Find change over an interval

Water in a tank

The curve shows the volume of water in a tank, in liters, from hour 0 to hour 8. The orange dots mark the start, at hour 1, and the end, at hour 3.

The tank holds 75 L at the start and 115 L at the end, so the change in volume is 115 − 75 = 40 L. A faint dashed line carries the starting height across, and the vertical segment from it to the end dot shows the change.

Drag the end to the right. The segment is green while the end is higher than the start, because the change is positive. After hour 4 the curve falls, so the change shrinks.

After hour 7 the end is lower than the start, so the change is negative and the segment turns red.

Any quantity

The curve can show any quantity \(f\) against an input \(x\), with the start at \(a\) and the end at \(b\). Drag the end, or use the button to move it below the start.

The vertical segment from the starting height to the ending height shows the change. Between the two dots the curve can rise and fall many times. The change does not depend on that path, because it is always the ending height minus the starting height.

Notation

1. The start is at the input \(a\), and the end is at the input \(b\). The amounts there are \(f(a)\) and \(f(b)\).

   The function must have a value at both inputs.

2. The Greek letter \(\Delta\) (delta) means "the change in", so the change in the input is the end minus the start.

   \[\Delta x = b - a\]

3. In words, the change in the output is the ending amount minus the starting amount.

   \[\cg{\Delta f} = f(b) - f(a)\]

   Subtract in this order, so that the sign of the result gives the direction.

4. The change has the units of the output, such as liters. A positive change means the quantity rose, and a negative change means it fell.

When it fails

If you subtract in the other order, you find \(f(a) - f(b)\), the negative of the change.

Here the quantity rose, but this order gives a negative number, so the red dashed segment points down, the wrong way. Subtract the start from the end.

Example 1: the tank gains water

A tank holds \(V(t) = 40 + 40t - 5t^2\) liters after \(t\) hours. Find the change in volume from \(t = 1\) to \(t = 3\).

1. Find the starting amount. At \(t = 1\) the volume is 75 L.

   \[V(1) = 40 + 40 - 5 = 75\]

2. Find the ending amount. At \(t = 3\) the volume is 115 L.

   \[V(3) = 40 + 120 - 45 = 115\]

3. Subtract the start from the end. The result is positive, so the tank gained 40 L.

   \[\cg{115 - 75 = 40 \text{ L}}\]

Example 2: the tank loses water

Use the same tank. Find the change in volume from \(t = 4\) to \(t = 7\).

1. Read the two amounts. At \(t = 4\) the volume is 120 L, and at \(t = 7\) it is 75 L.

   \[V(4) = 120, \quad V(7) = 75\]

2. A tempting wrong move: Subtract the end from the start.

   \[120 - 75 = 45 \text{ L}\]

   This says the tank gained 45 L, so the green dashed segment points up. But the end dot is lower than the start dot, so the tank lost water and the red segment points down.

3. Subtract the start from the end. The result is negative, so the tank lost 45 L.

   \[\cr{75 - 120 = -45 \text{ L}}\]

Practice

Use the same tank. Find the change in volume from \(t = 5\) to \(t = 8\).

Move the end to hour 8. Is the end dot higher or lower than the start dot? The answer gives the sign of the change.

Questions

The graph shows the temperature of a room, in degrees Celsius, from hour 0 to hour 12.

1. What is the change in temperature from hour 2 to hour 6?

   • 4 °C
   • −4 °C
   • 14 °C
   Answer

   4 °C

2. What is the change in temperature from hour 8 to hour 12?

   • −8 °C
   • 8 °C
   • 5 °C
   Answer

   −8 °C

3. A pool holds \(P(t) = 200 + 30t - 2t^2\) gallons after \(t\) hours. What is the change in volume from \(t = 2\) to \(t = 9\)?

   • 56 gal
   • −56 gal
   • 308 gal
   • 312 gal
   Answer

   56 gal

Summary

You subtract the starting amount from the ending amount, and you keep the sign and the units.

This works because the change depends only on the two heights of the curve, not on the path between them.

Use the change when a problem asks how much a quantity gained or lost between two inputs.

This is one of four skills in topic 1.1 of Which & Why: AP Calculus.