Unit 6 · Topic 6.1 Beta

Accumulating Change

When you know a rate, the change in the amount is the area between the rate graph and the horizontal axis.

Practice 2: Connecting RepresentationsPractice 4: Communication and Notation

Question set for this topic

Part 1 · Hook

Why this matters

A fitness watch never measures how far you ran. It measures how fast you are going, many times a second, and adds up speed times time. Every running app, water meter and electricity bill works the same way: it knows a rate, and it builds up the total. That building up is called accumulation, and on a graph it is an area.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Water flows into a tank at 12 liters per minute. What are the units of the rate of change of the rate, if it changes?

  1. liters per minute per minute
  2. liters per minute
  3. liters
  4. minutes per liter
Show the answer

A rate of a rate divides by time twice: liters per minute, per minute.

  • Correct: liters per minute per minute:
  • liters per minute:
  • liters:
  • minutes per liter:

2. A trapezoid has parallel sides of lengths 3 and 7 and a width of 4 between them. What is its area?

  1. 20
  2. 40
  3. 28
  4. 12
Show the answer

Average of the parallel sides times the width: 3+72⋅4=20\frac{3 + 7}{2} \cdot 4 = 20.

  • Correct: 20:
  • 40:
  • 28:
  • 12:

3. A car's position is s(t)s(t) meters. What is s′(t)s'(t)?

  1. its velocity, in meters per second
  2. its acceleration
  3. its distance from home
  4. its average speed over the trip
Show the answer

The derivative of position is velocity, the rate of change of position.

  • Correct: its velocity, in meters per second:
  • its acceleration:
  • its distance from home:
  • its average speed over the trip:

Part 4 · See it

See it first

Graph of the rate r(t) at which water flows into a tank, in liters per minute, for t from 0 to 12 minutes. The region above the t-axis from 0 to 8 has area 34 (34 liters flow in); the region below the axis from 8 to 12 has area 9 (9 liters drain out). The net change is 25 liters.
Water flows into a tank at rate r(t). The area above the t-axis is water added; the area below is water drained. The net change is the difference. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A rate is constant over a stretch of timethe change in the amount is rate times time, the area of a rectangle under the rate graph
  2. The rate changes from moment to momentthe change is still the area under the rate graph, now a triangle, a trapezoid or a curved region
  3. The rate is negative over part of the timethat region lies below the axis and its area is subtracted: the amount decreases there
  4. You know the starting amount and the net changethe amount at the end is the starting amount plus the net change

Part 6 · Key ideas

Key ideas

  • The area between a rate graph and the horizontal axis is the accumulated change in the amount.
  • Units of the area = units of the rate times units of the input: (liters per minute) times (minutes) = liters.
  • Area above the axis adds to the amount; area below the axis subtracts. The result is the net change.
  • Amount at the end = starting amount + net change.

Part 7 · Misconception

A common mistake

The wrong idea: Area is always positive, so the area under a rate graph always means the amount went up.

What actually happens: A region below the axis comes from a negative rate: the amount goes down by that area. Add the areas above the axis and subtract the areas below to get the net change.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The velocity of a toy train

A toy train moves back and forth on a straight track. Its velocity v(t)v(t), in feet per second, is positive when it moves forward. The graph is made of line segments.
0123456789101112−3−2−112345t (seconds)v(t) (feet per second)v

1. By how many feet does the train's position change from t=0t = 0 to t=6t = 6 seconds?

No calculator. Type the exact value (a fraction, π, e or a root is fine), then its unit.

Show the answer

The change in position is the area under the velocity graph. Triangle from 0 to 2: 12(2)(4)=4\frac{1}{2}(2)(4) = 4. Rectangle from 2 to 6: 4⋅4=164 \cdot 4 = 16. Total: 4+16=204 + 16 = 20 feet.

  • Answer: 20 feet

2. Find the net change in the train's position, in feet, from t=0t = 0 to t=12t = 12 seconds.

No calculator. Type the exact value (a fraction, π, e or a root is fine), then its unit.

