# How Do You Find The Instantaneous Rate Of Change

# How Do You Find The Instantaneous Rate Of Change

**Instantaneous Rate Of Change: **Imagine that you drive to a grocery store 10 miles away from your house, and it takes you 30 minutes to get there. That means that you traveled 10 miles in 1/2 hour, at an average speed of 20 miles per hour. (10 miles divided by 1/2 hour = 20 miles per hour). The speed of your car is a great example of a **rate of change**.

A rate of change tells you how quickly something is changing, such as the location of your car as you drive. You can also measure how quickly your hair grows, how much money your business makes each month, or how much water flows over a dam. All of these, and many more can be represented by calculating the average rate of change of a quantity over a certain amount of time.

One easy way to calculate the rate of change is to make a graph of the quantity that is changing versus time. Then you can calculate the rate of change by finding the slope of the graph, like this one. The slope is found by dividing how much the *y* values change by how much the *x* values change. Let’s look at a graph of position versus time and use that to determine the rate of change of position, more commonly known as speed.

## Instantaneous Rate Of Change Calculator

We see changes around us everywhere. When we project a ball upwards, its position changes with respect to time and its velocity changes as its position changes. The height of a person changes with time. The prices of stocks and options change with time. The equilibrium price of a good changes with respect to demand and supply. The power radiated by a black body changes as its temperature changes. The surface area of a sphere changes as its radius changes. This list never ends. It is amazing to measure and study these changes.

These changes depend on many factors; for example, the power radiated by a black body depends on its surface area as well as temperature. We shall be looking at cases where only one factor is varying and all others are fixed. Then we can model our system as $y=f(x),$ where $y$ changes with regard to $x$.

The **derivative** tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it “instantaneous rate of change”). This concept has many applications in electricity, dynamics, economics, fluid flow, population modeling, queuing theory and so on.

Wherever a quantity is always changing in value, we can use **calculus** (differentiation and integration) to model its behavior.

In this section, we will be talking about events at certain times, so we will be using* *Δ*t* instead of the Δ*x* that we saw in the last section Derivative from First Principles.

**Note: **This section is part of the introduction to differentiation. We learn some (much easier) rules for differentiating in the next section, Derivatives of Polynomials.

### Velocity

We learned before that velocity is **distance divided by time**. But this only works if the velocity is **constant**. We need a new method if the velocity is changing all the time.

If we have an expression for *s* (displacement) in terms of *t* (time), then the velocity at **any particular instant** *t* is given by:

\displaystyle{v}=\lim_{{\Delta{t}

to{0}}}\frac{{\Delta{s}}}

$v=t→$

$lim ts $

To make the algebra simple, we will use *h* for Δ*t* and write:

\displaystyle{v}=\lim_{{{h}\to{0}}}

frac{{ f{{\left({t}+{h}\right)}}

$v=h→$

$lim hf(t+h)−f(t) $

## Instantaneous Rate Of Change Formula

You can find the instantaneous rate of change of a function at a point by finding the derivative of that function and plugging in the x-value of the point.

The instantaneous rate of change of a function is represented by the slope of the line, it tells you by how much the function is increasing or decreasing as the x-values change.

Before moving on let’s do a quick review of just what we did in the above example. We wanted the tangent line to f(x) at a point x=a. First, we know that the point P=(a,f(a)) will be on the tangent line. Next, we’ll take a second point that is on the graph of the function, call it Q=(x,f(x)) and compute the slope of the line connecting P and Q as follows,

mPQ=f(x)−f(a)x−a

We then take values of x that get closer and closer to x=a (making sure to look at x’s on both sides of x=a and use this list of values to estimate the slope of the tangent line, m.

The tangent line will then be,

## How To Find Instantaneous Rate Of Change

The next problem that we need to look at is the rate of change problem. As mentioned earlier, this will turn out to be one of the most important concepts that we will look at throughout this course.

Here we are going to consider a function, f(x), that represents some quantity that varies as x varies. For instance, maybe f(x) represents the amount of water in a holding tank after x minutes. Or maybe f(x) is the distance traveled by car after x hours. In both of this example we used x to represent time. Of course, x doesn’t have to represent time, but it makes for examples that are easy to visualize.

What we want to do here is determine just how fast f(x) is changing at some point, say x=a. This is called the **instantaneous rate of change** or sometimes just the **rate of change** of f(x) at x=a.

As with the tangent line problem all that we’re going to be able to do at this point is to estimate the rate of change. So, let’s continue with the examples above and think of f(x) as something that is changing in time and x is the time measurement. Again, x doesn’t have to represent time but it will make the explanation a little easier. While we can’t compute the instantaneous rate of change at this point we can find the average rate of change.

