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Rates of Change

Rates of Change

Did you know that one of the greatest political campaign words used is 'change'?

When an individual gets infected with Covid-19, you can determine the rate at which the virus spreads given a specific period of time.

In this article, you shall understand the rate of change and its applications.

Rates of change meaning

The rate of change is defined as the relationship linking the change that occurs between two quantities.

It is known as the gradient or slope when changes occur during the comparison of two quantities.

The concept of rate of change has been widely used to derive many formulas like that of velocity and acceleration. It tells us the extent of activity when there are alterations in the quantities that make up such activities.

Suppose a car covers a distance of A meters in n seconds.

From point A it covers another distance B at the mth second, we notice then that there are changes between the distance A and B as well as differences between the nth and mth second.

The quotient of these differences gives us the rate of change.

What is a change in mathematics?

In mathematics, a change takes place when the value of a given quantity has been either increased or reduced.

This implies that change can be either positive or negative. There is a zero change when the value of a quantity does not change.

Imagine you have 5 oranges right now and later in the day you have 8 oranges. What just happened? Is there a change? Surely, there is a change because your total number of oranges just increased by 3 oranges. As a matter of fact, this is a positive change.

In contrast, consider you have 5 oranges at the moment and much later in the day you have an orange left. This suggests that you have experienced a reduction of 4 oranges. Thus, we say you have experienced a negative change.

This suffices to note that change is basically the difference in quantities calculated as,

where

is the change in quantity,

is the initial value of the quantity,

is the final value of the quantity.

Whenever ΔQ is positive it means there is a positive change, however, when ΔQ is negative it implies a negative change.

Since you know what a change is, we are now ready to calculate the rate of change.

Rates of change formula

To calculate the rate of change, we calculate the quotient between the changes in the quantities. This means,

Further to the derivation of this formula, we shall take the directions on a graph as a guide. Let us consider that changes are made in both the horizontal direction (x-axis) and the vertical direction (y-axis).

In the horizontal direction, a change will imply

where,

is the change in the horizontal direction (x-axis),

is the initial position on the x-axis,

is the final position on the x-axis.

Likewise, in the vertical direction, a change will imply,

where,

is the change in the vertical direction (y-axis),

is the initial position on the y-axis,

is the final position on the y-axis.

Therefore, the rate of change formula becomes,

If the value of a quantity at the start recorded 5 units horizontally and 3 units vertically, thereafter, it recorded 8 units horizontally and 4 units vertically, what is the rate of change?

Solution

From the information given, we have

is 5, is 8

is 3, is 4

Thus,

Rates of change of a function

The rate of change of a function is the rate at which a function of a quantity changes as that quantity itself changes.

Let w be a function of u, expressed as

.

The rate of change of the function w tells us the rate at which w changes and u changes, knowing that w is an expression of u.

The change in is expressed as

where,

is the change in the value of ,

is the initial value of ,

is the final value of ,

Similarly, the change in is given by

But,

thus we have,

Therefore the rate of change of a function formula would be,

The formula used in calculating the rate of change of a function is,

where,

is the change in the horizontal direction (x-axis),

is the initial position on the x-axis,

is the final position on the x-axis,

is the change in the vertical direction (y-axis),

is the function of the initial position on the x-axis,

is the function of the final position on the x-axis.

Rates of change on a graph

Representing rates of change on a graph requires representing quantities on a graph. Ideally, there are three types of graphs that are based on three different scenarios. They are the zero, positive and negative rate of change graphs as would be explained below.

Zero rates of change

The zero rates of change occur when the quantity in the numerator changes and it does cause any change to the second quantity. This takes place when

.

The graph below illustrates the zero rate of change.

Rates of Changes An illustration of zero rate of change when no change occurs in the y-direction StudySmarterAn illustration of zero rates of change when no change occurs in the y-direction - StudySmarter Originals

We notice that the arrow is pointing rightwards horizontally, this suggests that there is a change in the x-values but the y-values are unchanged. So the y-values are not affected by changes in x and as such the gradient is 0.

Positive rates of change

Positive rates of change occur when the quotient of the changes between both quantities is positive. The steepness of the slope is dependent on which quantity experiences a greater change relative to the order quantity.

This means that if the change in the y-values is greater than that of the x-values, then the slope will be gentle. In contrast, when the change in x-values is greater than that of the y-values, then the slope would be steep.

Note that the direction of the arrow pointing upwards reveals that the rate of change is indeed positive. Give a quick look at these figures below to understand much better.

Rates of Changes An illustration of a gentle sloped positive rate of changes StudySmarterAn illustration of a gentle sloped positive rate of change - StudySmarter Originals

Rates of Changes An illustration of a steep sloped positive rate of change StudySmarter

An illustration of a positive steep-sloped rate of change - StudySmarter Originals

Negative rates of change

Negative rates of change occur when the quotient of the changes between both quantities gives a negative value. For this to occur, one of the changes must produce a negative change while the other must give a positive change. Beware that when both changes produce negative values, then the rate of change is positive and not negative!

