Maths Advanced Q&A by dot point

Describe the rate of change of a quantity, distinguishing the average rate of change over an interval (the gradient of the chord joining two points on the curve) from the instantaneous rate of change at a point (the gradient of the tangent there), estimate an instantaneous rate of change, read and interpret rates from a graph, and recognise that the gradient of the tangent at each point defines a new function, the derivative

Graph exponential and logarithmic functions, identify their asymptotes, use the reflection in the line y=x that makes them inverse functions, apply translations, reflections and dilations, and state the domain and range of each

Apply the index laws to expressions with rational indices: use zero, negative and fractional indices, simplify and evaluate index expressions, solve simple index equations, and write numbers in scientific notation

Define logarithms as indices, convert between index form and logarithmic form, apply the logarithm laws (product, quotient, power), use the logarithms of 1 and of the base, change the base, and work with common logarithms

Define a radian as the angle whose arc length equals the radius, convert between degrees and radians using pi rad = 180 deg, know the exact radian values of the common angles, and use the radian formulas for arc length L = r theta, sector area A = (1/2) r^2 theta and the area of a segment as the sector minus the triangle

Define Euler's number e as the base for which y=e^x has gradient 1 at the y-intercept, work with the natural logarithm ln x = log_e x as the inverse of e^x, sketch y=e^x and y=ln x as reflections in y=x, transform y=e^x, and solve e^x=k and ln x=k by converting between the two forms

Use algebraic techniques to expand, factor and simplify expressions and algebraic fractions, and to solve linear and simultaneous equations

Find the natural domain of a function (avoiding division by zero and square roots of negatives), determine the range from a sketch, work with restricted domains, and read the range from a graph using horizontal lines

Define and use function notation, distinguish a function from the more general relation using the vertical line test, and classify relations as one-to-one, many-to-one, one-to-many or many-to-many using the horizontal line test

Sketch the graphs of power functions $y = x^n$ and contrast even and odd powers, sketch a cubic or higher polynomial that is factored into linear factors using a sign table, sketch circles $x^2 + y^2 = r^2$ centred at the origin and recognise shifted circles, and sketch the rectangular hyperbola $y = \frac{1}{x}$ with its asymptotes

Use interval notation and number-line graphs, solve linear and quadratic inequalities, and work with the absolute value definition to solve equations and inequalities of the form |x| < k and |x| > k

Work with the gradient of a line as rise over run and as the tangent of its angle of inclination, write the equation of a line in gradient-intercept, point-gradient and general form, use the parallel and perpendicular gradient conditions, and find the length and midpoint of an interval

Sketch a parabola by finding its intercepts by factoring, complete the square to write a quadratic in vertex form and read off the turning point and axis of symmetry, use the quadratic formula and the discriminant to find and classify the roots, and find the maximum or minimum value of a quadratic

Work with the real number system and surds: simplify surds, add, multiply and expand surdic expressions, rationalise single- and binomial-surd denominators, and apply the index laws to expressions with integer indices

Translate a known graph vertically and horizontally and reflect it in the $x$-axis and the $y$-axis, recognise and test even functions (symmetric about the $y$-axis, $f(-x) = f(x)$) and odd functions (symmetric about the origin, $f(-x) = -f(x)$), sketch absolute-value graphs as transformations of $y = |x|$, and form composite functions $f(g(x))$ and determine their domain

Use the trigonometric ratios sine, cosine and tangent to find unknown sides and angles in right-angled triangles, including the exact ratios of 30, 45 and 60 degrees, and apply them to angles of elevation and depression and to compass and true bearings

Establish and apply the sine rule $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ (including the ambiguous case when finding an angle), the cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$ to find a side and its rearrangement to find an angle, and the area rule $\text{Area} = \dfrac12 ab\sin C$, to solve problems involving non-right-angled triangles

Solve three-dimensional problems involving right-angled triangles, including finding the relevant right-angled triangle inside a solid such as a rectangular prism or pyramid, the angle between a line and a plane, and problems that combine right-triangle results across different planes

