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NCEA Level 3 Calculus formula sheet (formulae and tables booklet)

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Formula sheet

The card below opens today's Level 3 Calculus paper with the sheet in the drawer. The contents follow in our own layout, grouped like the booklet's own headings. This is the 2025 booklet, the latest NZQA has published. NZQA's 2026 specification says a Formulae and Tables Booklet will be provided again, so expect the same layout and check it in the exam.

Formula sheet

NCEA Level 3 Calculus

Open formula sheet

Algebra

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}Quadratic formula, for ax² + bx + c = 0
y=\log_bx\Leftrightarrow x=b^yDefinition of a logarithm
\log_b(xy)=\log_bx+\log_byLog of a product
\log_b\left(\frac{x}{y}\right)=\log_bx-\log_byLog of a quotient
\log_b\left(x^n\right)=n\log_bxLog of a power
\log_bx=\frac{\log_ax}{\log_ab}Change of base

Complex numbers

z=x+iy=r\ \text{cis}\ \theta=r(\cos\theta+i\sin\theta)Forms of z
\overline{z}=x-iy=r\ \text{cis}(-\theta)=r(\cos\theta-i\sin\theta)Forms of the conjugate
r=|z|=\sqrt{z\overline{z}}=\sqrt{x^2+y^2}Modulus
\theta=\arg z,\ \cos\theta=\frac{x}{r},\ \sin\theta=\frac{y}{r}Argument
(r\ \text{cis}\ \theta)^n=r^n\ \text{cis}(n\theta)De Moivre's theorem, for any integer n

Coordinate geometry

y-y_1=m(x-x_1)Straight line through (x₁, y₁)
(x-a)^2+(y-b)^2=r^2Circle, centre (a, b) and radius r
y^2=4axParabola: focus (a, 0), directrix x = −a
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1Ellipse
e=\frac{c}{a}Ellipse: foci (c, 0) and (−c, 0), with b² = a² − c²; eccentricity
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1Hyperbola
y=\pm\frac{b}{a}x,\ e=\frac{c}{a}Hyperbola: asymptotes; foci (c, 0) and (−c, 0), with b² = c² − a²; eccentricity

Differentiation and integration

Both booklets print tables for the standard derivatives and integrals

\frac{d}{dx}\ln x=\frac{1}{x}Derivative of ln x
\frac{d}{dx}e^{ax}=ae^{ax}Derivative of e^(ax)
\frac{d}{dx}\sin x=\cos xDerivative of sin x
\frac{d}{dx}\cos x=-\sin xDerivative of cos x
\frac{d}{dx}\tan x=\sec^2xDerivative of tan x
\frac{d}{dx}\sec x=\sec x\tan xDerivative of sec x
\frac{d}{dx}\text{cosec}\ x=-\text{cosec}\ x\cot xDerivative of cosec x
\frac{d}{dx}\cot x=-\text{cosec}^2xDerivative of cot x
\int x^ndx=\frac{x^{n+1}}{n+1}+c\ \ (n\neq-1)Integral of a power
\int\frac{1}{x}dx=\ln|x|+cIntegral of 1/x
\int\frac{f'(x)}{f(x)}dx=\ln|f(x)|+cIntegral of f′(x) over f(x)
\frac{dy}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}\ \ (y=uv)Product rule
\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\ \ \left(y=\frac{u}{v}\right)Quotient rule
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\ \ (y=f(u),\ u=g(x))Chain rule
\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}Parametric first derivative
\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot\frac{dt}{dx}Parametric second derivative

Numerical methods

In both rules, h = (b − a) ÷ n and y_r = f(x_r)

\int_a^bf(x)dx\approx\frac{1}{2}h\left[y_0+y_n+2(y_1+y_2+\ldots+y_{n-1})\right]Trapezium rule
\int_a^bf(x)dx\approx\frac{1}{3}h\left[y_0+y_n+4(y_1+y_3+\ldots+y_{n-1})+2(y_2+y_4+\ldots+y_{n-2})\right]Simpson's rule, with n even

