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Mathematics involves numbers, shapes, algebra and many more topics, but it’s much more than that. It is the study of the ‘language of the universe’. The logic and examples that underpin many mathematical concepts existed long before mathematics was created, and we can use them to help us understand how things work. Maths is essential for understanding the sciences. For…

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Math

- 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
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- 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
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- 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
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- 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
- 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
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- 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
- Angles in Polygons
- Approximation and Estimation
- 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
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- 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
- 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
- Eigenvalues and Eigenvectors
- 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 Trigonometric 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
- L'Hopital's Rule
- 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 Calculations
- 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
- 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
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Products
- Scalar Triple Product
- 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
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- 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
- Conducting a Survey
- 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
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Product Moment Correlation Coefficient
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- The Power Function
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

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Jetzt kostenlos anmeldenMathematics involves numbers, shapes, algebra and many more topics, but it’s much more than that. It is the study of the ‘language of the universe’. The logic and examples that underpin many mathematical concepts existed long before mathematics was created, and we can use them to help us understand how things work. Maths is essential for understanding the sciences. For example, physics can tell us how objects should behave due to gravity, which is supported by the diverse tools that mathematics provides.

Mathematics is one of the subjects many students find most challenging during their studies. The key to mastering mathematics is to learn where topics come from and how they are linked and complete lots of practice questions to know how to approach all kinds of problems.

At StudySmarter, you will find mathematics revision notes covering every aspect needed for your exams, complete with flashcards to help you practice and gain a solid grasp of the key concepts!

We can split Mathematics into many subtopics; however, there are three main areas: Pu**re maths****, ****mechanics**** **and **statistics****. **

**Pure maths** is dedicated to the study of maths independent of its application, but many of the concepts can still be implemented in everyday life.

Mathematics, especially pure mathematics, tries to extract the main concepts behind operations and numbers and utilises these concepts using symbols and relationships. The work done can then be applied to different fields such as social sciences, logic, engineering, biology, chemistry, and physics. For instance, calculus is the basis of many engineering courses. Many of the topics within pure maths, such as vectors, geometry, algebra, and coordinate spaces relate to each other.

Within pure maths, you can find the following topics on StudySmarter:

**Proof**– the area of maths that includes the verification of math theorems and laws.**Algebra**– the topic comprising the study of the types of numbers and mathematical operations.**Functions**– comprises the understanding of the relationships between two groups of numbers, the function input (independent variable) and the function output (the dependent variable). The relationship is created by applying operations on the input to obtain the output.**Coordinate geometry**– is the study of geometrical objects using a system to locate them in a space. An excellent example is Cartesian geometry which is the coordinate system you would use to know the positions of objects in a 3D space when studying triangles and other shapes.**Sequences and series**– sequences are lists of numbers such as n,n2,n3,n4,n5… where all n numbers relate to each other. Series are the sums of sequences.**Trigonometry**– examines geometrical objects in a coordinate space, strongly related to coordinate geometry.**Exponentials and logarithms**– these are special functions in maths. They are different from the ones you study in ‘Functions’ as they introduce something called a limit. When this limit is reached, the changes to the function input (independent variable) will produce little to almost zero changes in the function output (dependent variable). These functions are widely used to model different mechanisms in biology, physics and other areas of science.**Differentiation**– the area of pure maths used to measure the rate of change of one function. The modern basis of differentiation and integration was created by Isaac Newton and Wilhelm Leibniz. Differentiation is widely used in many areas of science and engineering, providing solutions for many practical problems such as heat transfer, fluid mechanics, and dynamics of systems in movement.**Integration**– this is the inverse of differentiation, used for finding the area under a graph. You can use it to obtain many physical properties such as energy, work, area, and so on.**Numerical methods**– this area uses mathematical approximations to solve problems that are difficult to solve using the standard mathematical tools. It is used in conjunction with differentiation and integration to solve many practical problems in engineering and science.**Vectors**– vectors are representations used to model quantities that have a direction. A common example is the representation of wind. The vector will have two main properties which are a magnitude (value) and a direction. In the wind example, the magnitude is its speed and the direction where it is blowing is the vector direction.

