# Assessment Report

Level 3 Calculus 2018

### Part A: Commentary

The 2019 papers followed a similar format to recent years’ papers. Overall candidate performance was also of a similar standard, with a wide range of responses and knowledge on display.

There is a strong positive correlation between correct, logical notation/setting out and the final grade for a question. Candidates who presented their work in a clear, concise manner had a considerable advantage when attempting questions which require a chain of reasoning.

Reliable algebra skills remains the overriding pre-requisite for success in this subject. Both in setting up a problem and manipulating the results of any differentiation/integration to obtain the solution, algebraic competency is required. Without it there is little prospect of success.

A number of candidates appeared to attempt to gain an Excellence grade by only responding to questions that they interpreted to be ‘excellence’ questions. This is not to be recommended as errors in the few questions they attempted, with no ‘back-up’ answers, frequently resulted in a disappointing overall grade.

## 91577:  Apply the algebra of complex numbers in solving problems

Candidates who were awarded Achievement commonly:

• manipulated surds successfully
• manipulated complex numbers successfully
• solved a quadratic equation with complex roots
• used the remainder or factor theorems successfully
• solved an equation by using completing the square method.

Candidates whose work was assessed as Not Achieved commonly:

• could not manipulate fractions, complex numbers or surds
• could not use the quadratic formula correctly
• had little understanding of the real and imaginary parts of a complex number
• could not simplify surds
• failed to accurately divide polar form complex numbers
• could not manipulate rectangular form complex numbers

Candidates who were awarded Achievement with Merit commonly:

• were accurate in their algebra when solving equations or manipulating expressions
• understood the meaning of modulus and argument and were able to express statements using these features correctly
• understood how to use De Moivre’s theorem and could apply it correctly
• knew the difference between the factors and solutions of an equation and understood the algebra around this
• understood the meaning of “purely real” or purely imaginary” complex numbers and could form and solve the equations that resulted.

Candidates who were awarded Achievement with Excellence commonly:

• solved problems that involved several stages of reasoning
• had the foresight to rearrange an equation involving complex numbers so that all the constant terms were together before performing any other manipulation
• worked logically and set out their reasoning clearly

The award of merit and excellence requires evidence of relational thinking and extended abstract thinking. This means that answers copied directly of the Graphic calculator will generally not be sufficient.

Candidates need to take care with the set-up of their answer for questions requiring the application of De Moivre’s theorem. Careful thought needs to be given to setting up a general solution with a correct initial angle.

The requirement to show an answer in terms of a specific variable or in in a specific form is not understood or ignored by some candidates. Requirements of these types are common and should be well practised by candidates.

The need for multiplying by a conjugate fraction was required several times in this examination. It should be a skill that is well practised by candidates in its various forms.

Candidates should realise that an explanation at the end of a calculation or a short concluding statement may be required to finish an answer.

An appropriate graph or diagram is often of assistance when answering questions. For example candidates should sketch an argand diagram to ensure they have the correct argument for a given complex number.

## 91578: Apply differentiation methods in solving problems

Candidates who were awarded Achievement commonly:

• used the chain rule correctly
• used the product and/or quotient rules correctly
• differentiated exponential and logarithmic functions correctly
• differentiated trigonometric functions correctly
• evaluated trigonometric functions correctly
• differentiated surd functions correctly, successfully manipulating negative and fractional indices
• found the first derivative of parametric functions, using the chain rule correctly
• demonstrated an understanding of the properties of graphs such as concepts of gradient, limits and differentiability
• displayed an understanding of kinematics and knew to differentiate velocity to get acceleration
• were able to make a start working on more complex problems, demonstrating a basic understanding of differentiation skills.

Candidates whose work was assessed as Not Achieved commonly:

failed to correctly apply the chain rule, product rule or quotient rule in combination with power functions, trigonometric functions, exponential functions and logarithmic functions

demonstrated poor algebraic skills especially in their use of fractions, negative and fractional powers

confused integration and differentiation techniques, often putting a “+C” onto their differentiated answer

made errors simplifying their answers when the question clearly stated that they did not need to simplify their answers

could not evaluate reciprocal trigonometric functions such as 2sec 2x

could not manipulate functions involving surds correctly. For example: They said that the square root of (36 – x^2) was (6 – x)

lacked understanding of the concepts of gradients, differentiability and limits

made careless errors when solving problems.

Candidates who were awarded Achievement with Merit commonly:

• understood what is meant by a “decreasing” function
• formed correct related rates equations to use when solving problems
• worked correctly with trigonometric functions, setting up a correct trigonometric model to solve problems
• displayed a good understanding of how to use the product, quotient and chain rules when differentiating
• successfully set up models for optimisation questions, in terms of 1 variable, before correctly differentiating and using their model to consistently solve problems
• had good algebraic skills and the ability to solve exponential equations using logarithms
• showed a good understanding of the relationship between the gradient of a curve at a point and the gradient of the tangent to the curve at that point.

Candidates who were awarded Achievement with Excellence commonly:

• demonstrated excellent algebraic skills when solving problems, working carefully and systematically
• communicated their thoughts in a clear and logical manner
• set up appropriate models and used these models correctly to solve problems
• found the correct first and second derivatives of exponential functions involving trigonometric functions and did not make careless errors when using them to complete a proof
• set up and used correct expressions to solve related rates of change questions
• were able to “think outside the box” when solving problems
• correctly completed a proof showing that LHS=RHS using logical, clear working
• displayed the ability to find the equation of a tangent to a curve from a point not on the curve and then find its point of intersection with the curve.

Many candidates seemed unfamiliar with reciprocal trig functions. It was surprising how few changed 4/sin x to 4 cosec x before differentiating.

Only the best-performing candidates could evaluate the reciprocal trigonometric function – indeed many wrote comments about how there was no secant function option on their calculator.

It was also surprising how few candidates could solve the inequation of question 1d using algebraic methods. Also many candidates struggled to solve the exponential equation of 3d using algebraic methods.

Candidates who avoid using Leibniz notation are at a disadvantage when attempting problems with multiple variables. They often made mistakes as to which variable they were differentiating with respect to.

Many candidates left answer spaces blank, choosing to only attempt the “excellence” parts of questions. A mistake in one of these parts can then leave them at merit overall with E8 + E8 + N2 = 18 rather than excellence if they had attempted some of the merit parts of questions also such as E8 + E8 + M5 = 21.

## 91579:  Apply integration methods in solving problems

Candidates who were awarded Achievement commonly:

• integrated polynomials, exponential and trigonometric functions correctly
• successfully found the constant of integration given the necessary information
• successfully applied the Trapezium Rule
• recognised that integration treats any area under the x–axis as negative.

Candidates whose work was assessed as Not Achieved commonly:

• could not find the correct value of h when applying the Trapezium Rule
• incorrectly integrated exponential and trigonometric functions
• could not write a term in surd form using fractional exponents
• forgot to include the arbitrary constant or assumed incorrectly it was zero
• failed to integrate and just used the GC to calculate answers for definite integrals
• attempted to use the product or quotient rule when integrating
• failed to understand the difference between finding the integral and finding the area.

Candidates who were awarded Achievement with Merit commonly:

• correctly used algebraic techniques to rewrite a function so that they could integrate the function
• correctly used trigonometric identities to rewrite a function to find a definite integral or area under a curve
• solved differential equations, including calculating the value of the constant of integration
• integrated a rational function correctly either by using long division or integration by substitution.

Candidates who were awarded Achievement with Excellence commonly:

• used correct techniques to demonstrate a proof
• developed and used a strategy to successfully solve a problem involving areas between curves
• successfully interpreted function notation and could apply this to solving a problem involving an unfamiliar function
• recognised and used similar triangles to establish a relationship between variables in a problem.

Many candidates were able to select the correct techniques to solve a problem, however poor algebraic manipulation skills prevented them arriving at the correct solution.

Many candidates showed little or no use of integration notation. A number of candidates used differentiation notation when the working indicated they were attempting to integrate. Correct notation is to a candidate’s advantage as it aids the logical progression through a problem.

There are candidates who seem to have no concept of the signed area property of integration.

### Mathematics and Statistics subject page

Previous years' reports

2018 (PDF, 125KB)

2017 (PDF, 48KB)

2016 (PDF, 255KB)