To compute the average rate of change of f(x) at x=a all we need to do is to choose another point, say x, and then the average rate of change will be,

A.R.C.=change in f(x)change in x=f(x)−f(a)x−a

Then to estimate the instantaneous rate of change at x=a all we need to do is to choose values of x getting closer and closer to x=a (don’t forget to choose them on both sides of x=a) and compute values of A.R.C. We can then estimate the instantaneous rate of change from that.

## Instantaneous Rate Of Change Calculus

In this section, we are going to take a look at two fairly important problems in the study of calculus. There are two reasons for looking at these problems now.

First, both of these problems will lead us to the study of limits, which is the topic of this chapter after all. Looking at these problems here will allow us to start to understand just what a limit is and what it can tell us about a function.

Secondly, the rate of change problem that we’re going to be looking at is one of the most important concepts that we’ll encounter in the second chapter of this course. In fact, it’s probably one of the most important concepts that we’ll encounter in the whole course. So, looking at it now will get us to start thinking about it from the very beginning.

#### Tangent Lines

The first problem that we’re going to take a look at is the tangent line problem. Before getting into this problem it would probably be best to define a tangent line.

A tangent line to the function f(x) at the point x=a is a line that just touches the graph of the function at the point in question and is “parallel” (in some way) to the graph at that point. Take a look at the graph below.

In this graph, the line is a tangent line at the indicated point because it just touches the graph at that point and is also “parallel” to the graph at that point. Likewise, at the second point shown, the line does just touch the graph at that point, but it is not “parallel” to the graph at that point and so it’s not a tangent line to the graph at that point.

At the second point shown (the point where the line isn’t a tangent line), we will sometimes call the line a **secant line**.

We’ve used the word parallel a couple of times now and we should probably be a little careful with it. In general, we will think of a line and a graph as being parallel at a point if they are both moving in the same direction at that point. So, in the first point above the graph and the line are moving in the same direction and so we will say they are parallel at that point. At the second point, on the other hand, the line and the graph are not moving in the same direction so they aren’t parallel at that point.

Okay, now that we’ve gotten the definition of a tangent line out of the way let’s move on to the tangent line problem. That’s probably best done with an example.

### Example 1 Find the tangent line to f(x)=15−2×2 at x=1.

There are a couple of important points to note about our work above. First, we looked at points that were on both sides of x=1. In this kind of process, it is important to never assume that what is happening on one side of a point will also be happening on the other side as well. We should always look at what is happening on both sides of the point. In this example, we could sketch a graph and from that guess that what is happening on one side will also be happening on the other, but we will usually not have the graphs in front of us or be able to easily get them.

Next, notice that when we say we’re going to move in close to the point in question we do mean that we’re going to move in very close and we also used more than just a couple of points. We should never try to determine a trend based on a couple of points that aren’t really all that close to the point in question.

The next thing to notice is really a warning more than anything. The values of mPQ in this example were fairly “nice” and it was pretty clear what value they were approaching after a couple of computations. In most cases, this will not be the case. Most values will be far “messier” and you’ll often need quite a few computations to be able to get an estimate. You should always use at least four points, on each side to get the estimate. Two points are never sufficient to get a good estimate and three points will also often not be sufficient to get a good estimate. Generally, you keeping picking points closer and closer to the point you are looking at until the change in the value between two successive points is getting very small.

Last, we were after something that was happening at x=1 and we couldn’t actually plug x=1 into our formula for the slope. Despite this limitation, we were able to determine some information about what was happening at x=1 simply by looking at what was happening around x=1. This is more important than you might at first realize and we will be discussing this point in detail in later sections.

## Is the derivative the instantaneous rate of change?

The **Derivative** as an **Instantaneous Rate of Change**. The **derivative** tells us the **rate of change** of one quantity compared to another at a particular instant or point (so we call it “**instantaneous rate of change**“).

## Is instantaneous velocity the same as the instantaneous rate of change?

**Velocity** is one kind of **rate of change**. It is the **rate of change** of position with respect to time. The rate** of change** is more general and includes **velocity** as one example. The word “**instantaneous**” has does not alter any of this.

## Is the instantaneous rate of change a limit?

The **instantaneous rate of change**, i.e. the derivative, is expressed using a **limit**. You need the **limit** notation on the left of all of your expressions, i.e. The **instantaneous rate of change** of a function f(x) at x=a is simply given by its derivative at x=a, i.e., f′(a).