Instantaneous 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 they 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
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 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.
In this graph, you can see how the blue function can have its instantaneous rate of change represented by a red line tangent to the curve. To find the slope of this line, you must first find the derivative of the function.
Using the power rule for derivatives, we end up with 4x as the derivative. Plugging in our point’s x-value, we have
This tells us that the slope of our original function at (1,6) is 4, which also represents the instantaneous rate of change at that point.
If we also wanted to find the equation of the line that is tangent to the curve at the point, which is necessary for certain applications of derivatives, we can use the Point-Slope Form
with m = slope of the line.
Plugging in our x,y, and slope value, we have
Which simplifies to
Problems involving the instantaneous rate of change of a function require you to use the derivative, though it could be disguised in a way you may not be familiar with, such as the velocity of an object after a certain amount of time. Practicing a similar rate of change problems will help you get a feel for the practical uses of derivatives.
Instantaneous Rate of Change Formula
The rate of change at one known instant is the Instantaneous rate of change, and it is equivalent to the value of the derivative at that specific point. So it can be said that, in a function, the slope, m of the tangent is equivalent to the instantaneous rate of change at a specific point. One more method to comprehend this concept clearly is with the difference quotient and limits. The average rate of change of y with respect to x is the difference quotient. Now if one looks at the difference quotient and lets Delta x->0, this will be the instantaneous rate of change. In guileless words, the time interval gets lesser and lesser.
The Instantaneous Rate of Change Formula provided with limit exists in,
When y = f(x), with regards to x, when x = a.
Is the derivative the instantaneous rate of change?
Is instantaneous velocity the same as the instantaneous rate of change?
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).