Wemazi

Topic 4.1

Identify the quantity and its rate

Builds on differentiating a function more than once. 6 minutes.

A phone charges

A phone is charging, and the top curve shows its charge \(B\), in percent, after \(t\) minutes.

Drag the orange point along the curve. Its height is the charge, so at 10 minutes the phone holds 42 percent.

The orange line is the tangent at the point, and its slope is how fast the charge changes at that moment.

At 10 minutes the tangent rises 2 percent per minute, and the lower graph shows this slope as the height of the orange bar.

The 42 percent is how much charge there is, and the 2 percent per minute is how fast the charge grows.

The amount and its rate

Without numbers, the curve shows an amount \(y\) that depends on an input \(x\).

Drag the point. Its height is the amount \(y\), and the slope of the tangent is the rate at which \(y\) changes.

The height and the slope are two different numbers, because one measures the amount and the other measures how fast it changes.

Notation

  1. First name the amount and the input. Here they are the charge \(B\), in percent, and the time \(t\), in minutes.

    \[\cy{B} = B(t)\]

  2. The derivative is the rate of change of the amount with respect to the input. It is the slope of the tangent, which the lower graph shows as the orange bar.

    \[\cy{B'(t)} = \dfrac{dB}{dt}\]

  3. The units of the rate are the units of the amount divided by the units of the input, here percent per minute.

    \[\dfrac{\text{percent}}{\text{minute}}\]

  4. Read backward, the units name the amount. If a hose delivers 12 liters per minute, the amount is a volume of water in liters.

Example 1: A bakery’s sales

A bakery has taken \(S(h) = 50h + 30h^2 - 2h^3\) dollars in total, \(h\) hours after opening. Say what \(S(3)\) and \(S'(3)\) mean.

  1. The amount \(S\) is the dollars taken so far, and the input \(h\) is the hours since opening.

  2. The amount is the height of the point at \(h = 3\), so substitute 3 into \(S\).

    \[S(3) = 150 + 270 - 54 = \cy{366}\]

  3. The slope of the tangent is the rate, so differentiate \(S\). Then substitute 3 into \(S'\).

    \[\begin{aligned} S'(h) &= 50 + 60h - 6h^2 \\ S'(3) &= \cy{176} \end{aligned}\]

  4. At 3 hours the bakery’s total is 366 dollars, and it takes in money at 176 dollars per hour.

Example 2: A savings account

A savings account holds \(A(y) = 500 + 40y + 2y^2\) dollars, \(y\) years after it opens. Find the amount and the rate at \(y = 3\).

  1. The amount is the height of the point, so substitute 3 into \(A\).

    \[A(3) = 500 + 120 + 18 = \cy{638}\]

  2. A tempting wrong move: The rate at year 3 is 638 dollars per year.

    \[A'(3) = 638\]

    The number 638 is the height of the point, an amount in dollars. The red line has slope 638 and is almost vertical, but the orange tangent rises far less steeply.

  3. The rate is the slope of the tangent, not the height of the point, so differentiate first.

    \[\begin{aligned} A'(y) &= 40 + 4y \\ A'(3) &= \cy{52} \end{aligned}\]

  4. At year 3 the account holds 638 dollars, and it grows by about 52 dollars per year.

Practice

A botanist measures a sunflower. Its height is \(H(t) = 5.1t - 0.015t^2\) centimeters, \(t\) days after planting. Find \(H(20)\) and \(H'(20)\). Say which one is the amount and which one is the rate.

The amount is the height of the point at \(t = 20\), and the rate is the slope of the tangent there.

Solution

  1. The amount is the height of the point, so substitute 20 into \(H\).

    \[H(20) = 102 - 6 = \cy{96}\]

  2. The rate is the slope of the tangent, so differentiate \(H\). Then substitute 20 into \(H'\).

    \[\begin{aligned} H'(t) &= 5.1 - 0.03t \\ H'(20) &= \cy{4.5} \end{aligned}\]

  3. On day 20 the sunflower is 96 centimeters tall, and it grows by about 4.5 centimeters per day.

Questions

Questions 1 and 2 use the number of people \(P\) in a park, \(h\) hours after it opens. The top curve shows \(P\), and the lower graph shows its derivative.

  1. The point is at \(h = 5\), at height 150. What does the height of the point tell?

    • There are 150 people in the park.
    • The park gains 150 people each hour.
    • The park gains 20 people each hour.
    • There are 20 people in the park.
    Answer

    There are 150 people in the park.

  2. At \(h = 8\) the tangent has slope 8. What does this tell?

    • The number of people is rising by about 8 people each hour.
    • There are 8 people in the park.
    • The number of people is rising by 8 percent.
    • The park has been open for 8 hours.
    Answer

    The number of people is rising by about 8 people each hour.

  3. A solar panel has stored \(E(t) = 8t - 0.25t^2\) kilowatt-hours after \(t\) hours. Which statement is correct at \(t = 4\)?

    • The panel has stored 28 kilowatt-hours, and the stored energy grows by about 6 kilowatt-hours each hour.
    • The panel has stored 6 kilowatt-hours, and the stored energy grows by about 28 kilowatt-hours each hour.
    • The panel has stored 28 kilowatt-hours each hour.
    • The stored energy grows by about 8 kilowatt-hours each hour.
    Answer

    The panel has stored 28 kilowatt-hours, and the stored energy grows by about 6 kilowatt-hours each hour.

Summary

The height of a point on the curve is the amount, and the slope of the tangent there is the rate.

The rate is the derivative of the amount, so its units are the units of the amount divided by the units of the input.

You can now tell whether a number gives how much there is or how fast that amount changes.

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