Rates of Change: Key Definitions, Formula & Examples

Are you aware that the word 'change' is among the most influential in political campaigns?

Upon contracting Covid-19, it's possible to calculate how quickly the virus spreads over a certain time frame.

Through this article, the concept of rate of change and its practical uses will be made clear.

Understanding rates of change

The rate of change refers to the ratio that exists between changes in two different quantities.

Often referred to as the slope or gradient, this occurs when comparing alterations across two quantities.

Rate of change is a fundamental concept used in creating various formulas, including those for velocity and acceleration, indicating the level of activity amidst variations in contributing quantities.

Consider a vehicle traveling A meters in n seconds.

Moving from point A, it traverses an additional B distance by the mth second, showing variances between distances A to B and time from nth to mth second.

Dividing these variances yields the rate of change.

Defining change in mathematics

In the realm of mathematics, change occurs with any increase or decrease in a quantity's value.

This suggests that the change might be positive, negative, or even zero if the quantity remains unchanged.

If you currently possess 5 oranges and later find you have 8, has there been a change? Indeed, as the count of oranges has risen by 3, indicating a positive change.

Conversely, if you start with 5 oranges and end up with just one, this implies a loss of 4 oranges, representing a negative change.

It's important to realize that change is essentially the differential in quantities, calculated as,

$\Delta Q=Q_f-Q_i$

where

$∆Q$ indicates the change in quantity,

$Q_i$ signifies the initial quantity,

$Q_f$ represents the quantity's final value.

A positive ΔQ indicates a positive change, whereas a negative ΔQ suggests a decrease.

Now that the concept of change is clear, let's proceed to evaluate the rate of change.

Formula for rate of change

To determine the rate of change, we divide the alterations in quantities, implying,

$\mathrm{rate}of\mathrm{change}=\frac{\mathrm{change}\mathrm{in}\mathrm{quantity}}{\mathrm{change}\mathrm{in}\mathrm{quantity}}$

Building on this formula, we'll use graph directions for guidance. Assume modifications happen along the x-axis (horizontal) and y-axis (vertical).

In the x-direction, we describe a change as

$\Delta x=x_f-x_i$

where,

$∆x$ denotes the variance in the x-direction,

$x_i$ indicates the starting point,

$x_f$ marks the endpoint on the x-axis.

For changes in the y-direction, we note as,

$\Delta y=y_f-y_i$

where,

$∆y$ characterizes the shift in the y-direction,

$y_i$ is the initial vertical position,

$y_f$ is the final point on the y-axis.

So, we articulate the rate of change formula as,

$$\begin{gathered} rateofchange=\frac{\Delta y}{\Delta x}=\frac{y_f-y_i}{x_f-x_i} \\ rateofchange=\frac{y_f-y_i}{x_f-xi} \end{gathered}$$

If initially a value was observed at 5 units along x and 3 units along y, and later changed to 8 units x-wise and 4 units y-wise, what's the rate of change?

Solution

Given the data, we find

$x_i$ as 5, $x_f$ as 8

$y_i$ as 3, $y_f$ as 4

Thus,

$\mathrm{rate}\mathrm{of}\mathrm{change}=\frac{y_f-y_i}{x_f-x_i}=\frac{1}{3}$

Function rate of change

The function's rate of change measures how the outcome of a function varies with the change in the input quantity.

Let's assign u as a function of w, depicted as

$w=f\left(u\right)$.

This rate demonstrates how w shifts in response to changes in u, provided w depends on u.

The variation in $u$ is defined as

$\Delta u=u_f-u_i$

where,

$∆u$ represents the alteration in $u$,

$u_i$ marks the original value of $u$,

$u_f$ is the resulting value of $u$,

Furthermore, the modification in $w$ is articulated by

$\Delta w=w_1-w_0$

However,

$w=f\left(u\right)$

leading us to,

$f\left(\Delta u\right)=f(u_1-u_0)=f\left(u_1\right)-f\left(u_0\right)$

Hence, the formula for a function's rate of change is,

$\frac{\Delta w}{\Delta u}=\frac{f\left(\Delta u\right)}{\Delta u}=\frac{f(u_f-u_i)}{u_f-u_i}$

The formula to assess a function's change rate is:

$\frac{\Delta y}{\Delta x}=\frac{f\left(x_f\right)-f\left(x_i\right)}{x_f-x_i}$

where:

$∆x$ is the shift along the x-axis,

$x_i$ is the x-axis starting point,

$x_f$ is the endpoint on the x-axis,

$∆y$ is the vertical shift,

$f\left(x_i\right)$ is the initial x-position's function,

$f\left(x_f\right)$ is the concluding x-position's function.

Graphical Rates of Change

Displaying rates of change graphically involves charting quantities on a graph, with three scenarios: zero, positive, and negative rate changes.

Instances of zero rate changes

A zero rate change is seen when there's no effect on the second quantity despite a change to the first. This occurs when

$y_f-y_i=0$.

The graphical representation below showcases the zero rate of change.

Observation of the horizontal orientation suggests x-values shift but y-values remain fixed, rendering the gradient as negligible.

Scenarios with positive rate changes

A positive rate change manifests when the quotient of changes yields a positive outcome. The slope's angle depends on the relative magnitude of change between the pair of quantities.

This infers a lenient slope if y-value alterations surpass those of x-values; conversely, a sharper gradient results when x-value changes exceed those of y-values.

The upward slope detected confirms the positive nature of the rate change. Inspect the images below to comprehend the concept further.

Instances of negative rate changes

Negative rate changes are identified when the quotient of variations results in a negative figure. This scenario emerges when one modification is negative and the other positive. Note, if both alterations yield negative figures, the rate change is considered positive, not negative.

The slope’s steepness again reflects the relative magnitude of change between the quantities, indicating a milder incline if y-value changes outweigh those of x-values, and a sharper descent if x-value changes overshadow those of y-values.

The downward direction of the slope solidifies the rate of change as negative. Examine these illustrations for a better grasp.

To compute the rate of change between two points (1,2) and (5,1):

a. Ascertain the variety of rate change.

b. Determine if the slope is mild or steep.

Solution

With $x_i=1,y_i=2,x_f=5,y_f=1$,

To graphically plot, coordinates are marked within the plane.

Applying the calculation formula for rate change,

$rateofchange$

a. Given our rate of change as -4, it categorizes as a negative rate change.

b. Observing a significant change towards the y-direction (4 positive steps) against a minor x-direction change (1 negative step), the graphed slope emerges as mild, as represented.

Real-world examples of rate changes

Various real-life applications involve rate of change, notably in calculating velocity. This example illuminates the concept:

To determine a vehicle’s average velocity, apply:

Average Velocity = Total Traversed Distance / Total Duration

With the vehicle covering 300m in the first 30 seconds and 500m in the subsequent 70 seconds, the total distance comes to 300m + 500m = 800m.

The summed time spans 30 seconds + 70 seconds = 100 seconds.

Thus, the car's average velocity computes as:

Average Velocity = 800m / 100s = 8 m/s

Key takeaways on rate of change

• Rate of change is established as the ratio between modifications among two quantities.
• Change is observed when a specific quantity either increases or decreases in value.
• The mechanism to figure the rate of change adopts the formula; Rate of change = (yf - yi) / (xf - xi)
• The rate of change concerning a function reflects how a function's output varies as its input does.
• To visually represent rates of change, one maps quantities onto a graph.