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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 5. Calculus / 5.7 Basic Limits & Continuity / 5.7.1 Basic Limits & Continuity


5.7.1 Basic Limits & Continuity


Limits

What are limits in mathematics?

  • When we consider a limit in mathematics we look at the tendency of a mathematical process as it approaches, but never quite reaches, an ‘end point’ of some sort
  • We use a special limit notation to indicate this
    • For example limit as x rightwards arrow 3 of invisible function application f open parentheses x close parentheses denotes ‘the limit of the function f(x) as x goes to (or approaches) 3’
      • I.e., what value (if any) f(x) gets closer and closer to as x takes on values closer and closer to 3
      • We are not concerned here with what value (if any) f(x) takes when x is equal to 3 – only with the behaviour of f(x) as x gets close to 3
  • The sum of an infinite geometric sequence is a type of limit
    • When you calculate S subscript infinity for an infinite geometric sequence, what you are actually finding is limit as n rightwards arrow infinity of invisible function application S subscript n
      • I.e., what value (if any) the sum of the first n terms of the sequence gets closer and closer to as the number of terms (n) goes to infinity
      • The sum never actually reaches S subscript infinity, but as more and more terms are included in the sum it gets closer and closer to that value
  • In this section of the IB course we will be considering the limits of functions
    • This may include finding the limit at a point where the function is undefined
      • For example, space f left parenthesis x right parenthesis equals fraction numerator sin x over denominator x end fraction is undefined when x = 0, but we might want to know how the function behaves as x gets closer and closer to zero
    • Or it may include finding the limit of a function f(x) as x gets infinitely big in the positive or negative direction
      • For this type of limit we write limit as x rightwards arrow infinity of invisible function application f open parentheses x close parentheses or limit as x rightwards arrow negative infinity of invisible function application f open parentheses x close parentheses (the first one can also be written as limit as x rightwards arrow plus infinity of invisible function application f open parentheses x close parentheses to distinguish it from the second one)
    • These sorts of limits are often used to find the asymptotes of the graph of a function

How do I find a simple limit?

  • STEP 1: To find limit as x rightwards arrow a of invisible function application f open parentheses x close parentheses begin by substituting a into the function f(x)
    • If f(a) exists with a well-defined value, then that is also the value of the limit
    • For example, for space f left parenthesis x right parenthesis equals fraction numerator x minus 1 over denominator x end fraction we may find the limit as x approaches 3 like this:

limit as x rightwards arrow 3 of invisible function application f open parentheses x close parentheses equals limit as x rightwards arrow 3 of invisible function application fraction numerator x minus 1 over denominator x end fraction equals fraction numerator 3 minus 1 over denominator 3 end fraction equals 2 over 3

      • In this case,  is simply equal to f(3)
  • STEP 2: If f(a) does not exist, it may be possible to use algebra to simplify f(x) so that substituting a into the simplified function gives a well-defined value
    • In that case, the well-defined value at x = a of the simplified version of the function is also the value of the limit of the function as x goes to a
    • For example, space f left parenthesis x right parenthesis equals x squared over x is not defined at x = 0, but we may use algebra to find the limit as x approaches zero:

limit as x rightwards arrow 0 of invisible function application f open parentheses x close parentheses equals limit as x rightwards arrow 0 of invisible function application x squared over x equals limit as x rightwards arrow 0 of invisible function application x over 1 blank open parentheses cancelling blank the blank x to the power of straight apostrophe straight s close parentheses equals 0 over 1 equals 0

      • Note that space f left parenthesis x right parenthesis equals x squared over x and space g left parenthesis x right parenthesis equals x are not the same function!
      • They are equal for all values of x except zero
      • But for x = 0, g(0) = 0 while f(0) is undefined
      • However f(x) gets closer and closer to zero as x gets closer and closer to zero
  • If neither of these steps gives a well-defined value for the limit you may need to consider more advanced techniques to evaluate the limit
    • For example l’Hôpital’s Rule or using Maclaurin series

How do I find a limit to infinity?

