ARC: The change in a quantity's value divided by the time elapsed. This is the change in the y-value divided by the change in the x-value at **two distinct points** on the graph for a function. For example, the average rate of change for the equation \frac{dy}{dx} with x as its independent variable and y as **its dependent variable** is \frac{dy}{dx}.

To find the average rate of change for an equation, you would take **the quotient rule** to find the derivative of the equation expression with respect to x, then divide it by the derivative of x with respect to x. For example, the average rate of change for the equation \sin(x) + \cos(x) is \frac{\sin(x) + \cos(x) - \sin(x) - \cos(x)}{1} = 0.

Average rates of change are useful in determining whether a function is increasing or decreasing.

- What is the average rate of change in math?
- What is a constant rate of change graph?
- What is the rate of change for Y?
- How do you find the instantaneous rate of change between two points?
- What is constant change in math?
- What is the instantaneous rate of change?
- Is the instantaneous rate of change the same as the derivative?

Definition No. 2 (constant rates of change) The slope of a linear function y = f (x) is its (constant) rate of change with respect to the variable x. If x and f have units in Definition 2, then the rate of change is defined as f divided by x. For example, if x has meters and f has centimeters, then the rate of change is 100 cm/m.

Example Let's say that a car travels at a constant speed of 60 miles per hour for the first 10 minutes and then slows down to a final speed of 30 miles per hour. What was the original distance traveled by the car?

Solution Start with a knowledge base point: A car traveling at a constant speed will cover a certain distance. In this case, the distance is the product of the speed times the time. So, the answer is that the car traveled 60 mi (97 km) in the first 10 minutes and then slowed down to **a final speed** of 30 mi (48 km).

Now, let's put **our knowledge base** into practice. We know that the initial speed was 60 mi (97 km), so we can divide 97 by 10 to find that the car started out driving **9.7 miles** (15.5 km) per hour. After slowing down, the final speed was 30 mi (48 km), so we can multiply 48 by 10 to find that the car stopped moving after going 150 miles (241 km).

If two values, x and y, are connected by the equation y = f(x), the derivative f'(xo) gives the rate of change of y with regard to x at **the position xo**. It is measured in **y units** or x units. A positive rate of change is also known as a rate of rise. The larger the number, the faster the value is increasing or rising.

Here are some examples:

The rate of change of y with respect to x is positive if f'(x) > 0. It means that if you move up the curve, y will be going up too. In **this case**, the value of y is increasing.

The rate of change of y with respect to x is negative if f'(x) < 0. In this case, the value of y is decreasing.

The rate of change of y with respect to x is 0 if f'(x) = 0. This means that the curve is flat at **this point**. If you go further up or down the curve, the value of y will not change anymore.

The rate of change of y with respect to x is undefined if f'(x) is either negative or positive but not zero.

You can calculate **the instantaneous rate** of change of a function at a point by taking its derivative and filling in the point's x-value. For example, if you want to find the rate of change of f(x) = x^3 at x = 2, then you would take 3*2^2 = 12 as the answer.

A constant rate of change is one in which something changes by **the same amount** at **equal intervals**. A line is a graph with a constant rate of change, and the rate of change is the same as the slope of the line. A line with a slope of 1 has a constant rate of change if it passes through the origin.

A linear equation describes a situation where an object's size or quantity increases or decreases at a constant rate. The equation "size = size * 10" describes an object that grows by 10 percent every time it gets resized. It is linear because the percentage increase (or decrease) is always the same. An equation with a different number for each instance of **increasing size** would be non-linear.

The phrase "a constant rate" makes sense when you think about how much stuff there is on Earth. If the total mass of all the people, animals, plants, etc. on Earth increased at a constant rate, then there would be more stuff than there is now after a certain point. This isn't true, though; instead, the total mass increases but not at a constant rate. At some points in time there are more stars in the sky than others, for example, because over time galaxies merge together and form **larger ones**.

This means that you can't predict exactly when something will run out of space to grow into.

The slope of the tangent line at a location is the instantaneous rate of change. As the "run" of each secant line approaches zero, the slope at a point P (the slope of the tangent line) may be estimated. Using this method, we can estimate the rate of change of any function at any point.

Instantly means right now or in this case. The speedometer on your car measures your car's instant rate of change by calculating how many miles per hour you are going and then displaying that number on its LCD screen. If the car is traveling north and the display reads 60 mph, you are driving exactly six miles per hour below **the legal limit**.

Function refers to a mathematical term for what we call an action or behavior. In **other words**, it describes something that can be measured. Speed is a function of time. More precisely, it is the rate of change of time with respect to distance traveled. This means that if you know one thing about the vehicle, you can calculate the other. For example, if you know that it has a fixed suspension system, then you can assume that it will have **the same speed** at **a given location** regardless of whether it is going up or down a hill.

You should understand that the speedometer only shows you the rate of change of distance vs time.

The change in the rate at a single moment is known as **the instantaneous rate** of change, and it is the same as the change in **the derivative value** at a specific point. The instantaneous rate of change at **a certain location** on a graph is the same as the tangent line slope. These two values are equal because a positive increase in x will result in a increase (or decrease) in y and therefore a parallel shift to the left (or right). As you can see, the instantaneous rate of change is not a constant; instead, it depends on the position along the curve.

Examples: If the function f(x) = x^3 then its derivative is 3x^2. At any point where f'(x) exists, the derivative equals the instantaneous rate of change. So, at time 1, the rate of change is 10, and at time 2 it's -5. The total change in the rate of change is therefore 5. This means that the curve increases its rate of change by five at a time interval of one second.

Notes: The instantaneous rate of change is the same as the derivative at a specific point. However, they are different concepts with different applications. The derivative tells us how the function changes over an interval while the instantaneous rate of change tells us how quickly the function changes at a particular point.