Reading Two Graphs as One Story
A rate graph and its accumulated quantity describe different aspects of the same process. The rate tells how quickly the quantity is changing at each moment; the cumulative graph shows the quantity itself as the process unfolds. Reading them together lets you interpret not only whether the total rises or falls, but also how its steepness changes.
For example, an inventory can be increasing even while its rate of increase slows. The inventory graph still rises, but it becomes less steep. If the rate becomes negative, the inventory falls. If that negative rate becomes more negative, the inventory falls more steeply. These distinctions are visible when the two graphs are connected precisely.
Definitions and Conditions
Let \(t\) denote time, let \(r(t)\) be the rate at time \(t\), and let \(A(t)\) be the accumulated quantity. If the process begins at time \(t_0\), with initial quantity \(A(t_0)\), then the accumulation model is
This relationship requires that the rate be integrable over the interval in question. When \(r\) is continuous, the Fundamental Theorem of Calculus gives \(A'(t)=r(t)\) at interior points of the interval. Thus the sign of \(r\) determines the direction of change in \(A\).
The height of the rate graph is not the height of the cumulative graph. Instead, it is the slope of that graph. A rate of \(4\) units per hour means the cumulative quantity is rising at \(4\) units per hour at that moment. A rate of \(-4\) units per hour means it is falling at that rate.
If \(r\) is differentiable, then differentiating \(A'(t)=r(t)\) gives \(A''(t)=r'(t)\). Therefore an increasing rate makes the cumulative graph concave up, while a decreasing rate makes it concave down. This describes a change in steepness, not necessarily a change in whether the quantity rises or falls.
Why the Rate Gives the Cumulative Slope
The link between the graphs follows from the definition of accumulation. Suppose \(r\) is continuous near \(t\). For a small nonzero change \(h\), the change in the cumulative quantity is the accumulated rate over that short interval:
Divide by \(h\). The resulting difference quotient is the average rate over the interval from \(t\) to \(t+h\):
To see why this approaches \(r(t)\), continuity means that for any \(\varepsilon>0\), \(r(s)\) can be made to differ from \(r(t)\) by less than \(\varepsilon\) whenever \(s\) is sufficiently close to \(t\). For such \(h\), the average also differs from \(r(t)\) by less than \(\varepsilon\): the difference between the average and \(r(t)\) is the average of \(r(s)-r(t)\), whose absolute value is less than \(\varepsilon\). Taking the limit as \(h\) approaches zero gives
So the rate is the instantaneous slope of the cumulative graph, not merely a general indication of its movement.
Sign, Size, and Changing Steepness
Use the rate graph in three related ways. First, its sign tells the direction of the cumulative graph. Second, its distance from zero tells the magnitude of the cumulative graph’s slope. Third, whether the rate is rising or falling tells whether that slope is increasing or decreasing.
| Rate graph | Cumulative graph |
|---|---|
| \(r(t)>0\) | Increasing |
| \(r(t)<0\) | Decreasing |
| \(r(t)=0\) | Horizontal tangent, if the derivative exists |
| \(r(t)\) increasing | Slope is increasing; the graph is concave up if \(r\) is differentiable |
| \(r(t)\) decreasing | Slope is decreasing; the graph is concave down if \(r\) is differentiable |
Combine sign and trend rather than relying on either one alone. For instance, if \(r\) is positive but decreasing, \(A\) rises while flattening. If \(r\) is negative and decreasing, \(A\) falls while becoming steeper downward. If \(r\) is negative but increasing toward zero, \(A\) still falls, but it flattens.
A rate of zero gives a horizontal tangent to \(A\), but it does not by itself prove that \(A\) has a maximum or minimum. For a turning point, the rate must change sign. If \(r\) remains positive on both sides of a zero, \(A\) can keep increasing through a moment when its slope is zero.
Worked Examples
Worked Example: A Rate Changes from Positive to Negative
Suppose the rate, in units per hour, is \(r(t)=6-2t\) for \(0\leq t\leq 5\), and the initial quantity is \(A(0)=10\) units. Interpret the rate and cumulative graphs together.
First find where the rate is zero:
For \(0\leq t<3\), \(r(t)>0\), so \(A\) increases. For \(3<t\leq5\), \(r(t)<0\), so \(A\) decreases. The rate is decreasing throughout because \(r'(t)=-2\), so the cumulative graph is concave down throughout.
