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SOUTH AFRICAN QUALIFICATIONS AUTHORITY 
REGISTERED UNIT STANDARD: 

Find the derivatives and integrals of a range of functions including the trigonometric functions and apply these to problems 
SAQA US ID UNIT STANDARD TITLE
7481  Find the derivatives and integrals of a range of functions including the trigonometric functions and apply these to problems 
ORIGINATOR
SGB Math Literacy, Math, Math Sciences L 2 -4 
PRIMARY OR DELEGATED QUALITY ASSURANCE FUNCTIONARY
-  
FIELD SUBFIELD
Field 10 - Physical, Mathematical, Computer and Life Sciences Mathematical Sciences 
ABET BAND UNIT STANDARD TYPE PRE-2009 NQF LEVEL NQF LEVEL CREDITS
Undefined  Regular-Fundamental  Level 4  NQF Level 04 
REGISTRATION STATUS REGISTRATION START DATE REGISTRATION END DATE SAQA DECISION NUMBER
Reregistered  2018-07-01  2023-06-30  SAQA 06120/18 
LAST DATE FOR ENROLMENT LAST DATE FOR ACHIEVEMENT
2024-06-30   2027-06-30  

In all of the tables in this document, both the pre-2009 NQF Level and the NQF Level is shown. In the text (purpose statements, qualification rules, etc), any references to NQF Levels are to the pre-2009 levels unless specifically stated otherwise.  

This unit standard does not replace any other unit standard and is not replaced by any other unit standard. 

PURPOSE OF THE UNIT STANDARD 
This unit standard will be useful to people who aim to achieve recognition at level 4 in Further Education and Training or to meet the Fundamental requirement of a wide range of qualifications registered on the National Qualifications Framework.

People credited with this unit standard are able to:
  • Use radian measure in working with trigonometric functions
  • Express the relationships involved between a function, its derivative and integral in terms of numerical, graphical, verbal and symbolic approaches
  • Sketch the graphs of a range of functions
  • Analyse and represent mathematical and contextual situations using derivatives and integrals
  • Use mathematical models involving derivatives and integrals to deal with problems that arise in real and abstract contexts. 

  • LEARNING ASSUMED TO BE IN PLACE AND RECOGNITION OF PRIOR LEARNING 
    The credit value is based on the assumption that people starting to learn towards this unit standard are competent in Unit Standard Math 4002B. 

    UNIT STANDARD RANGE 
    This unit standard includes the requirement to:
  • Work with trigonometric functions where the angle is in radian measure
  • Use more complex rules of differentiation to differentiate functions
  • Determine the integrals of suitable functions
  • Use differentiation to determine maxima and minima
  • Sketch the graphs of suitable functions
  • Determine areas between curves
  • Determine instantaneous rates of change. 

  • Specific Outcomes and Assessment Criteria: 

    SPECIFIC OUTCOME 1 
    Use radian measure in working with trigonometric functions. 
    OUTCOME RANGE 
    This outcome includes the requirement to:
  • Work with angle in terms of radian measure
  • Define the trigonometric functions in terms of a real variable (angle in radians) and the unit circle
  • Work with radians in trigonometric identities such as the squares (Pythagorean) identities, co-function identities, reciprocal identities; reduction formulae; compound angle identities; double angle and half angle identities; product in terms of sum and difference of angles
  • Define and use the inverse trigonometric functions: arcsinq, arccosq, arctanq.
  • Use the basic properties of the inverse functions such as sin (arcsinq) = q and arccos (cosq) = q
  • Find the general solutions of trigonometric equations. 

  • ASSESSMENT CRITERIA
     

    ASSESSMENT CRITERION 1 
    1. Angles are expressed correctly in terms of radians. 

    ASSESSMENT CRITERION 2 
    2. Acceptable proofs of identities are given. 

    ASSESSMENT CRITERION 3 
    3. Equations are solved correctly and solutions are given in radians. 

