From a Rate Table to an Accumulated Change
A rate describes how quickly a quantity changes. A table of rates can therefore tell us more than what is happening at a few individual moments: it can help us estimate how much the quantity changes over an entire time interval.
The central calculation is simple: multiply a rate by the amount of time for which it applies. Do this on each interval in the table, then add the contributions. The important work is interpreting the context correctly: choose the right elapsed-time widths, use the rate values as instructed, keep track of signs, and attach the resulting units.
Rate, Time, and Units
Suppose \(Q\) is a quantity that depends on time \(t\), and \(r(t)\) is its rate of change. If \(Q\) is measured in liters and \(t\) in minutes, then \(r(t)\) has units of liters per minute. Multiplying by an elapsed time measured in minutes gives liters:
For a table with time values \(t_0,t_1,\ldots,t_n\), the elapsed-time width of the \(i\)th interval is \(t_i-t_{i-1}\), not \(t_i\). If \(r_i\) is the rate used to represent the rate on that interval, its estimated contribution to the change is
Adding the contributions gives an estimate of the net change. The rate \(r_i\) may be a rate explicitly given for the whole interval or a rate sampled at a time in the interval. When the table gives rates only at particular times, the problem’s stated method determines which value to use for each interval. A numerical estimate depends on that choice.
Why Multiplying and Adding Works
First suppose the rate is constant on a time interval. By the meaning of a constant rate, each unit of time produces the same amount of change. A rate of \(r\) units per minute over \(\Delta t\) minutes therefore produces a change of \(r\Delta t\) units.
For a rate that varies, divide the time span into shorter intervals. On each interval, use a representative rate as an approximation to the changing rate. The rate-times-width product estimates the change on that interval. Since the change across consecutive intervals is the sum of the separate changes, adding those products estimates the total change:
This argument explains both the calculation and its limitation: unless the rate is constant on each interval or the table provides exact interval rates, the products generally estimate rather than determine the exact change.
Worked Example: Estimating Water Added to a Tank
Worked Example: Estimating Water Added to a Tank
At time \(t=0\), a tank contains \(120\) liters of water. The inflow rate is measured in liters per minute. Use the left-endpoint rate on each interval to estimate the amount of water added from \(t=0\) to \(t=12\) minutes, and estimate the final amount.
| Time (min) | Inflow rate (L/min) |
|---|---|
| 0 | 3.0 |
| 4 | 2.4 |
| 9 | 1.2 |
| 12 | 0.8 |
The time intervals have widths \(4-0=4\), \(9-4=5\), and \(12-9=3\) minutes. The left-endpoint rates are \(3.0\), \(2.4\), and \(1.2\) liters per minute, respectively.
On the first interval, the estimated amount added is \(3.0(4)=12\) liters. On the second, it is \(2.4(5)=12\) liters. On the third, it is \(1.2(3)=3.6\) liters. Thus the estimated total added is
Adding this change to the initial amount gives an estimated final amount of \(120+27.6=147.6\) liters. As a check, the three interval contributions sum to \(27.6\) liters, the estimated net change used in the final-amount calculation. The result has units of liters, as required for an amount of water.
Signed Rates and Net Change
A rate can be positive or negative. A positive rate means the quantity is increasing according to the chosen direction or convention; a negative rate means it is decreasing. The sign must be retained when calculating each contribution.
For example, velocity is a signed rate of change of position. A negative velocity contributes negative displacement. The sum of signed contributions is net change in position, not necessarily total distance traveled. Likewise, for a quantity such as a bank balance, deposits and withdrawals may have opposite signs, and the signed sum represents the net change.