Extend the definitions of sine, cosine and tangent to any angle using the unit circle and the four quadrants, use the ASTC rule for the signs of the ratios, find the related acute angle, determine exact values of the trigonometric functions at angles around the circle, and find one ratio given another together with the quadrant, all in degrees

Sketch the graphs of $y = \sin x$, $y = \cos x$ and $y = \tan x$ in degrees over one or more periods, identify their amplitude (where it exists) and period and the key maximum, minimum, zero and intercept points, and solve trigonometric equations of the form $\sin x = k$, $\cos x = k$ and $\tan x = k$ for all solutions in a given domain using the graph together with the related acute angle and the ASTC rule

Prove and apply the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ and its rearrangements, and the ratio identity $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, to prove further trigonometric identities by transforming one side to the other and to simplify trigonometric expressions

Use the first and second derivatives to find stationary points, points of inflection, and to solve optimisation and related rates problems

Calculate the area under a curve, the area between two curves, and the volume of a solid of revolution about the $x$ or $y$ axis

Apply the product, quotient and chain rules, and differentiate exponential, logarithmic and trigonometric functions

Establish and solve differential equations of the form $\frac{dN}{dt} = k N$ and $\frac{dT}{dt} = k(T - T_a)$ and apply them to growth, decay and Newton's law of cooling

Solve problems involving related and general rates of change, including integrating a given rate dQ/dt to recover a quantity with an initial condition, and interpreting rate graphs

Find antiderivatives of standard functions, apply integration by substitution and evaluate definite integrals using the Fundamental Theorem of Calculus

Find derivatives and integrals of $e^x$ and $\ln x$, including composed forms, and apply them to modelling problems

Apply calculus to motion in a straight line, with displacement, velocity and acceleration as derivatives and integrals with respect to time

Use the trapezoidal rule to estimate areas and definite integrals, and determine whether the estimate is an over- or under-estimate

Find derivatives and integrals of $\sin$, $\cos$ and $\tan$ (with linear inside arguments) and apply them to model simple harmonic and periodic motion

Derive and use the future value formula for an annuity to find the accumulated value of a series of equal regular contributions

Use the formulas for the nth term and the sum of n terms of an arithmetic sequence in financial and modelling contexts

Use the formulas for the nth term and the sum of n terms of a geometric sequence, and the limiting sum, in financial contexts

Use recurrence relations and the present value of an annuity to find loan repayments, outstanding balances and total interest paid

Use simple and compound interest formulas to find future values, present values, interest rates and time periods

Sketch graphs of sums, differences, products, quotients, squares and reciprocals of two known functions

Sketch and interpret graphs of exponential and logarithmic functions, including transformations, and use the inverse relationship between them

Form composite functions, determine when a function has an inverse, find and graph the inverse, and use restriction of domain to invert non-one-to-one functions

Apply translations, reflections and dilations to the graph of a function and identify the resulting equation

Construct scatter plots, calculate and interpret Pearson's correlation coefficient, and fit and use the least-squares regression line

Use probability density functions and cumulative distribution functions to find probabilities, medians, modes, means and variances of continuous random variables

Define a discrete random variable by its probability distribution, and calculate the expected value, variance and standard deviation

Apply tree diagrams, conditional probability, independence and complementary events to solve multi-step probability problems

Use the normal distribution, z-scores, the empirical rule and the standard normal table to find probabilities and percentiles

Display and summarise univariate data using frequency and cumulative-frequency tables, histograms, ogives, the mean and standard deviation, the five-number summary, box plots and outliers

Sketch and interpret graphs of $y = a \sin(b x + c) + d$, $y = a \cos(b x + c) + d$ and $y = a \tan(b x + c) + d$, identifying amplitude, period, phase shift and vertical shift

Use radian measure to find arc length, the area of a sector, and the area of a segment of a circle

Solve trigonometric equations over a given interval using exact values, the unit circle, and identities to reduce to a single trig function

Solve problems using the sine rule, cosine rule and area rule, including angles of elevation and depression, bearings and three-dimensional applications

Use Pythagorean, ratio, double angle and complementary identities to simplify expressions and prove equalities