    SPECIFIC OUTCOME 2 
    Analyse and represent mathematical and contextual situations using derivatives and curve sketching. 
    OUTCOME RANGE 
    This outcome includes the requirement to:
  • Determine limits from the left and from the right
  • Deal with continuity at point. The theorem (without proof):
    "A function that is differentiable at a point is continuous at that point", and examples to indicate that the converse is not valid
  • Work with composite functions
  • Determine the derivatives of trigonometric functions and their inverses from first principles where necessary
  • Confirm and use the product, quotient and function of a unction (chain) rules for differentiation on suitable forms f polynomial, rational and trigonometric functions
  • Work with higher order derivatives. 

  • ASSESSMENT CRITERIA
     

    ASSESSMENT CRITERION 1 
    1. The definition of continuity at a point is used correctly. 

    ASSESSMENT CRITERION 2 
    2. Sketches are used appropriately to illustrate issues relating to continuity and differentiability. 

    ASSESSMENT CRITERION 3 
    3. The basic derivatives for the trigonometric functions are found by correct application of first principles. 

    ASSESSMENT CRITERION 4 
    4. The rules for differentiation are used efficiently and correctly. 

    ASSESSMENT CRITERION 5 
    5. Representations conform to the reality of the given situations. 
    ASSESSMENT CRITERION RANGE 
    Symbolic and graphical representations.
     

    SPECIFIC OUTCOME 3 
    Analyse and represent mathematical and contextual situations using integrals and curve sketching. 
    OUTCOME RANGE 
    This includes the requirement to:
  • Determine antiderivatives of functions related to the derivatives of trigonometric and inverse trigonometric functions
  • Use the substitution method to determine the
    antiderivatives of suitable functions
  • Work intuitively with the concept of Riemann sums to obtain approximations of areas under a curve
  • Define the definite integral as the limit of a
    Riemann sum and relate this to the area under a curve
  • Demonstrate knowledge of the fundamental theorem of the calculus (without proof)
  • Use the basic properties of indefinite integrals. 

  • ASSESSMENT CRITERIA
     

    ASSESSMENT CRITERION 1 
    1. Antiderivatives and integrals are found by using rules and simplifications correctly. 

    ASSESSMENT CRITERION 2 
    2. Explanations related to Riemann sums and the fundamental theorem of the calculus are correct and supported by diagrams, numerical work and symbols as appropriate. 

    ASSESSMENT CRITERION 3 
    3. Representations conform to the reality of the given situations. 
    ASSESSMENT CRITERION RANGE 
    Symbolic and graphical representations
     

    SPECIFIC OUTCOME 4 
    Use mathematical models involving derivatives and integrals to deal with problems. 
    OUTCOME NOTES 
    Use mathematical models involving derivatives and integrals to deal with problems that arise in real and abstract contexts. 
    OUTCOME RANGE 
    This includes the requirement to:
  • Use trigonometric and inverse trigonometric functions in the solution of practical problems
  • Solve optimisation problems involving maxima and minima
  • Solve problems in which areas need to be determined
  • Solve problems in which the volumes of solids of revolution need to be determined (Restricted to cases where the rotating region revolves around a boundary of the region.)
  • Apply knowledge of differentiation and antidifferentiation to solve problems involving rates
  • Model the problems using calculus methods
  • Verify the results of the calculus modeling by referring to the practical context. 

  • ASSESSMENT CRITERIA
     

    ASSESSMENT CRITERION 1 
    1. Calculus knowledge is applied correctly in modelling the practical problem. 

    ASSESSMENT CRITERION 2 
    2. Relevant mathematics is used correctly to solve the model. 

    ASSESSMENT CRITERION 3 
    3. The results of solving the model are verified in the practical context. 


    UNIT STANDARD ACCREDITATION AND MODERATION OPTIONS 
    Accreditation Option: Providers of learning towards this unit standard will need to meet the accreditation requirements of the GENFETQA.
    Moderation Option:
    The moderation option of the GENFETQA must be met in order to award credit to learners for this unit standard. 

