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Net Change Theorem

Net Change Theorem

Have you ever wondered why airplanes charge baggage fees? This is mainly because, as the weight of the airplane increases, it consumes more fuel. Hence, it makes sense that they charge an extra fee, as the airline needs to make up for the extra fuel the airplane uses!

Applying the Net Change Theorem excess baggage StudySmarterMost airlines will charge you an extra fee for any excess baggage - pixabay.com

However, as the fuel is consumed during the flight, the weight of the plane slowly decreases, so the rate of change of the fuel during the flight is not constant. This is more noticeable on long flights!

In this article, we will explore how to use integrals to model circumstances where the rate of change is not constant. We are talking about Applying the Net Change Theorem!

Applying the Net Change Theorem

Let's say you are driving at a constant speed of 60 mph. This means that in one hour, you will drive 60 miles. In half an hour, you will drive half of 60 miles, which is 30 miles. We can easily find this since the speed is constant.

But what happens if the speed is variable? We need to integrate.

What is the Net Change Theorem?

The Net Change Theorem is a formula for obtaining the new value of a changing quantity. It considers the integral of the rate of change of a function.

Let be a differentiable function whose initial value is known. The Net Change Theorem gives us a formula for finding the new value of the function.

The integral involved in the formula for the net change theorem is also known as the Net Change of the function over the interval .

The net change theorem is closely related to The Fundamental Theorem of Calculus, more specifically to the Evaluation Theorem. In fact, we can use The Fundamental Theorem of Calculus to prove The Net Change Theorem!

Let be a differentiable function on the interval . Its derivative is denoted as .

Now, because is the derivative of we can also think of this as if is the antiderivative of , so we can use The Fundamental Theorem of Calculus to relate and by means of a definite integral.

From here, we just isolate and we are good to go!

Let's see how this works with an example.

Find the value of knowing that and .

Apply the Net Change Theorem with and .

Substitute .

We will now focus on evaluating the definite integral. We begin by finding the antiderivative of the integrand.

Use The Power Rule to find the antiderivative of

Use the Fundamental Theorem of Calculus to evaluate the definite integral.

Now that we know all the values involved in the net change theorem formula, we can substitute everything back and find .

Substitute and back into the net change theorem formula.

You might be wondering: Why don't we just find by integrating? Let's see why.

Suppose we are trying to find given the same information as before. Since we know that we can just integrate and find , right?

Next, we would be tempted to evaluate the function at 5 to find

Wait, what do we do with the C? If we do not do any definite integral, we will have that C stuck with us, hence, we should use the net change theorem.

You can actually work the function further given the fact that F (1)=4, and you will find out that C=1. But given such trouble, you should stick to the net change theorem.

The above example is abstract as we used just mathematical functions and values without meaning, so we will now look at some application examples.

Applications of the Net Change Theorem

A prime example of two quantities related by derivatives is velocity and position. Both are functions of time, and velocity is the derivative of position with respect to time. If the speed is variable, we can find net displacement by using the net change theorem.

A car that starts from rest accelerates in a way that its velocity (in meters per second) is described as a function of time (in seconds) by the following function:

Find the displacement of the car during the first 5 seconds after accelerating.

In this case, we want to find the displacement (in meters) of the car 5 seconds after accelerating

Since the car starts from rest, at it wouldn't have moved at all, so . We can now use the net change theorem!

Use The Power Rule to find the antiderivative of .

Use the Fundamental Theorem of Calculus to evaluate the definite integral.

Substitute and back into the net change theorem formula.

Therefore, the car advances 37.5 meters in 5 seconds.

Let's see an example involving fuel!

A fishing boat consumes fuel at a rate (in gallons per hour) given by the following function:

Where are hours. Suppose that the boat goes on a fishing trip lasting 4 hours. At half the trip, the boat has 54 gallons left of fuel. How much fuel will be left at the end of the fishing trip? How much fuel was consumed during the second half of the trip

Letbe the amount of fuel that the boat has at a time . We can conclude from the given information that and we want to find . Let's use the net change theorem!

We are given so we will focus on evaluating the definite integral. This will tell us how much fuel was consumed during the second half of the trip!

Use The Power Rule to find the antiderivative of .

Use The Fundamental Theorem of Calculus to evaluate the definite integral.

The above result tells us that 48 gallons of fuel were consumed during the second half of the trip. Let's now find how much fuel was left on the boat.

Substitute andback into the net change theorem formula.

The boat will have 6 gallons of fuel left after the trip! Close enough!

The net change theorem can also be used to find the increase in population!

A sanctuary of wolves was funded with 1000 specimens. The population grows at a rate (of wolves per year) described by the following function:

Where represents years. How many wolves will be in the sanctuary after 10 years?

Let be the population of wolves at the year of the sanctuary's foundation. Since the sanctuary was founded with 1000 specimens, we have that , and we want to find . Let's use the net change theorem!

Integrate the exponential function.

Use the Fundamental Theorem of Calculus to evaluate the definite integral.

Substitute back and into the net change formula.

Evaluate with a calculator. Round down to the nearest integer.

We rounded down because we cannot have a fraction of an individual!

Summary

Net Change Theorem - Key takeaways

  • The Net Change Theorem is a formula for finding the new value of a changing quantity.
    • The net change theorem formula is the following:
  • You can prove the net change theorem by using the evaluation part of The Fundamental Theorem of Calculus.
  • There are many real-life applications of the net change theorem!
    • It can be used to find the displacement of an object.
    • Fuel consumption can also be modeled using the net change theorem.
    • It even works for modeling population growth!

Frequently Asked Questions about Net Change Theorem

The net change theorem is a formula for finding the new value of a changing quantity given you know its initial value and the rate of change of the quantity.

You can calculate the net change of a function by finding the definite integral of the derivative of a function over an appropriate interval.

To find the net change of position you must integrate the velocity function over the duration of the motion.

Final Net Change Theorem Quiz

Question

Which of the following can be used to prove the net change theorem?

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Answer

The Fundamental Theorem of Calculus

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Question

Suppose you know \(F(b)\) and the value of \(\int_a^b F'(x) \, \mathrm{d}x\). Can you find \(F(a)\) using this information?

Show answer

Answer

Yes.

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