Again, the steepness of the slope is dependent on which quantity experiences a greater change relative to the order quantity. This means that if the change in y-values is greater than that of the x-values, then the slope will be gentle. In contrast, when the change in x-values is greater than that of the y-values, then the slope would be steep.

Note that the direction of the arrow pointing downwards reveals that the rate of change is indeed negative. Take a quick check on these figures below to understand much better.

Rates of Change An illustration of a gentle sloped negative rate of change StudySmarter

An illustration of a negative gentle-sloped rate of change - StudySmarter Originals

Rates of Changes An illustration of a steep sloped negative rate of change StudySmarter

An illustration of a negative steep-sloped negative rate of change - StudySmarter Originals

Calculate the rate of change between two coordinates (1,2) and (5,1) and determine

a. The type of rate of change.

b. Whether the slope is steep or gentle.

Solution

We have ,

In order to sketch the graph, we plot the points in the coordinate plane.

Now, in order to calculate the rate of change, we apply the formula,

a. Since our rate of change is -4, thus, it has a negative rate of change.

b. We notice that the change towards the y-direction (4 positive points) is greater than the change in the x-direction (1 negative step), therefore, the slope when plotted on a graph would be gentle as shown in the figure.

Rates of change examples

There are practical applications of rates of change. A good application is in the determination of speed. An illustration below would elaborate better.

A car starts from rest and arrives at a point J which is 300m from where it started in 30 seconds. At the 100th second, it reaches a point F which is 500m from his starting point. Calculate the average speed of the car.

Solution

Below is a sketch of the journey of the car.

The average speed of the car is equivalent to the rate of change between the distance travelled by the car and the time it took.

Thus;

Therefore, the average speed of the car is 2.86ms-1.

Rates of Change - Key takeaways

  • The rate of change is defined as the relationship linking the change that occurs between two quantities.
  • A change takes place when the value of a given quantity has been either increased or reduced.
  • The formula used in calculating the rate of change is;
  • The rate of change of a function is the rate at which a function of a quantity changes as that quantity itself changes.
  • Representing rates of change on a graph requires representing quantities with points on a graph.

Frequently Asked Questions about Rates of Change

The rate of change is defined as the relationship linking the change that occurs between two quantities.

rate of change = (y - yi) /( x- xi)

An example of rate of change would be when you buy 2 pies for £6 and much later you buy 4 of same pies for £12. Thus, the rate of change is (12 - 6)/(4-2) = £3 per unit of pie.

You graph the rate of change by representing quantities in relationship with points on a graph. 

The rate of change of a function is the rate at which a function of a quantity changes as that quantity itself changes.

Final Rates of Change Quiz

Question

What is simple interest?

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Answer

Simple interest is a way of calculating the interest on an amount of money. Simple interest is usually associated with borrowing or investing an amount of money.

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Question

What are the terms associated with Simple Interest?

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Answer

The terms associated with simple interest are Principal, Rate and Time.

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Question

What is interest?

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Answer

Interest is fee paid on borrowed money or on a loan.

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Question

What is the principal amount?


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Answer

Principal amount is the term used to describe the borrowed money, or the loan.

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Question

Interest rate is expressed in percentage. True or False.

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Answer

True

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Question

What are the types of interest?


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Answer

Simple interest and Compound interest.

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Question

What is simple interest?

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Answer

Simple interest is used to find the interest on a principal amount.

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Question

What is compound interest?


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Answer

Compound interest refers to the amount of interest that has been gathered or earned over time on an amount of money.

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Question

What is Compound interest?

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Answer

Compound interest is the accumulation or addition of interest to a principal amount. 

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Question

What is the difference between simple and compound interest?

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Answer

The difference between simple and compound interest is that simple interest has to do with one time interest on the principal amount while compound interest has to do with an accumulation of interest on the principal amount over a period of time.


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Question

What are the two ways of calculating compound interest?

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Answer

Compound Interest can be calculated by using a table or by using the compound interest formula. 

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Question

What is an example of Compound interest with example?

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Answer

Compound interest is the accumulation or addition of interest to a principal amount. An example is when money is deposited in a savings account. The money is expected to gain interest continuously over time. 


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Question

Which of the following is true about compound interest.

A. Compound interest is one time interest on the principal amount.

B. As time increases, money increases.

C. There are two ways to solve for compound interest.

D. Compound interest is the same as simple interest

Show answer

Answer

B and C

Show question

Question

Compound interest is the accumulation or addition of interest to a principal amount.

True or False

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Answer

True

Show question

Question

What is gradient?

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Answer

Gradient measures the steepness and inclination of a line and shows the rate of change over time. 

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Question

How do you find the gradient of a curve?

