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Rates of Change

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
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- Derivative of Logarithmic Functions
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- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
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- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
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- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
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- Limit of a Sequence
- Limits
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- Limits of a Function
- Linear Approximations and Differentials
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- Logarithmic Differentiation
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- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
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- Vectors in Calculus
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- Decision Maths
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- Area of a Kite
- Composition
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- Coordinate Systems
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- Distance and Midpoints
- Equation of Circles
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- Figures
- Fundamentals of Geometry
- Geometric Inequalities
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Isometry
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- Law of Cosines
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- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
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- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
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- Segment Length
- Similarity
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- Special quadrilaterals
- Squares
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- Symmetry
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- Triangle Inequalities
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- Using Similar Polygons
- Vector Addition
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- Volume of Cone
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- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
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- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
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- Power
- Projectiles
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- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
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- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
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- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
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- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
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- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
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- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
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- Representation of Complex Numbers
- Rewriting Formulas and Equations
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- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
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- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
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- Simultaneous Equations
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- Small Angle Approximation
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- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
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- Solving and Graphing Quadratic Inequalities
- Special Products
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- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
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- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
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- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
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- Trigonometry
- Turning Points
- Types of Functions
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- Unit Circle
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- Statistics
- Bias in Experiments
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- Bivariate Data
- Box Plots
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- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
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- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
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- Data Analysis
- Data Interpretation
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- Distributions
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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.

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.

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.

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,

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.

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.

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.

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 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.

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

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.

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

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.

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}.

- 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.

rate of change = (y_{f } - y_{i})_{ }/( x_{f }- x_{i})

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

More about Rates of Change

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