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
  • D43 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?

  • A1310 m
  • B2 m
  • C110 m
  • D12 m
  • E710 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𝑓(𝑑)π‘‘οŠ§οŠ¦d
  • B𝑓′(𝑑)π‘‘οŠ¬οŠ¦οŠ§d
  • C𝑓′(𝑑)π‘‘οŠ§οŠ¦d
  • D𝑓′(𝑑)π‘‘οŠ¬οŠ¦οŠ¦d
  • E𝑓(𝑑)π‘‘οŠ¬οŠ¦οŠ¦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
  • B133 m
  • C443 m
  • D223 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
  • B143 m
  • C263 m
  • D233 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?

  • A254 m
  • B14 m
  • C32 m
  • D6 m
  • E314 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?

  • A292 m
  • B35 m
  • C4 m
  • D10 m
  • E412 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?

  • A172 m
  • B20 m
  • C2 m
  • D4 m
  • E232 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
  • Bβˆ’5 m
  • C10 m
  • D0 m
  • Eβˆ’10 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𝑃′(𝑑)𝑑οŠͺd
  • B𝑃(𝑑)𝑑οŠͺd
  • C𝑃′(𝑑)π‘‘οŠ§οŠ«οŠͺd
  • D𝑃(𝑑)π‘‘οŠ¨οŠ«οŠ¦d
  • E𝑃′(𝑑)π‘‘οŠ¨οŠ«οŠ¦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.

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?

  • Aβˆ’6∘C
  • B6∘C
  • C8∘C
  • Dβˆ’2∘C
  • E2∘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.

Q20:

The growth rate for the number of bacteria in a sample is given by dd𝑁𝑑=10π‘’οŠ¨ο, where 𝑑 is the time in hours. Assuming no bacteria die, which of the following is an expression for the change in the number of bacteria, 𝑁, from 𝑑=1 to 𝑑=2?

  • A10ο€Ήπ‘’βˆ’π‘’ο…οŠͺ
  • B5π‘’οŠ¨
  • C10𝑒+𝑒οŠͺ
  • D20ο€Ήπ‘’βˆ’π‘’ο…οŠͺ
  • E5ο€Ήπ‘’βˆ’π‘’ο…οŠͺ

Q21:

If 𝑀(𝑑) is the rate of growth of a child in grams per week, what does 𝑀(𝑑)π‘‘οŠ«οŠ§d represent?

  • AIt represents the child’s weight (in grams) at the age of 3 weeks.
  • BIt represents the child’s weight (in grams) at the age of 5 weeks.
  • CIt represents the child’s weight (in grams) at the age of 1 week.
  • DIt represents the increase in the child’s weight (in grams) during the last week.
  • EIt represents the net change in the child’s weight (in grams) between the ages of 1 and 5 weeks.

Q22:

If oil leaks from a tank at a rate of π‘Ÿ(𝑑) gallons per minute at time 𝑑, which of the following does ο„Έπ‘Ÿ(𝑑)π‘‘οŠ§οŠ¨οŠ¦οŠ¨οŠ¦d represent?

  • AThe number of gallons of oil left in the tank after 120 minutes
  • BThe number of gallons of oil leaked between the 20th and 120th minutes
  • CThe number of gallons of oil left in the tank after 20 minutes
  • DThe number of gallons of oil left in the tank after 70 minutes
  • EThe number of gallons of oil leaked during the last minute

Q23:

A honeybee population starts with 100 bees and increases at a rate of π‘š(𝑑) bees per week. What does 100+ο„Έπ‘š(𝑑)π‘‘οŠ§οŠ¬οŠ§d represent?

  • AIt represents the total population in week 16.
  • BIt represents the net change in population between week 1 and week 16.
  • CIt represents the net change in population between week 1 and week 8.
  • DIt represents the total population in week 1.
  • EIt represents the total population in week 8.

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