Worked Example: Position from Signed Velocity Data
Worked Example: Position from Signed Velocity Data
An object is at position \(40\) meters at time \(t=0\). Use the stated representative velocities to estimate its position at \(t=8\) minutes. The velocity is in meters per minute, and each listed velocity represents its associated interval.
| Time interval (min) | Representative velocity (m/min) |
|---|---|
| \([0,2]\) | 5 |
| \([2,5]\) | 2 |
| \([5,8]\) | -1 |
The interval widths are \(2-0=2\), \(5-2=3\), and \(8-5=3\) minutes. The estimated displacements are \(5(2)=10\) meters, \(2(3)=6\) meters, and \((-1)(3)=-3\) meters. Therefore, the estimated net change in position is
The estimated final position is \(40+13=53\) meters. Check the interpretation: the object moved forward by a net \(13\) meters, so its estimated position is \(13\) meters greater than its starting position. This calculation does not give total distance traveled; it gives signed net displacement.
When the Table Gives Rates at Times
Sometimes a table lists a rate at each time rather than stating a rate for each interval. A method such as a left-endpoint sum tells you to use the rate at the start of each interval; a right-endpoint sum uses the rate at the end. A midpoint sum uses a rate at the interval’s midpoint when such data are provided. Use the specified rule, and make sure that each rate is paired with the width of the interval it represents.
If no sampling rule is specified, do not assume that a rate measured at one instant is automatically the exact rate throughout a whole interval. The table may require a particular approximation method, or a problem may explicitly provide an interval rate. State which interpretation you use when it matters.
Worked Example: A Left-Endpoint Estimate with Unequal Time Gaps
Worked Example: A Left-Endpoint Estimate with Unequal Time Gaps
A machine uses fuel at the rates shown. The rates are in liters per hour, and the left-endpoint rate is to be used on each interval. Estimate the fuel consumed between hour \(0\) and hour \(4\).
| Time (h) | Fuel-use rate (L/h) |
|---|---|
| 0 | 6 |
| 1 | 5 |
| 3 | 3 |
| 4 | 2 |
The intervals are \([0,1]\), \([1,3]\), and \([3,4]\). Their widths are \(1\), \(2\), and \(1\) hour. Using the left-endpoint rates, the estimated fuel use on these intervals is \(6(1)=6\) liters, \(5(2)=10\) liters, and \(3(1)=3\) liters. The estimate for the whole period is
The units check because \((\text{L/h})(\text{h})=\text{L}\). The unequal interval from hour \(1\) to hour \(3\) lasts \(2\) hours, so its contribution is \(10\) liters, not \(5\) liters.
A Common Mistake: Using the Time Value Instead of the Interval Width
A frequent error is multiplying each rate by the time printed at the right side of an interval rather than by the time elapsed during that interval. In the fuel example, this mistake would use the time values \(1\), \(3\), and \(4\) as widths, giving
This is incorrect because those numbers are not the three elapsed-time widths. The interval lengths are \(1-0=1\), \(3-1=2\), and \(4-3=1\) hour. The correct calculation is
A useful check is to write the endpoints of every interval and subtract the earlier time from the later time before multiplying.
Interpreting the Result in Context
Before reporting an accumulated quantity, identify exactly what the sum represents. A rate of inflow gives an estimate of volume added; it does not, by itself, give the final volume unless the initial volume is also known. A rate of fuel use gives fuel consumed; finding fuel remaining also requires the starting amount. A signed velocity gives displacement; total distance requires accounting for motion in each direction without cancellation.
Check the units as well as the arithmetic. If the rate is in people per year and the interval width is in years, the contribution is in people. If rate and time units do not combine to the units of the desired quantity, revisit the setup. Correct units are a practical way to catch mismatches between a rate and an interval.
Check Your Understanding
For each question, show the interval widths and include units in your answer.
- A rate of \(7\) liters per minute applies for \(3\) minutes. What change in volume does it produce?
- A quantity has rates \(4\) units per hour on \([0,2]\) and \(-1\) unit per hour on \([2,5]\). Estimate the net change over the full interval.
- Why is a rate measured at one time not necessarily the exact rate over the following interval?
- A table lists times \(1\), \(4\), and \(6\) minutes. What are the widths of the intervals between consecutive listed times?
- An object starts at position \(12\) meters and has an estimated displacement of \(-5\) meters. What is its estimated final position, and what does the negative sign indicate?