Statistics involves the collection, analysis, organisation and presentation of data. It allows you to obtain meaningful results and conclusions from data collected through experiments, surveys, interviews or observations. As a student, statistics provides other important areas of application for your knowledge of mathematics.Statistics is widely used in social sciences, economics, the natural sciences and engineering. There are many different techniques and models in statistics, and you’ll need to learn how and when to apply them.StudySmarter will help you understand statistics and covers sampling, data interpretation, distributions, and hypothesis testing.

The only way of ensuring data is accurate is to test every item or individual in a population.

Sampling is the method of extracting a subset of the population to make general statements or predictions about the dataset.

StudySmarter will help you to understand the basics of sampling and its methods.

Once data has been collected, it needs to be interpreted to analyse it effectively and draw conclusions. This process uses key information from your data analysis.

It’s also helpful to represent the data in a graph or chart, so it’s easier to understand and identify patterns.

At StudySmarter, we will teach you how to extract key information such as the measures of central tendency. You will learn how to present this data using plots, graphs, histograms and so on.

Statistical outcomes can be modelled in different ways depending on the event. For example, some events are independent, meaning the likelihood of it occurring is unaffected by previous results, while some are dependent - one result affects another.

For this reason, different probability distributions can be used in different scenarios, such as normal, binomial and Poisson distributions.

StudySmarter will guide you through learning how to analyse data and derive information from its distribution.

Before collecting and analysing data, you often predict the expected result. This is known as a hypothesis.

The process of hypothesis testing allows you to quantify the accuracy of your prediction. You will need to know how to test hypotheses for different probability distributions, but the general method is the same.

On StudySmarter, you will learn all about the procedure of hypothesis testing and when to apply them.

Mechanics studies the relationships between force, matter and motion. To do so, we use the tools of calculus, which was invented for this purpose.Mechanics involves using examples of **systems in movement**, using real-life simplified scenarios for more straightforward calculations. Examples include a box on a slope or a weight on a rope pulley.On StudySmarter, Mechanics is divided as shown below:

Quantities, Units and Assumptions

Kinematics

Forces and Newton’s Laws

This subtopic covers the fundamental knowledge you will need to solve problems involving mechanics. This includes common assumptions you make and their implications for your answer – e.g. if you assume there is no air resistance on a moving object, you will get a higher theoretical velocity than would be the case in practice. You will also learn key units and how to convert between them.

Kinematics is the mechanics of motion, including acceleration (both constant and variable) and parabolic motion (projectiles).

Within this subtopic, you will also learn how to use the equations of uniform acceleration (constant acceleration), a set of equations that link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). We refer to these equations as suvat equations.

A force is a ‘push’ or ‘pull’ acting on an object, which might cause a change in its motion.

An example is gravity, which is always acting on Earth at a force of approximately 9.8 N on a unit mass object.

Forces appear in a variety of situations. On StudySmarter, you will learn how to calculate forces when applied parallel to the object in motion or at an angle to the object in motion (component forces).

An example of when forces have components is when a box slides over a slope.

Newton’s Laws are a set of three laws that define the interaction between forces, acceleration and movement. The third law can be summarised as ‘every action has an equal and opposite reaction’. We know that gravity is a force acting downwards, but in order to keep us in **equilibrium**, there must be some other force acting upwards, too - otherwise, we would sink into the ground!

To gain a solid understanding of mathematics, it’s important to complete lots of practice questions to know which techniques you need to use in different situations and how to approach different types of questions. StudySmarter provides notes, flashcards, quizzes and other comprehensive revision materials to help you get top grades in your exams - completely free!

In addition, the StudySmarter app:

Creates an intelligent learning plan just for you.

Tracks your progress and motivates you with badges and awards.

Lets you quickly create notes and flashcards and share them with other students in your class. You can also join a wider learning community of other maths students.

By using our resources, you will be well prepared and confident for every level of exam.

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