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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!
Most 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 aboutApplying 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 tointegrate.
TheNet Change Theoremis a formula for obtaining the new value of a changing quantity. It considers theintegralof therate of changeof a function.
Let
be a differentiable function whose initial value
is known. TheNet Change Theoremgives 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 theNet Changeof the function
over the interval
.
The net change theorem is closely related toThe Fundamental Theorem of Calculus, more specifically to theEvaluation Theorem. In fact, we can use The Fundamental Theorem ofCalculusto 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 useThe Fundamental Theorem of Calculusto 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.
UseThe Power Ruleto find the antiderivative of
Use the Fundamental Theorem ofCalculusto 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 thatF(1)=4, and you will find out thatC=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.
A prime example of two quantities related byderivativesis 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!
UseThe Power Ruleto find the antiderivative of
.
Use the Fundamental Theorem ofCalculusto 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
Let
be 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!
UseThe Power Ruleto find the antiderivative of
.
UseThe Fundamental Theorem of Calculusto 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
and
back 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 ofCalculusto 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!
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.
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