    UNIT STANDARD ESSENTIAL EMBEDDED KNOWLEDGE 
    The following essential embedded knowledge will be assessed through assessment of the specific outcomes in terms of the stipulated assessment criteria. Candidates are unlikely to achieve all the specific outcomes, to the standards described in the assessment criteria, without knowledge of the listed embedded knowledge. This means that the possession or lack of the knowledge can be inferred directly from the quality of the candidate`s performance against the standards.
  • Radian measure and the trigonometric functions of a real variable
  • Differentiation more generally but specifically with respect to the trigonometric functions
  • Integration as the inverse of differentiation
  • The increasing, decreasing or stationary nature of a function at a point. 


  • Critical Cross-field Outcomes (CCFO): 

    UNIT STANDARD CCFO IDENTIFYING 
  • Identify and solve a variety of problems using critical and creative thinking:
    Solving a variety of problems based on differentiation and antidifferentiation. 

  • UNIT STANDARD CCFO COLLECTING 
  • Collect, analyse, organise and critically evaluate information:
    Interpret information in order develop a corresponding mathematical model of the context. 

  • UNIT STANDARD CCFO COMMUNICATING 
  • Communicate effectively:
    Use everyday language and mathematical language and symbols to describe processes and in solving problems. 

  • REREGISTRATION HISTORY 
    As per the SAQA Board decision/s at that time, this unit standard was Reregistered in 2012; 2015. 

    QUALIFICATIONS UTILISING THIS UNIT STANDARD: 
      ID QUALIFICATION TITLE PRE-2009 NQF LEVEL NQF LEVEL STATUS END DATE PRIMARY OR DELEGATED QA FUNCTIONARY
    Fundamental  66370   Further Education and Training Certificate: Roof Truss Technology  Level 4  NQF Level 04  Passed the End Date -
    Status was "Reregistered" 
    2015-06-30  FPMSETA 
    Fundamental  48399   Further Education and Training Certificate: Sugar Processing  Level 4  NQF Level 04  Reregistered  2023-06-30  AgriSETA 
    Fundamental  14854   National Certificate: Agric Sales and Services  Level 4  NQF Level 04  Reregistered  2023-06-30  AgriSETA 
    Fundamental  20893   National Certificate: Human Resources Management and Practices Support  Level 4  NQF Level 04  Passed the End Date -
    Status was "Registered" 
    2005-02-13  Was SABPP until Last Date for Achievement 
    Fundamental  21792   National Diploma: Contact Centre Management  Level 5  NQF Level 05  Passed the End Date -
    Status was "Reregistered" 
    2020-07-30  SERVICES 


    PROVIDERS CURRENTLY ACCREDITED TO OFFER THIS UNIT STANDARD: 
    This information shows the current accreditations (i.e. those not past their accreditation end dates), and is the most complete record available to SAQA as of today. Some Primary or Delegated Quality Assurance Functionaries have a lag in their recording systems for provider accreditation, in turn leading to a lag in notifying SAQA of all the providers that they have accredited to offer qualifications and unit standards, as well as any extensions to accreditation end dates. The relevant Primary or Delegated Quality Assurance Functionary should be notified if a record appears to be missing from here.
     
    1. Balemi Consulting Pty Ltd 
    2. Cedara College of Agriculture 
    3. ELSENBURG AGRICULTURAL COLLEGE 
    4. Mitek Industries SA ( Pty) Ltd 
    5. NWK Beperk 
    6. RCL Foods-Sugar & Milling (MP) 
    7. Sekhukhune FET College - Central Office 
    8. Senwes Beperk 
    9. Suidwes Beleggings Eiendoms Beperk 
    10. VKB LANDBOU (PTY) LTD 



    All qualifications and part qualifications registered on the National Qualifications Framework are public property. Thus the only payment that can be made for them is for service and reproduction. It is illegal to sell this material for profit. If the material is reproduced or quoted, the South African Qualifications Authority (SAQA) should be acknowledged as the source.