# Rates of Change

## Key Questions

As below.

#### Explanation:

Slope is the ratio of the vertical and horizontal changes between two points on a surface or a line.
The vertical change between two points is called the rise, and the horizontal change is called the run.
The slope equals the rise divided by the run: .
This simple equation is called the slope formula.

If $y = f \left(x + h\right) = 3 {\left(x + h\right)}^{2}$, (Just plug x + h in for x). So, you get this:

The instantaneous rate of change, or derivative, can be written as dy/dx, and it is a function that tells you the instantaneous rate of change at any point.

$y ' = f ' \left(x + h\right) = \left(\frac{d}{\mathrm{dx}}\right) \left(3 \cdot {\left(x\right)}^{2}\right) = 6 x \cdot 1 = 6 x$

. For example, if x = 1, then the instantaneous rate of change is 6.

Rate of Change Formula helps us to calculate the slope of a line if the coordinates of the points on the line are given. ... If coordinates of any two points of a line are given, then the rate of change is the ratio of the change in the y-coordinates to the change in the x-coordinates.

Hope this helps.

• Rate of change is a number that tells you how a quantity changes in relation to another.
Velocity is one of such things. It tells you how distance changes with time.
For example: 23 km/h tells you that you move of 23 km each hour.

Another example is the rate of change in a linear function.

Consider the linear function: $y = 4 x + 7$
the number 4 in front of $x$ is the number that represent the rate of change. It tells you that every time $x$ increases of 1, the corresponding value of $y$ increases of 4.
If you get a negative number it means that the $y$ value is decreasing.
If the number is zero it means that you do not have change, i.e you have a constant!

Examples:

• Average Rate of Change

The average rate of change of a function $f \left(x\right)$ on an interval $\left[a , b\right]$ can be found by

$\left(\text{Average Rate of Change}\right) = \frac{f \left(b\right) - f \left(a\right)}{b - a}$

Example

Find the average rate of change of $f \left(x\right) = {x}^{2} + 3 x$ on $\left[1 , 3\right]$.

$f \left(3\right) = {\left(3\right)}^{2} + 3 \left(3\right) = 18$

$f \left(1\right) = {\left(1\right)}^{2} + 3 \left(1\right) = 4$

$\left(\text{Average Rate of Change}\right) = \frac{f \left(3\right) - f \left(1\right)}{3 - 1} = \frac{18 - 4}{2} = \frac{14}{2} = 7$

I hope that this was helpful.