  • As x goes to plus infinity or negative infinity, a typical function f(x) may converge to a well-defined value, or it may diverge to plus infinity or negative infinity
    • Other behaviours are possible – for example limit as x rightwards arrow infinity of invisible function application sin invisible function application x is simply undefined, because sin x continues to oscillate between 1 and -1 as x gets larger and larger
  • There are two key results to be used here:
    • limit as x rightwards arrow plus-or-minus infinity of invisible function application k over x to the power of n  converges to 0 for all n >0 and all k element of straight real numbers
    • limit as x rightwards arrow plus infinity of invisible function application x to the power of n diverges to plus infinity for all n > 0
      • limit as x rightwards arrow negative infinity of invisible function application x to the power of n for n > 0 will need to be considered on a case-by-case basis, because of the differing behaviour of xn for different values of n when x is negative
  • STEP 1: If necessary, use algebra to rearrange the function into a form where one or the other of the key results above may be applied
  • STEP 2: Use the key results above to evaluate your limit
  • For example:

limit as x rightwards arrow infinity of invisible function application fraction numerator 3 x squared minus 2 x plus 1 over denominator 4 x squared minus x plus 2 end fraction equals limit as x rightwards arrow infinity of invisible function application fraction numerator 3 minus 2 over x plus 1 over x squared over denominator 4 minus 1 over x plus 2 over x squared end fraction equals fraction numerator 3 minus 0 plus 0 over denominator 4 minus 0 plus 0 end fraction equals 3 over 4

  • Or:

limit as x rightwards arrow plus infinity of invisible function application fraction numerator x squared plus 5 x minus 2 over denominator 32 x plus 3 end fraction equals limit as x rightwards arrow plus infinity of invisible function application fraction numerator x plus 5 minus 2 over x over denominator 32 plus 3 over x end fraction equals fraction numerator open parentheses plus infinity close parentheses plus 5 minus 0 over denominator 32 plus 0 end fraction equals plus infinity

    • I.e., the limit diverges to plus infinity (because begin inline style fraction numerator x squared plus 5 x minus 2 over denominator 32 x plus 3 end fraction end style it gets bigger and bigger without limit as x gets bigger and bigger)
  • Remember that neither 0 over 0 nor fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction has a well-defined value!
    • If you attempt to evaluate a limit and get one of these two forms, you will need to try another strategy
    • This may just mean different or additional algebraic rearrangement
    • But it may also mean that you need to consider using l’Hôpital’s Rule or Maclaurin series to evaluate the limit
  • It is also worth remembering that if limit as x rightwards arrow infinity of invisible function application f open parentheses x close parentheses equals infinity, then limit as x rightwards arrow infinity of invisible function application fraction numerator k over denominator f open parentheses x close parentheses end fraction equals 0 for any non-zero k element of straight real numbers
    • This can be useful for example when evaluating the limits of functions containing exponentials
      • limit as x rightwards arrow infinity of invisible function application straight e to the power of p x end exponent equals infinity for any p > 0, so we immediately have limit as x rightwards arrow infinity of invisible function application straight e to the power of negative p x end exponent equals limit as x rightwards arrow infinity of 1 over straight e to the power of p x end exponent equals 0 for p > 0
      • See the worked example below for a more involved version of this

Do limits ever have ‘directions’?

  • Yes they do!
  • The notation limit as x rightwards arrow a to the power of plus of invisible function application f open parentheses x close parentheses means ‘the limit of f(x) as x approaches a from above’
    • I.e., this is the limit as x comes ‘down’ towards a, only considering the function’s behaviour for values of x that are greater than a
  • The notation limit as x rightwards arrow a to the power of minus of invisible function application f stretchy left parenthesis x stretchy right parenthesis means ‘the limit of f(x) as x approaches a from below’
    • I.e., this is the limit as x comes ‘up’ towards a, only considering the function’s behaviour for values of x that are less than a
  • One place these sorts of limits appear is for functions defined piecewise
    • In this case the limits ‘from above’ and ‘from below’ may well be different for values of x at which the different ‘pieces’ of the function are joined
  • But also be aware of a situation like the following, where the limits from above and below may also be different:
    • limit as x rightwards arrow 0 to the power of plus of invisible function application 1 over x equals plus infinity (because 1 over x greater than 0 for x > 0, with 1 over x becoming bigger and bigger in the positive direction as x gets closer and closer to zero ‘from above’)
    • limit as x rightwards arrow 0 minus of invisible function application 1 over x equals plus infinity (because 1 over x less than 0 for x < 0, with 1 over x becoming bigger and bigger in the negative direction as x gets closer and closer to zero ‘from below’)
    • The graph of y equals 1 over x shows this limiting behaviour as x approaches zero from the two different directions

Worked Example

a)
Consider the function
 space f open parentheses x close parentheses equals fraction numerator 3 minus 4 x minus 5 x to the power of 4 over denominator 2 x to the power of 4 plus x cubed plus 7 end fraction,
find limit as x rightwards arrow infinity of f left parenthesis x right parenthesis.