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Answer

To find the gradient of a curve you

  1. Select a point on the curve (the point to be selected may be given).
  2. Draw a tangent at that point.
  3. Select two points on the tangent and note their coordinates. You can decide to take the point at which the tangent is drawn as the first point and select another point as the second. You may also want to draw a right angle triangle to highlight the points you are using.
  4. Use the gradient formula to find the gradient. 

Show question

Question

How to find instantaneous rate of change without derivatives?

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Answer

An alternative way of finding the instantaneous rate of change at a point is by calculating the tangent at that point. 

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Question

What is an example of instantaneous rate of change?

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Answer

Suppose a car has covered 40 km in 20 minutes. To find the rate of change of distance relative to time at any point over the interval of that time would be a typical example. 

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Question

How to calculate instantaneous rate of change?

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Answer

The instantaneous rate of change is calculated by finding the derivative of the curve and evaluating it at that point. 

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Question

State the difference between instantaneous and average rate of change.

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Answer

The instantaneous rate of change of a function is the rate of change at a certain point (at a certain instant) whereas the average rate of change of the same function is taken over a big interval.

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Question

From a graphical view, how can the instantaneous rate of change can be represented on a curve at a specific point?

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Answer

On a curve, the instantaneous rate of change at a certain point can be represented by drawing a tangent at that point and measuring its slope.

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Question

State the rate of change of the line y=x intuitively.

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Answer

Since this is a line passing through the origin with a slope of 1, the rate of change of y w.r.t. x is 1.

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Question

Define average rate of change.

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Answer

The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.

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Question

State the difference between instantaneous and average rate of change.

Show answer

Answer

The instantaneous rate of change of a function is the rate of change at a certain point (at a certain instant) whereas the average rate of change of the same function is taken over a big interval.

Show question

Question

What is an example of average rate of change?

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Answer

An example is the average distance covered in a certain time interval by a car. 

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Question

How to graph the average rate of change?

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Answer

To graph the graph of average rate of change between two points, plot the two endpoints and draw a line joining them. The slope of the line will be the average rate of change.

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Question

For any straight line, what will be the rate of change of y w.r.t x between any two points on the line?

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Answer

The average rate of change between any two points will be equal to the slope of that line.

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Question

For any curve, what will be the rate of change between two points on the curve?

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Answer

The average rate of change between any two points on a curve will be equal to the slope of the line passing through those points.

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Question

From a graphical view, how can one describe the rate of change between two points?

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Answer

The gradient between the two points on any curve acts as the average rate of change between them.

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Question

For which kind of curve, the rate of change of y w.r.t. x is the same as the rate of change of x w.r.t. y?

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Answer

In the case of a straight line with a gradient of 1, the rate of change of both quantities w.r.t. each other are the same.

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Question

Growth  is defined as a process that involves the reduction in quantity, size, or value.

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Answer

FALSE

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Question

Growth is defined as a process that involves an increase in quantity, size, or value. 


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Answer

TRUE

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Question

If a certain organism has a growth rate of 20% every year, and its current length is 25cm, find its length after 3 years.

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Answer

43.2cm

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Question

Everyday the value of a certain currency increases by 0.4. If its current value is 12 units, what is its expected value in a week?

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Answer

126.5 units

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Question

If the brain cells of a man reduces its efficiency by 12% every decade after the age of 40, and Nonny's brain efficiency at 40 is 80%, what would be the efficiency of his brain at 60?

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Answer

61.95%

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Question

Brume's food decays at the rate of 15% every hour. If it is currently about 25 spoons left, what would be the size in 5 hours time?

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Answer

11.1 spoons

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Question

The population of a bacteria is reduced by a certain antibiotic by half every 2 hours. If the current population in an individual is 5000, what would the population of the bacteria be in 6 hours time?

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Answer

78

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Question

The population of a bacteria is reduced by a certain antibiotic by half every 2 hours. If the current population in an individual is 5000, in what time would be the bacteria be totally eradicated from the individual assuming only a minimum of 1 bacterium has to be left before it is said to be eradicated?

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Answer

12.3 hours

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Question

The decay constant of a substance per second  is 1.32. If the substance is 13 units presently, what would be its value in 3 seconds time?

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Answer

29.9 units

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Question

If a car worth 2000 pounds depreciates yearly by 20%, what is its value 4 years after.

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Answer

819.2 pounds

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Question

If a car worth 2000 pounds appreciates yearly by 20%, what is its value 3 years after.

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Answer

3456 pounds

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Question

Find the amount after 3 years compounded thrice yearly at 5% for an initial investment of 5000 pounds

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Answer

5802 pounds

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Question

The amount after 2 years of investment  compounded quarterly is 2500 pounds. If the initial investment made is 2000 pounds, find the rate. 

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Answer

11.31%

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Question

Appreciation and depreciation are the same

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Answer

FALSE

Show question

60%

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