To locate the cumulative values, use the initial value and accumulate the rate:
At the rate’s zero, \(A(3)=10+18-9=19\) units. At \(t=5\), \(A(5)=10+30-25=15\) units. Thus the quantity rises from \(10\) to \(19\), then falls to \(15\). Check the formula by differentiation: \(A'(t)=6-2t=r(t)\), and \(A(0)=10\), as specified.
Worked Example: Increasing Throughout, but First Flattening and Then Steepening
Let \(r(t)=(t-2)^2\) units per hour for \(0\leq t\leq4\), with \(A(0)=3\) units. Describe the shape of the cumulative graph.
Because a square is nonnegative, \(r(t)\geq0\) on the interval. It is zero only at \(t=2\), so \(A\) increases on either side and has a horizontal tangent at \(t=2\). The rate decreases for \(t<2\) and increases for \(t>2\), as \(r'(t)=2(t-2)\). Hence \(A\) is concave down before \(2\) and concave up after \(2\).
The accumulated quantity is
At \(t=2\), \(A(2)=17/3\) units. Differentiating the result gives \(A'(t)=(t-2)^2=r(t)\), and substituting \(t=0\) gives \(A(0)=(-8)/3+17/3=3\). The cumulative graph does not turn at \(t=2\): it continues increasing, changing from flattening to steepening.
Worked Example: A Negative Rate That Becomes Less Negative
Let \(r(t)=-(t-2)^2-1\) units per hour on \(0\leq t\leq4\), and suppose \(A(0)=20\) units. Describe the direction and steepness of \(A\).
Since \((t-2)^2\geq0\), \(r(t)\leq-1<0\) throughout. Therefore \(A\) decreases throughout. The rate rises until \(t=2\) and falls after \(t=2\): indeed, \(r'(t)=-2(t-2)\), which is positive before \(2\) and negative after. Consequently, \(A\) is concave up before \(2\), where its negative slope is becoming less negative, and concave down after \(2\), where its slope becomes more negative.
Expanding the rate gives \(r(t)=-t^2+4t-5\). Accumulating it yields
At \(t=2\), \(A(2)=20-8/3+8-10=46/3\) units. At \(t=4\), \(A(4)=20-64/3+32-20=32/3\) units. Both values are below the initial quantity, consistent with the negative rate. The check is \(A'(t)=-t^2+4t-5=-(t-2)^2-1\), and \(A(0)=20\).
A Common Misreading
A frequent mistake is to confuse the rate’s sign with whether the rate itself is increasing or decreasing. For example, if a rate is negative and falling, it is incorrect to conclude that the cumulative quantity is increasing because “the graph is going down.” The fact that the rate graph goes down describes a decreasing rate; its negative values mean the cumulative quantity is decreasing.
If the rate is also decreasing, then \(A''(t)=r'(t)<0\), so the cumulative graph is concave down: it falls with an increasingly negative slope. Always distinguish the rate graph’s vertical position, which controls the cumulative direction, from the rate graph’s trend, which controls the cumulative curvature.
Why Paired Graphs Matter
Many models are easier to understand when the rate and cumulative quantity are considered together. A stock balance, inventory, reservoir level, or total production can change direction only as its net rate changes sign. Meanwhile, the rate’s magnitude indicates how quickly that change is occurring. The paired view can therefore reveal whether a total is rising, falling, or changing steepness without treating the two graphs as unrelated pictures.
Keep units attached to each graph. If \(A\) is measured in kilograms and time in hours, then \(r=A'\) is measured in kilograms per hour. The slope of the cumulative graph has the same units as the rate, while the cumulative graph’s height has units of kilograms. This dimensional distinction is another way to remember that rate height and cumulative height are not interchangeable.
Check Your Understanding
Use the rate-to-accumulation relationship to answer each question.
- If \(r(t)>0\) but is decreasing, what happens to the cumulative quantity and to its steepness?
- If \(r(t)<0\) and is increasing toward zero, does the cumulative quantity increase or decrease? What happens to its steepness?
- What does a zero of the rate guarantee about the cumulative graph, and what additional information is needed to establish a turning point?
- If the rate is increasing on an interval, what does that imply about the cumulative graph’s concavity, assuming the rate is differentiable?
- A rate is negative and decreasing. Describe the cumulative graph’s direction and curvature.
