Interpret a changing rate

An app’s downloads

The top curve shows \(D\), the total downloads of a new app in thousands, \(t\) days after launch. The lower curve shows \(D'\), the rate at which downloads arrive, in thousands per day.

Drag the point. The top curve is green everywhere because every day adds downloads, so the total never falls.

The lower curve is red because it falls, since each day brings fewer new downloads than the day before. The rate is 50 at day 1, 20 at day 4 and 10 at day 9.

The total rises while its rate falls, and both are true at once.

Two separate questions

Without numbers, the picture answers two separate questions.

Is the amount rising? The upper curve is green where it rises, which is where the rate \(f'\) is positive.

Is the rate rising? The lower curve is red where it falls, because its own slope is negative there. Press “Show the slope of the rate” to draw that slope.

The direction of the amount and the direction of its rate are separate facts, so neither one tells you the other.

Notation

1. The sign of \(f'\) tells whether the amount \(f\) rises or falls.

   \[f'(x) > 0 \ \Rightarrow\ f \text{ rises}\]

2. The rate \(f'\) is also a function, so it has its own rate of change, the second derivative \(f''\). It is the slope of the small orange tangent to the lower curve.

   \[\cy{f''} = (f')'\]

3. The sign of \(f''\) tells whether the rate rises or falls. If \(f''\) is negative, the rate falls, so the lower curve is red.

   \[f''(x) < 0 \ \Rightarrow\ f' \text{ falls}\]

4. The two signs give four cases. If \(f'\) and \(f''\) have the same sign, the amount rises or falls faster. If they have opposite signs, it rises or falls more slowly.

Example 1: A draining tub

A tub holds \(W(t) = \tfrac{1}{2}(12 - t)^2\) liters of water, \(t\) minutes after the plug is pulled, for \(0 \le t \le 12\). Describe the water and its rate at \(t = 2\).

1. Differentiate twice.

   \[W'(t) = t - 12, \qquad W''(t) = 1\]

2. At \(t = 2\), \(W'\) is negative and the upper curve is red, so the water falls by about 10 liters each minute.

   \[W'(2) = \cr{-10}\]

3. The second derivative is positive, so the rate rises toward 0. At \(t = 8\) the water still falls, but by only about 4 liters each minute, so the tub drains more slowly.

   \[W'(8) = \cr{-4}, \qquad W''(t) = 1 > 0\]

Example 2: Course enrollment

A course has enrolled \(E(w) = w^3 - 12w^2 + 60w\) students, \(w\) weeks after it opens. Describe the enrollment and its rate at \(w = 2\) and at \(w = 6\).

1. Differentiate twice.

   \[\begin{aligned} E'(w) &= 3w^2 - 24w + 60 \\ E''(w) &= 6w - 24 \end{aligned}\]

2. At \(w = 2\) the rate is positive and the second derivative is negative.

   \[E'(2) = \cg{24}, \qquad E''(2) = \cr{-12}\]

3. A tempting wrong move: \(E''(2)\) is negative, so the number of students falls at week 2.

   The red line falls through the point, but the curve of \(E\) is green there, so the number rises. The negative \(E''\) describes the lower curve, the rate, not the number of students.

4. \(E'(2)\) is positive, so the number rises by about 24 students each week. \(E''(2)\) is negative, so the rate falls and the number rises more slowly.

5. At \(w = 6\) the rate is again 24, but \(E''(6)\) is positive. The number rises by about 24 students each week, and it rises faster.

   \[E'(6) = \cg{24}, \qquad E''(6) = \cg{12}\]

Practice

A generator's tank holds \(F(h) = 60 - 0.5h^2\) liters of fuel, \(h\) hours after it starts. At \(h = 4\), does the fuel fall faster or more slowly as time passes?

The color of the upper curve gives the direction of the fuel, and the color of the lower curve gives the direction of its rate.

Questions

Questions 1 and 2 use the number of members \(N\) of a club, \(y\) years after 2020. The top curve shows \(N\), and the lower curve shows \(N'\).

1. At \(y = 5\) the upper curve is green and the lower curve falls. Which statement is correct?

   • The membership rises, and it rises more slowly each year.
   • The membership falls.
   • The membership rises faster each year.
   • The membership rises at a constant rate.
   Answer

   The membership rises, and it rises more slowly each year.

2. At \(y = 10\) the lower curve is at its lowest point, at height 30. What does this tell you about the membership?

   • The membership still rises, by about 30 members each year.
   • The membership stops rising.
   • The membership is at its smallest value.
   • The membership falls by about 30 members each year.
   Answer

   The membership still rises, by about 30 members each year.

3. A bicycle is worth \(V(y) = 900 - 120y + 6y^2\) dollars after \(y\) years. Which statement is correct at \(y = 2\)?

   • The value falls by about 96 dollars each year, and it falls more slowly.
   • The value falls by about 96 dollars each year, and it falls faster.
   • The value rises by about 12 dollars each year.
   • The value rises, and it rises faster.
   Answer

   The value falls by about 96 dollars each year, and it falls more slowly.

Summary

The sign of \(f'\) gives the direction of the amount, and the sign of \(f''\) gives the direction of the rate.

The amount and its rate are two functions, so each one can rise or fall without the other.

With both signs, you can say whether a quantity grows faster, grows more slowly, falls faster or falls more slowly.

This is one of four skills in topic 4.1 of Which & Why: AP Calculus.