One of the most common questions we get here at Calculus Basics is “how do I find the rate of change?” We’re here to help with that!
There are a few things we need to know before we can answer that question though. The first is what a function is. A function is a set of ordered pairs (x y) where each x corresponds to a unique y. In other words a function is a way of pairing up things so that we can put them in a specific order.
The second thing we need to know is what the domain and range of a function are. The domain is the set of all x-values that a function can take while the range is the set of all y-values that the function can output.
So armed with that information let’s talk about how to find the rate of change of a function. There are two ways to do this: using the slope formula or taking the derivative of the function.
We’ll start with the slope formula which is:
rate of change = (change in y-values) / (change in x-values)
You can use this formula to find the rate of change between any two points on a graph. Just plug in the corresponding x- and y-values for each point and you’ll be all set!
For example let’s say we have the following graph:
We can see from the graph that the y-values are increasing as the x-values increase. So the rate of change is positive. If we want to find the actual rate of change we can pick any two points and use the slope formula. Let’s say we pick the points (1 2) and (2 4). We plug these values into the formula and get:
rate of change = (4-2) / (2-1) = 2
This tells us that for every 1 unit increase in x there is a corresponding 2 unit increase in y. In other words the function is increasing at a rate of 2 units per 1 unit increase in x.
Now let’s talk about finding the rate of change using derivatives. The derivative of a function at a specific point is the slope of the tangent line to the graph of the function at that point.
So to find the rate of change at a specific point we need to find the slope of the tangent line at that point. We can do this using the following formula:
derivative = (change in y-values) / (change in x-values)
Just like with the slope formula we can plug in the corresponding x- and y-values for the point we’re interested in to find the derivative.
For example let’s say we want to find the derivative of the following function at the point (1 2):
We can see from the graph that the tangent line at (1 2) is parallel to the x-axis which means the slope of the tangent line is 0. We can confirm this by plugging the x- and y-values into the derivative formula:
derivative = (2-2) / (1-1) = 0
This tells us that the rate of change of the function at (1 2) is 0.
So those are two ways to find the rate of change of a function: using the slope formula or taking the derivative. If you have any questions about this process or anything else related to calculus feel free to leave a comment below or contact us at Calculus Basics!
How do you find the rate of change of a function?
The rate of change of a function is the derivative of the function.
How do you find the derivative of a function?
The derivative of a function is the slope of the tangent line to the graph of the function at a point.
What is the slope of the tangent line?
The slope of the tangent line is the difference in the y-coordinates divided by the difference in the x-coordinates.
What is the difference quotient?
The difference quotient is the difference in the y-coordinates divided by the difference in the x-coordinates.
What is the instantaneous rate of change?
The instantaneous rate of change is the derivative of the function.
What is the definition of the derivative?
The derivative is the limit of the difference quotient as the difference in the x-coordinates approaches zero.
How do you find the limit of the difference quotient?
You find the limit of the difference quotient by taking the limit of the numerator and the limit of the denominator separately and then dividing them.
What is the limit of the numerator?
The limit of the numerator is the difference in the y-coordinates.
What is the limit of the denominator?
The limit of the denominator is the difference in the x-coordinates.
What is the difference in the y-coordinates?
The difference in the y-coordinates is the y-coordinate of the point minus the y-coordinate of the tangent line.
What is the difference in the x-coordinates?
The difference in the x-coordinates is the x-coordinate of the point minus the x-coordinate of the tangent line.
What is the y-coordinate of the point?
The y-coordinate of the point is the function evaluated at the x-coordinate of the point.
What is the x-coordinate of the point?
The x-coordinate of the point is the point at which the derivative is being evaluated.
What is the y-coordinate of the tangent line?
The y-coordinate of the tangent line is the derivative of the function evaluated at the x-coordinate of the point.
What is the x-coordinate of the tangent line?
The x-coordinate of the tangent line is the x-coordinate of the point.
