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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 5. Calculus / 5.1 Differentiation / 5.1.2 Applications of Differentiation


5.1.2 Applications of Differentiation


Increasing & Decreasing Functions

What are increasing and decreasing functions?

  • A function,space f left parenthesis x right parenthesis, is increasing ifbold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold greater than bold 0
    • This means the value of the function (‘output’) increases asspace x increases
  • A function,space f left parenthesis x right parenthesis, is decreasing ifbold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold less than bold 0
    • This means the value of the function (‘output’) decreases asspace x increases
  • A function,space f left parenthesis x right parenthesis, is stationary ifbold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold 0

Incr Decr Illustr 1

How do I find where functions are increasing, decreasing or stationary?

  • To identify the intervals on which a function is increasing or decreasing 
STEP 1
Find the derivative f'(x)

STEP 2
Solve the inequalities
bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold greater than bold 0 (for increasing intervals) and/or
bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold less than bold 0 (for decreasing intervals)

  • Most functions are a combination of increasing, decreasing and stationary
    • a range of values ofspace x (interval) is given where a function satisfies each condition
    • e.g.  The functionspace f begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px x to the power of size 16px 2 has derivativespace f to the power of size 16px apostrophe begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px 2 size 16px x so
      • space f left parenthesis x right parenthesis is decreasing for x less than 0
      • space f left parenthesis x right parenthesis is stationary at x equals 0
      • space f left parenthesis x right parenthesis is increasing for x greater than 0

Worked Example

space f stretchy left parenthesis x stretchy right parenthesis equals x squared minus x minus 2

a)
Determine whetherspace f left parenthesis x right parenthesis is increasing or decreasing at the points where x equals 0 and x equals 3.

oT8b2gMg_5-1-2-ib-sl-ai-as-we1-soltn-a

b)
Find the values of x for whichspace f left parenthesis x right parenthesis is an increasing function.
mHWByvDO_5-1-2-ib-sl-ai-as-we1-soltn-b

 

Tangents & Normals

What is a tangent?

  • At any point on a curve (the graph of a (non-linear) function), the tangent is the straight line that passes through that point and has the same gradient as the curve at that point

Grad Tang Norm Illustr 2

How do I find the equation of a tangent?

  • The equation of the tangent to the function y equals f left parenthesis x right parenthesis at the point left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis is

bold space bold italic y bold minus bold italic y subscript bold 1 bold equals bold italic f to the power of bold apostrophe stretchy left parenthesis bold italic x subscript 1 stretchy right parenthesis bold left parenthesis bold italic x bold minus bold italic x subscript bold 1 bold right parenthesis

What is a normal?

  • At any point on a curve (the graph of a (non-linear) function), the normal is the straight line that passes through that point and is perpendicular to the tangent at that point

Grad Tang Norm Illustr 3

How do I find the equation of a normal?

  • The equation of the normal to the function size 16px y size 16px equals size 16px f size 16px left parenthesis size 16px x size 16px right parenthesis at the point left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis is

bold space bold italic y bold minus bold italic y subscript bold 1 bold equals fraction numerator bold minus bold 1 over denominator bold italic f bold apostrophe bold left parenthesis bold italic x subscript bold 1 bold right parenthesis end fraction bold left parenthesis bold italic x bold minus bold italic x subscript bold 1 bold right parenthesis

Exam Tip

  • The equations of a tangent and a normal are not in the formula booklet
    • However both can be derived from the equation of a straight line space y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses
    • This is given in the formula booklet

Worked Example

The functionspace f left parenthesis x right parenthesis is defined by

 space f stretchy left parenthesis x stretchy right parenthesis equals 2 x to the power of 4 plus 3 over x squared blank x not equal to 0

a)
Find an equation for the tangent to the curve y equals f left parenthesis x right parenthesis at the point where x equals 1, giving your answer in the form y equals m x plus c.

5-1-2-ib-sl-ai-aa-we2-soltn-a

b)
Find an equation for the normal to the curve y equals f left parenthesis x right parenthesis at the point where x equals 1, giving your answer in the form a x plus b y plus d equals 0, where a, b and d are integers.
5-1-2-ib-sl-ai-aa-we2-soltn-b

 

Local Minimum & Maximum Points

What are local minimum and maximum points?

  • Local minimum and maximum points are two types of stationary point
    • The gradient function (derivative) at such points equals zero
      i.e. space f apostrophe left parenthesis x right parenthesis equals 0
  • A local minimum point, left parenthesis x comma space f left parenthesis x right parenthesis right parenthesis spacewill be the lowest value ofspace f left parenthesis x right parenthesis in the local vicinity of the value of x
    • The function may reach a lower value further afield
  • Similarly, a local maximum point, left parenthesis x comma space f left parenthesis x right parenthesis right parenthesis spacewill be the greatest value ofspace f left parenthesis x right parenthesis in the local vicinity of the value of x
    • The function may reach a greater value further afield
  • The graphs of many functions tend to infinity for large values of x
    (and/or minus infinity for large negative values of x)
  • The nature of a stationary point refers to whether it is a local minimum or local maximum point

How do I find the coordinates and nature of stationary points?

  • The instructions below describe how to find local minimum and maximum points using a GDC on the graph of the function y equals f left parenthesis x right parenthesis.
 STEP 1
 Plot the graph of y equals f left parenthesis x right parenthesis
  
Sketch the graph as part of the solution

 STEP 2
 Use the options from the graphing screen to “solve for minimum”
 The GDC will display the x and y coordinates of the first minimum point
 Scroll onwards to see there are anymore minimum points
 Note down the coordinates and the type of stationary point

 STEP 3
 Repeat STEP 2 but use “solve for maximum” on your GDC
 
  • In STEP 2 the nature of the stationary point should be easy to tell from the graph
    • a local minimum changes the function from decreasing to increasing
      • the gradient changes from negative to positive
    • a local maximum changes the function from increasing to decreasing
      • the gradient changes from positive to negative

Stationary Points incr decr min max

Worked Example

Find the stationary points ofspace f stretchy left parenthesis x stretchy right parenthesis equals x stretchy left parenthesis x squared minus 27 stretchy right parenthesis, and state their nature.

5-1-2-ib-sl-ai-only-we2-soltn



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 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
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse Functions
                                                                        • 2.5 Transformations of Graphs
                                                                          • 2.5.1 Translations of Graphs
                                                                            • 2.5.2 Reflections of Graphs
                                                                              • 2.5.3 Stretches of Graphs
                                                                                • 2.5.4 Composite Transformations of Graphs
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 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
                                                                                                    • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                      • 3.3.2 Non Right-Angled Trigonometry
                                                                                                        • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                        • 3.4 Further Trigonometry
                                                                                                          • 3.4.1 The Unit Circle
                                                                                                            • 3.4.2 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      Paul Freeman

                                                                                                                                                                                                                                                                                      Author: Paul

                                                                                                                                                                                                                                                                                      Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.


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