Trigonometry

\text{cosec}\ \theta=\frac{1}{\sin\theta}Cosecant
\sec\theta=\frac{1}{\cos\theta}Secant
\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}Cotangent
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}Sine rule
c^2=a^2+b^2-2ab\cos CCosine rule
\cos^2\theta+\sin^2\theta=1Pythagorean identity
\tan^2\theta+1=\sec^2\thetaTangent and secant identity
\cot^2\theta+1=\text{cosec}^2\thetaCotangent and cosecant identity
\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^n\alphaGeneral solution: sine
\cos\theta=\cos\alpha\Rightarrow\theta=2n\pi\pm\alphaGeneral solution: cosine
\tan\theta=\tan\alpha\Rightarrow\theta=n\pi+\alphaGeneral solution: tangent
\sin(A\pm B)=\sin A\cos B\pm\cos A\sin BCompound angle: sine
\cos(A\pm B)=\cos A\cos B\mp\sin A\sin BCompound angle: cosine
\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}Compound angle: tangent
\sin2A=2\sin A\cos ADouble angle: sine
\tan2A=\frac{2\tan A}{1-\tan^2A}Double angle: tangent
\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2ADouble angle: cosine
2\sin A\cos B=\sin(A+B)+\sin(A-B)Product to sum: sin cos
2\cos A\sin B=\sin(A+B)-\sin(A-B)Product to sum: cos sin
2\cos A\cos B=\cos(A+B)+\cos(A-B)Product to sum: cos cos
2\sin A\sin B=\cos(A-B)-\cos(A+B)Product to sum: sin sin
\sin C+\sin D=2\sin\frac{C+D}{2}\cos\frac{C-D}{2}Sum to product: sin plus sin
\sin C-\sin D=2\cos\frac{C+D}{2}\sin\frac{C-D}{2}Sum to product: sin minus sin
\cos C+\cos D=2\cos\frac{C+D}{2}\cos\frac{C-D}{2}Sum to product: cos plus cos
\cos C-\cos D=-2\sin\frac{C+D}{2}\sin\frac{C-D}{2}Sum to product: cos minus cos

Measurement

A=\frac{1}{2}ab\sin CArea of a triangle
A=\frac{1}{2}(a+b)hArea of a trapezium
A=\frac{1}{2}r^2\thetaArea of a sector
s=r\thetaArc length
V=\pi r^2hVolume of a cylinder
A=2\pi rhCurved surface area of a cylinder
V=\frac{1}{3}\pi r^2hVolume of a cone
A=\pi rlCurved surface area of a cone, slant height l
V=\frac{4}{3}\pi r^3Volume of a sphere
A=4\pi r^2Surface area of a sphere

Exact value triangles

The trigonometry page also prints two labelled triangles for exact values: a right triangle with angles π/6 and π/3 and sides 1, √3 and 2, and a right isosceles triangle with angles π/4 and sides 1, 1 and √2.

What is not on the sheet

  • The derivative of xⁿ. The differentiation table starts at ln x and has no power row. You must know that the derivative of x² is 2x without looking.
  • Most integrals. The integration table has three rows: powers of x, 1/x and f′(x)/f(x). Integrals of e^(ax), sin x, cos x and the rest are not on it, and nor are integration by substitution or by parts. Learn them as the reverse of the derivative table.
  • Derivative of aˣ and of inverse functions. Not in the table.
  • The binomial theorem and differentiation from first principles. These are printed in the Scholarship Calculus booklet, not this one.
  • Which formula to use. The sheet has the product, quotient and chain rules, but you decide which one a question needs.
  • Reasoning. Merit rewards linking steps and relating answers to the context, and Excellence rewards devising a strategy and justifying results. No formula does that for you.

NZQA's 2026 specification also says a graphing calculator is recommended, that you must show any derivatives you use, and that candidates without a graphing calculator will be disadvantaged. The booklet does nothing for that part.

How to use the sheet in the exam

  1. Name the rule the question is built on: product, quotient, chain, parametric, or one of the numerical methods.
  2. Find the line on the sheet and copy its pattern, writing the pieces u, v, du/dx and dv/dx out before you combine them.
  3. Substitute carefully and keep exact forms, such as e³, until the last line.
  4. Check units, radians for trigonometry and the sign of the answer.