5-7-1-ib-aa-hl-limits-a-we-solution

b)
Consider the function 

g open parentheses x close parentheses equals open curly brackets table row cell fraction numerator 1 minus 5 x over denominator x squared end fraction comma end cell cell x less than 5 end cell row cell x squared minus 4 x minus 6 comma end cell cell x greater or equal than 5 end cell end table close

find (i) limit as x rightwards arrow 5 to the power of minus of g left parenthesis x right parenthesis, and (ii) limit as x rightwards arrow 5 to the power of plus of g left parenthesis x right parenthesis.

5-7-1-ib-aa-hl-limits-b-we-solution

c)
Consider the function
h open parentheses x close parentheses equals fraction numerator 2 straight e to the power of 3 x end exponent minus 3 over denominator 4 minus 5 straight e to the power of 3 x end exponent end fraction  
find limit as x rightwards arrow infinity of space h left parenthesis x right parenthesis.

5-7-1-ib-aa-hl-limits-c-we-solution

Continuity & Differentiability

What does it mean for a function to be continuous at a point?

  • If a function is continuous at a point then the graph of the function does not have any ‘holes’ or any sudden ‘leaps’ or ‘jumps’ at that point
    • One way to think about this is to imagine sketching the graph
      • So long as you can sketch the graph without lifting your pencil from the paper, then the function is continuous at all the points that your sketch goes through
      • But if you would have to lift your pencil off the paper at some point and continue drawing the graph from another point, then the function is not continuous at any such points where the function ‘jumps’

5-7-1-ib-aa-hl-cont-_-diff_contin-exx

  • There are two main ways a function can fail to be continuous at a point:
    • If the function is not defined for a particular value of x then it is not continuous at that value of x
      • For example, space f left parenthesis x right parenthesis equals 1 over x  is not continuous at x = 0
    • If the function is defined for a particular value of x, but then the value of the function ‘jumps’ as x moves away from that x value in the positive or negative direction, then the function is not continuous at that value of x
      • This type of discontinuity can occur in a piecewise function, for example, where the different pieces of the function’s graph don’t ‘join up’
  • You can use limits to show that a function is continuous at a point
    • Let f(x) be a function defined at x = a, such that f(a) = b
      • If limit as x rightwards arrow a to the power of minus of invisible function application f open parentheses x close parentheses equals b and limit as x rightwards arrow a plus of invisible function application f open parentheses x close parentheses equals b, then f(x) is continuous at x = a
      • If either of those limits is not equal to b, then f(x) is not continuous at x = a
    • This is a slightly more formal way of expressing the ‘you don’t have to lift your pencil from the paper’ idea!

What does it mean for a function to be differentiable at a point?

  • We say that a function f(x) is differentiable at a point with x-coordinate x0, if the derivative f’(x) exists and has a well-defined value f’(x0) at that point
  • To be differentiable at a point a function has to be continuous at that point
    • So if a function is not continuous at a point, then it is also not differentiable at that point
  • But continuity by itself does not guarantee differentiability
    • This means that differentiability is a stronger condition than continuity
    • If a function is differentiable at a point, then the function is also continuous at that point
    • But a function may be continuous at a point without being differentiable at that point
    • This means there are functions that are continuous everywhere but are not differentiable everywhere
  • In addition to being continuous a point, differentiability also requires that the function be smooth at that point
    • ‘Smooth’ means that the graph of the function does not have any ‘corners’ or sudden changes of direction at the point
    • An obvious example of a function that is not smooth at certain points is a modulus function |f(x)| at any values of x where f(x) changes sign from positive to negative
      • At any such point a modulus function will not be differentiable

5-7-1-ib-aa-hl-cont-_-diff_smooth-exx

Exam Tip

  • On the exam you will not usually be asked to test a function for continuity at a point
    • You should however be familiar with the basic ideas about continuity outlined above
  • On the exam you will not be asked to test a function for differentiability at a point
    • You should however be familiar with the basic ideas about differentiability and its relationship with continuity as outlined above

Worked Example

Consider the function space f defined by

space f left parenthesis x right parenthesis equals open curly brackets table row cell x squared minus 2 x minus 1 comma end cell cell x less than 3 end cell row 2 cell x equals 3 end cell row cell fraction numerator x plus 2 over denominator 2 end fraction comma end cell cell x greater than 3 end cell end table close

a)
use limits to show that space f is not continuous at x equals 3.

5-7-1-ib-aa-hl-cont--diff-a-we-solution

b)
Hence explain why space f cannot be differentiable at x equals 3.

5-7-1-ib-aa-hl-cont--diff-b-we-solution



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Roger Bliss

                                                                                                                                                                                                                                                                                                        Author: Roger

                                                                                                                                                                                                                                                                                                        Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.


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