Differentiation and applications
Chapter 130 of 51 strong
RESTS ONEvaluate and analyse a general piecewise-defined function with non-linear branches, including branch selection and junction behaviourFOUNDATIONSSubstitute known values into a formula and evaluate it correctlyCH 1Solve two-step linear equations by inverse operations in reverse orderCH 1Connect zero remainder, P(a)=0, the linear factor corresponding to a, and the zero a, in both directionsCH 4Divide one polynomial by a lower-degree polynomial, maintaining descending-power columns and zero placeholders, to obtain an exact quotient and a remainder of lower degreeCH 4Multiply polynomials using complete distribution and exponent laws, then collect equal powersCH 4Multiply and divide fractionsCH 2Differentiate a constant multiple, sum or difference term by term while preserving each exact rational coefficient and signCH 12Solve equations with unknowns on both sides by collecting them onto one sideCH 1Check a claimed solution or point by substituting into the original equationCH 1Obtain the derivative of a low-degree polynomial from the limit definition and evaluate the resulting derivative at an exact input when askedCH 12State a polynomial's degree, leading coefficient, constant term, monic status and coefficient set after arranging its terms by descending powerCH 4Remove the greatest common numerical and algebraic factor from every term before considering any further factorisationCH 2Factor a simple quadratic with integer coefficients by finding factor data whose product and sum reproduce the constant and linear coefficientsCH 2Recognise and factor a difference of two squares, a perfect-square trinomial, and a sum or difference of two cubes, with the correct sign patternCH 2Simplify a fractionCH 2Write a line's equation from its y-intercept and slopeCH 6Find the equation of the line through two pointsCH 1Rearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutionsCH 4Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variableCH 4Use the parity and size of a root's multiplicity to determine whether the graph crosses or touches the horizontal axis, distinguish an ordinary crossing from a stationary crossing, and decide whether the sign changesCH 5Determine both end directions of a polynomial from degree parity and the sign of its leading coefficientCH 5Rewrite reciprocal, root and rational terms as a constant times x^n, then differentiate term-by-term with the power rule n a x^(n−1), using linearity for sums and differences and returning the result with positive or fractional indices as requiredFOUNDATIONSClassify a graphed line's slope as positive, negative, zero or undefinedCH 1Write a recurrence relation as a starting value and a ruleFOUNDATIONSEstablish that at least one real root lies in an interval by exact endpoint evaluation with opposite signs, without claiming uniqueness or claiming the converse when the signs agreeCH 5Write a linear model from a description with signed intercept and rateCH 1Multiplication and divisionCH 2Apply the unitary method to a rateFOUNDATIONSCompute the scale factor k of a similar pairFOUNDATIONSInterpret the derivative as the limit of a simplified difference quotient as the increment approaches zero, cancelling the common increment only after it has been factored and never treating the original quotient as defined at zeroCH 12Compute a slope from two coordinate pairs with the slope formulaCH 1
Limits, continuity and differentiability
Power functions and tangent geometry
Curve analysis and stationary points
Newton's method
Optimisation and rates of change
Rectilinear motion