Differentiation: product rule with an exponential

Differentiate y = x² e³ˣ and find the gradient at x = 1. Let u = x² and v = e³ˣ. The table gives the derivative of e^(ax) as ae^(ax), so dv/dx = 3e³ˣ, and du/dx = 2x. The product rule line then gives:

\frac{dy}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}=e^{3x}\times2x+x^2\times3e^{3x}=xe^{3x}(2+3x)

At x = 1 the gradient is 1 × e³ × 5 = 5e³ = 100.4 to 4 significant figures. Check it in the Derivative Calculator.

Numerical methods: trapezium rule and Simpson's rule

Estimate the integral of f(x) = x² + 1 from 0 to 2, using 4 strips. Here h = (2 − 0) ÷ 4 = 0.5, and the y values at x = 0, 0.5, 1, 1.5 and 2 are 1, 1.25, 2, 3.25 and 5. The trapezium line gives:

\frac{1}{2}h\left[y_0+y_n+2(y_1+y_2+y_3)\right]=\frac{1}{2}\times0.5\times\left[1+5+2(1.25+2+3.25)\right]=0.25\times19=4.75

Simpson's rule needs n even, and 4 is. The odd-numbered values are y₁ and y₃, and the even-numbered one is y₂:

\frac{1}{3}h\left[y_0+y_n+4(y_1+y_3)+2y_2\right]=\frac{1}{3}\times0.5\times\left[1+5+4(4.5)+2(2)\right]=\frac{28}{6}=4.667

The exact value is 8/3 + 2 = 4.667, because Simpson's rule is exact for quadratics, so the second estimate is a good check on the first. Practise recognising the rule from the question in our Calculus course, then press Mark to check your answer there:

Find f'(x), the stationary points, and classify each as a max or min.

Press Mark to have your answer marked in the Calculus course.

The Level 3 sheet and the others

The Level 2 Maths sheet is a single page with the quadratic formula, logarithms and the derivative of xⁿ, which the Level 3 table leaves out. The Scholarship Calculus booklet is this booklet plus the binomial theorem, first principles and a table of binomial coefficients. For how the three Level 3 Calculus standards work, see the NCEA Level 3 Calculus guide, and for one standard in detail see NCEA 91578.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the booklet open beside you, mark it, and the worked answers show which line each question wanted. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the sheet with today's paper

Uni-Sine publishes a free daily paper on the NCEA Level 3 Calculus track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. Open the sheet from the paper itself and practise finding the rule fast. For the standards the real exam assesses, use the NCEA Level 3 Calculus guide.

Common mistakes with the Level 3 Calculus sheet

  • Looking for the power rule in the table. It is not there. Know it.
  • Mixing up the quotient rule's order. The numerator is v du/dx − u dv/dx, with v first.
  • Wrong calculator mode. NZQA's 2025 assessment report says many candidates made radian errors when substituting into trigonometric functions.
  • Forgetting the + c on an indefinite integral.
  • Quoting a graphing calculator's answer. NZQA says you must show the derivative you used.

Common questions

Do you get a formula sheet in NCEA Level 3 Calculus?
Yes. NZQA provides a Formulae and Tables Booklet, L3-CALCF, with the exam for 91577, 91578 and 91579. You may keep it after the exam.
When is the 2026 Level 3 Calculus exam?
Monday 16 November 2026, in the morning session, which starts at 9.30 am. Sessions last 3 hours and you should arrive at least 20 minutes early.
Is the derivative of x^n on the Level 3 Calculus formula sheet?
No. The differentiation table lists ln x, e^(ax) and the trigonometric functions, but not powers of x.
What integrals are on the Level 3 Calculus formula sheet?
Three: x^n for n not equal to −1, 1/x, and f′(x)/f(x). Everything else you must know.
Is Simpson's rule on the Level 3 Calculus formula sheet?
Yes, with the trapezium rule, under Numerical Methods. Simpson's rule needs an even number of strips.

Revision cards

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