystyle {x}={0} to x=3\displaystyle {x}={3}.

                f(x)=x3\displaystyle f(x)=x^{3}

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  f(x)=x3\displaystyle f(x)=x^{3}

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    f(x)=x3\displaystyle f(x)=x^{3}

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      b

                      What do the heights tell about the function from x=0\displaystyle {x}={0} to x=3\displaystyle {x}={3}?

                      Exercise 5

                      A car sets off.
                      After t\displaystyle t seconds it has covered s(t)=3t2\displaystyle {s(t)}={3t^2} metres.

                      a

                      Work out the car’s average speed in the first 4 seconds.

                      s(t)=3t2\displaystyle s(t)=3t^{2}

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                        b

                        Work out its average speed from t=2\displaystyle {t}={2} to t=4\displaystyle {t}={4}.

                        s(t)=3t2\displaystyle s(t)=3t^{2}

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                          c

                          Why is the average speed from t=2\displaystyle {t}={2} to t=4\displaystyle {t}={4} greater than the average speed in the first 4 seconds?