Worksheet: The Net Change Theorem

In this worksheet, we will practice applying the net change theorem.

Q1:

True or False? The Net Change Theorem says that ο„Έβˆšπ‘₯π‘₯=√9βˆ’βˆš4οŠͺd.

  • Atrue
  • Bfalse

Q2:

True or False: If 𝑔(𝑑) represents the rate of change of a cat’s weight in kilograms per month when it is 𝑑 months old, then 𝑔(𝑑)𝑑οŠͺd equals the cat’s weight when it is 4 months old.

  • Atrue
  • Bfalse

Q3:

True or False: The net change theorem states that ο„Έπ‘₯βˆšπ‘‘π‘‘=√π‘₯βˆ’βˆšπ‘Žο—οŒΊddd.

  • ATrue
  • BFalse

Q4:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=2π‘‘βˆ’5π‘‘βˆ’3. What distance does it travel between 𝑑=2 and 𝑑=4?

  • A31 m
  • B2 m
  • C7 m
  • D 4 3 m

Q5:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=5π‘‘βˆ’3. What distance does it travel between 𝑑=0 and 𝑑=1?

  • A 1 3 1 0 m
  • B2 m
  • C 1 1 0 m
  • D 1 2 m
  • E 7 1 0 m

Q6:

Suppose that 𝐹(𝑑)=π‘‘οŠ¨. Evaluate 𝐹′(𝑑)π‘‘οŠ©οŠ§d using the net change theorem.

Q7:

An oil tanker leaks at the rate π‘Ÿ=𝑓(𝑑) liters per minute, where 𝑑 is the number of minutes after the accident. Write an integral that expresses the volume of oil leaked out in the first hour.

  • A ο„Έ 𝑓 ( 𝑑 ) 𝑑   d
  • B ο„Έ 𝑓 β€² ( 𝑑 ) 𝑑    d
  • C ο„Έ 𝑓 β€² ( 𝑑 ) 𝑑   d
  • D ο„Έ 𝑓 β€² ( 𝑑 ) 𝑑    d
  • E ο„Έ 𝑓 ( 𝑑 ) 𝑑    d

Q8:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=π‘‘βˆ’4𝑑+3. What distance does it travel between 𝑑=2 and 𝑑=5?

  • A60 m
  • B 1 3 3 m
  • C 4 4 3 m
  • D 2 2 3 m
  • E6 m

Q9:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=π‘‘βˆ’2π‘‘βˆ’3. What distance does it travel between 𝑑=1 and 𝑑=4?

  • A15 m
  • B 1 4 3 m
  • C 2 6 3 m
  • D 2 3 3 m
  • E3 m

Q10:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=4π‘‘βˆ’1. What distance does it travel between 𝑑=0 and 𝑑=2?

  • A 2 5 4 m
  • B14 m
  • C 3 2 m
  • D6 m
  • E 3 1 4 m

Q11:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=2π‘‘βˆ’3. What distance does it travel between 𝑑=0 and 𝑑=5?

  • A 2 9 2 m
  • B35 m
  • C4 m
  • D10 m
  • E 4 1 2 m

Q12:

A particle’s velocity in meters per second as a function of time is 𝑣(𝑑)=2π‘‘βˆ’3. What distance does it travel between 𝑑=0 and 𝑑=4?

  • A 1 7 2 m
  • B20 m
  • C2 m
  • D4 m
  • E 2 3 2 m

Q13:

A piston moves back and forth in one direction. The velocity of the piston (in m/s) can be modeled by the function 𝑣(𝑑)=5(𝑑)sin, where 𝑑 is the time in seconds. What is the change in the position of the piston from 𝑑=0 to 𝑑=πœ‹?

  • A5 m
  • B βˆ’ 5 m
  • C10 m
  • D0 m
  • E βˆ’ 1 0 m

Q14:

The rate of growth of a population is represented by the function 𝑃′(𝑑), where 𝑑 is the number of days after the start of the year. How might the difference in population from 𝑑=15 to 𝑑=40 be represented using an integral?

  • A ο„Έ 𝑃 β€² ( 𝑑 ) 𝑑 οŠͺ    d
  • B ο„Έ 𝑃 ( 𝑑 ) 𝑑 οŠͺ    d
  • C ο„Έ 𝑃 β€² ( 𝑑 ) 𝑑   οŠͺ  d
  • D ο„Έ 𝑃 ( 𝑑 ) 𝑑    d
  • E ο„Έ 𝑃 β€² ( 𝑑 ) 𝑑    d

Q15:

If π‘Ž(𝑑) is the acceleration of a car, what does ο„Έπ‘Ž(𝑑)π‘‘οŠ©οŠ¦d represent?

  • AThe total distance the car has traveled from 𝑑=0 to 𝑑=3.
  • BThe velocity of the car at 𝑑=3.
  • CThe position of the car at 3.
  • DThe change in the acceleration of the car from 𝑑=0 to 𝑑=3.
  • EThe change in the velocity of the car from 𝑑=0 to 𝑑=3.

Q16:

Water is poured in a tank at a rate of π‘Ÿ(𝑑)=0.01π‘‘βˆ’0.02𝑑+10 liters per hour. The tank contains a hole from which water leaks at a rate of 𝑠(𝑑)=√3.1𝑑 liters per hour. Given that the tank is initially empty, find the total amount of water in the tank after 3 hours. Give your answer to the nearest liter.

  • A30 liters
  • B6 liters
  • C24 liters
  • D20 liters
  • E36 liters

Q17:

True or False: The net change theorem states that 𝑒π‘₯=12ο€Ήπ‘’βˆ’1ο…οŠ¨οŠ¦οŠ¨ο—οŠͺd.

  • ATrue
  • BFalse

Q18:

The water temperature in a water bath changes at a rate of 𝐻′(𝑑)=𝑑+2, where 𝐻′(𝑑) is measured in degrees Celsius per hour and 𝑑 is the time in hours. What is the change in the water temperature from 𝑑=0 to 𝑑=2?

  • A βˆ’ 6 ∘ C
  • B 6 ∘ C
  • C 8 ∘ C
  • D βˆ’ 2 ∘ C
  • E 2 ∘ C

Q19:

A tank is filled with water at a rate of π‘Ÿ(𝑑)=π‘‘βˆ’π‘‘βˆ’6 cubic meters per minute. Find the amount of water that will be filled in the tank from 𝑑=2 to 𝑑=4 minutes. Round the answer to one decimal place.

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