Show the answer

Forward from 0 to 8: 4+16+4=244 + 16 + 4 = 24 feet. Backward from 8 to 12: triangle 12(2)(2)=2\frac{1}{2}(2)(2) = 2 plus rectangle 2⋅2=42 \cdot 2 = 4, so 6 feet. Net change: 24−6=1824 - 6 = 18 feet.

  • Answer: 18 feet

3. At what time in the interval from 0 to 12 seconds is the train farthest forward of where it started?

  1. t=4t = 4, in the middle of the time at top speed
  2. t=8t = 8, where vv turns from positive to negative
  3. t=6t = 6, where the velocity starts to decrease
  4. t=12t = 12, the right end of the time interval
Show the answer

The position grows while v(t)>0v(t) > 0 (from 0 to 8) and shrinks while v(t)<0v(t) < 0 (from 8 to 12). It is largest at t=8t = 8.

  • t=4t = 4, in the middle of the time at top speed: The flat part is where the speed is largest, but the train keeps moving forward after t = 4, so its position keeps growing.
  • Correct: t=8t = 8, where vv turns from positive to negative: Correct: forward motion adds to the position until t = 8, and after that every second takes some back.
  • t=6t = 6, where the velocity starts to decrease: From 6 to 8 the velocity is smaller but still positive, so the train is still moving forward and its position still grows.
  • t=12t = 12, the right end of the time interval: After t = 8 the velocity is negative, so the train backs up: its position at 12 is 6 feet less than at 8.

4. Tank A fills at a constant 5 liters per minute. Tank B fills at a rate that grows steadily from 0 to 10 liters per minute over the same 4 minutes. Which tank gains more water in those 4 minutes?

  1. Tank A, because its rate is larger at the start
  2. It depends on how much water each tank held at first
  3. They gain the same amount, 20 liters each
  4. Tank B, because its rate reaches 10 liters per minute
Show the answer

A: 5⋅4=205 \cdot 4 = 20 liters. B: a triangle, 12(4)(10)=20\frac{1}{2}(4)(10) = 20 liters. The amounts gained are equal.

  • Tank A, because its rate is larger at the start: A leads early, but B catches up. Compare whole areas, not rates at one time.
  • It depends on how much water each tank held at first: The question asks for the amount gained, which is the area under each rate graph, not the amount held.
  • Correct: They gain the same amount, 20 liters each: Correct: a 4 by 5 rectangle and a triangle with base 4 and height 10 both have area 20.
  • Tank B, because its rate reaches 10 liters per minute: B ends faster, but it starts at 0. Compare areas: B has a triangle of area 20, the same as the rectangle for A.

5. Water flows into or out of a tank at a rate r(t)r(t) liters per minute, positive when water enters. From t=0t = 0 to t=10t = 10, the regions between the graph of rr and the t-axis have area 14 above the axis and 5 below. The tank held 30 liters at t=0t = 0. How much does it hold at t=10t = 10?

  1. 9 liters
  2. 39 liters
  3. 49 liters
  4. 44 liters
Show the answer

Net change = area above minus area below =14−5=9= 14 - 5 = 9 liters. Amount at 10 = 30+9=3930 + 9 = 39 liters.

  • 9 liters: That is the net change alone. Add it to the 30 liters already in the tank.
  • Correct: 39 liters: Correct: 30 + 14 - 5.
  • 49 liters: This adds the 5 liters, but the region below the axis is water leaving the tank.
  • 44 liters: This ignores the 5 liters that drain out.

6. Oil leaks from a pipe at a rate that rises steadily from 2 liters per minute at t=0t = 0 to 8 liters per minute at t=6t = 6 minutes. How many liters leak in those 6 minutes?

No calculator. Type the exact value (a fraction, π, e or a root is fine), then its unit.

Show the answer

The rate graph is a line from (0,2)(0, 2) to (6,8)(6, 8); the region under it is a trapezoid with area 2+82⋅6=30\frac{2 + 8}{2} \cdot 6 = 30 liters.

  • Answer: 30 liters

Part 9 · Summary

Summary

When you know a rate, the change in the amount is the area between the rate graph and the horizontal axis. Its units are the rate's units times the input's units. Area above the axis adds and area below subtracts, giving the net change; add it to the starting amount to get the amount at the end.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections