Correcting and Classifying AS91262 Mini-Set Errors

Kia ora. You have now completed a timed AS91262 mini-set and recorded where you felt uncertain. This lesson turns that attempt into useful revision evidence.

The aim is not simply to copy correct answers onto your paper. You will identify the first point where your reasoning differed from a valid solution, check what evidence the marking schedule requires, and classify the issue precisely. That is how a vague thought such as “I need more calculus practice” becomes a specific next step such as “I can find stationary times, but I do not yet justify which one is the maximum.”

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Correcting like an assessor: preserve the original evidence

Keep your timed attempt unchanged. Use a different-coloured pen for all corrections. If you erase and rewrite everything, you lose the evidence that tells you what to revise.

For each part, use this sequence:

1. Read the question again. Underline what it asks for: a gradient, coordinate, equation, interval, maximum, duration, or contextual interpretation.
2. Compare your working with the marking schedule. Look for required mathematical evidence, not just the final number.
3. Mark each line:
   • ✓ correct and sufficiently shown;
   • △ mathematically promising, but incomplete or unsupported;
   • × first incorrect step;
   • ! different valid method that still reaches and justifies the answer.
4. Write one correction beginning at the first divergence. Do not copy a whole worked answer if everything after one error was just a consequence of that error.
5. Classify the error by both its topic and its reasoning gap.

A marking schedule does not mean every individual line receives a separate grade. It identifies the evidence NZQA can credit across the question. A correct final answer with no visible method may not show enough evidence; conversely, a correct derivative can still earn useful credit even if later arithmetic fails.

For today’s 2023 mini-set, use the official 2023 assessment schedule if you have it open. The worked-answer video below is a helpful check of the required method and conclusions, but the official schedule remains the final authority.

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Question 1: gradients, anti-derivatives, and increasing intervals

Watch these parts of NCEA Level 2 Calculus 2023 NZQA Exam – Worked Answers by infinityplusone only after you have marked your own first attempt. Pause after each part and annotate your paper rather than rewriting the solution from scratch.

1(a): A point with a stated gradient

The question asks for the point where the gradient is $4$. The required chain is:

$$f^{\prime }(x)=8x-12$$ $$8x-12=4$$ $$x=2$$

That gives the x-coordinate, but the question asks for a point. Return to the original function:

$$f(2)=4(2{)}^2-12(2)=-8$$

So the point is:

$$(2,-8)$$

A common reasoning gap is writing $f(2)=4$. The value $4$ is the gradient, so it belongs in $f^{\prime }(x)$, not in $f(x)$.

If your work shows...	Classify the issue as...	Write this correction note
You substituted $4$ into $f(x)$	Topic: derivative and gradient. Gap: confusing a function with its derivative.	“The stated gradient is a value of $f^{\prime }(x)$, so I should solve $f^{\prime }(x)=4$.”
You found $x=2$ but stopped	Topic: coordinates. Gap: incomplete response evidence.	“A point needs both coordinates; substitute $x=2$ into the original function.”
You used $f^{\prime }(2)$ to find the y-value	Topic: function notation. Gap: wrong expression selected.	“Use $f(2)$, not $f^{\prime }(2)$, for the y-coordinate.”

1(b): Recovering a function from its derivative

The derivative was given as:

$$f^{\prime }(x)=4x^3-6x^2-4x$$

Anti-differentiate every term and include the constant:

$$f(x)=x^4-2x^3-2x^2+c$$

The curve passes through $(2,-5)$, so $f(2)=-5$:

$$-5=2^4-2(2^3)-2(2^2)+c$$ $$-5=-8+c$$ $$c=3$$

Therefore:

$$f(x)=x^4-2x^3-2x^2+3$$

The key schedule-style distinction here is important:

• A correct anti-derivative including $+c$ shows the main calculus procedure.
• Using the point correctly to determine $c$ completes the unique function and shows the longer reasoning chain.

If your final function was wrong, find out whether the first issue was calculus, substitution, or algebra. For example, getting $c=-3$ is not an anti-differentiation error if your integrated expression was correct; it is an algebra or rearrangement error when solving for $c$.

1(c): Where the function is increasing

For this part, the derivative factorises to:

$$f^{\prime }(x)=x(x-6)(x+4)$$

The stationary values occur when:

$$f^{\prime }(x)=0$$ $$x=-4,x=0,x=6$$

These are boundaries, not the final answer. “Increasing” means:

$$f^{\prime }(x)>0$$

Testing the sign of the derivative in each interval gives:

Interval	Sign of $f^{\prime }(x)$	Function behaviour
$x<-4$	Negative	Decreasing
$-4<x<0$	Positive	Increasing
$0<x<6$	Negative	Decreasing
$x>6$	Positive	Increasing

Therefore, the function is increasing when:

$$-4<x<0$$

and when:

$$x>6$$

Do not write only $x=-4,0,6$. Those are where the gradient is zero; they do not answer where the function is increasing.

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Question 3(b): rates, maximum values, and contextual interpretation

Now correct the connected followers question. Notice that parts (i), (ii), and (iii) use the same derivative. Strong exam responses reuse earlier work accurately rather than starting again.

The follower model has derivative:

$$F^{\prime }(t)=15t^2-1680t+42180$$

3(b)(i): Rate on day 5

Evaluate the derivative at $t=5$:

$$F^{\prime }(5)=15(5{)}^2-1680(5)+42180$$ $$F^{\prime }(5)=34155$$

The interpretation matters:

On day $5$, the number of followers is increasing at $34155$ followers per day.

A positive rate means the account is gaining followers. A final answer of just $34155$ is incomplete because it does not state what the number measures.

3(b)(ii): Highest number of followers in the first 90 days

Stationary times occur when:

$$F^{\prime }(t)=0$$ $$15t^2-1680t+42180=0$$

This gives:

$$t=38,t=74$$

To identify the highest follower count, evaluate the original function at the relevant candidate times. Because the question specifies the first $90$ days, it is also mathematically secure to check the interval endpoints.

Time $t$	$F(t)$
$0$	$0$
$38$	$664240$
$74$	$547600$
$90$	$637200$

The maximum is:

$$664240$$

at day $38$.

The major reasoning point is this: solving $F^{\prime }(t)=0$ finds candidates for extrema. It does not, by itself, prove which candidate gives the highest value.

3(b)(iii): How long followers are decreasing

Followers decrease where the rate of change is negative:

$$F^{\prime }(t)<0$$

Since the derivative is negative between the roots $38$ and $74$, the account loses followers for:

$$38<t<74$$

The length of that period is:

$$74-38=36$$

So the contextual conclusion is:

The account loses followers for $36$ days, between day $38$ and day $74$.

At $t=38$ and $t=74$, the derivative is zero, so these are boundary times rather than times when the follower count is actually decreasing.

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Classify each error: topic first, then reasoning gap

An error label should tell you what to practise next. “Careless mistake” is usually too vague to act on. Use the following two-part classification for every △ or × on your paper.

Topic tag	What it covers	Typical mini-set example
DER	Differentiation	Missing the coefficient when differentiating a power
INT	Anti-differentiation	Omitting $+c$, or dividing by the old rather than new power
EVAL	Substitution and numerical evaluation	Finding $x=2$ correctly but calculating $f(2)$ incorrectly
STAT	Stationary points and extrema	Finding $F^{\prime }(t)=0$ but not deciding which value is the maximum
SIGN	Derivative signs and intervals	Listing roots instead of giving where $f^{\prime }(x)>0$ or $F^{\prime }(t)<0$
RATE	Rates and units	Giving a numerical derivative without followers-per-day units
CONTEXT	Conditions from the situation	Ignoring the first-$90$-days restriction

Then add one reasoning-gap label:

Reasoning gap	Meaning	Example
Method selection	You did not know whether to differentiate, anti-differentiate, or use derivative signs.	Using $F(t)$ rather than $F^{\prime }(t)$ for a rate.
Procedure	You chose the right method but applied a rule incorrectly.	Incorrectly differentiating $15t^3$.
Algebra or calculation	The calculus setup was correct but arithmetic, factorising, or rearranging failed.	Solving the stationary-time quadratic incorrectly.
Logical justification	You had values but did not establish the requested conclusion.	Naming $t=38$ as a maximum without comparison or derivative reasoning.
Communication or context	Your mathematics was sound but the answer lacked units, an interval, or an interpretation.	Writing $34155$ rather than “increasing at $34155$ followers per day.”
Missing evidence	You may have known the method but did not show enough working for credit.	Writing only $c=3$, without the integrated expression and substitution.

Use this record on your uncertainty log:

Part	First incorrect or missing evidence	Topic	Reasoning gap	Correct replacement	One next action
1(a)
1(b)
1(c)
3(b)(i)
3(b)(ii)
3(b)(iii)

For example:

Part	First incorrect or missing evidence	Topic	Reasoning gap	Correct replacement	One next action
3(b)(ii)	I found $t=38$ and $t=74$, then chose $38$ without evaluating $F(t)$.	STAT	Logical justification	Evaluate the original function at the candidates and compare values.	Practise one maximum question with a candidate-value table.

This is better than classifying the error as “stationary points” alone, because it identifies that the missing skill is the decision after finding them.

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Use examiner feedback to make the labels more accurate

The NZQA assessment report helps distinguish common content mistakes from missing reasoning. Read it after you have created your own error record, and refine any labels that are still too broad.

A useful final check is to look for patterns, not just individual wrong answers:

• Two or more DER errors mean your immediate priority is power-rule accuracy.
• Two or more INT errors, especially missing $+c$, mean practise reconstructing functions from gradients and conditions.
• Several STAT or SIGN errors mean practise the meaning of $f^{\prime }(x)=0$, $f^{\prime }(x)>0$, and $f^{\prime }(x)<0$.
• Several CONTEXT or missing-evidence errors mean your mathematics may be stronger than your written exam responses show. Build a final-answer routine: value, units or interval, then a sentence of interpretation.

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Finish with a targeted correction summary

At the bottom of your mini-set, write three short statements:

1. Strongest method today: one method you used correctly with sufficient working.
2. Highest-priority gap: the topic-plus-reasoning label that occurred most often.
3. Exam habit to use next time: one visible action, such as “For every maximum, I will evaluate the original function at my stationary times before choosing.”

You have now done more than mark answers right or wrong. You have separated derivative skills, anti-derivative skills, sign reasoning, extrema decisions, and contextual communication. That separation makes the next revision session efficient.

In the next lesson, you will use the Casio fx-9860 to check graphs, roots, and numerical values while still showing the algebraic and calculus working required for NCEA credit.