Rates and derivatives

Chapter 120 of 18 strong

RESTS ONCompute a slope from two coordinate pairs with the slope formulaCH 1 Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variableCH 4Apply the unitary method to a rateFOUNDATIONSClassify a graphed line's slope as positive, negative, zero or undefinedCH 1 Interpret f, f(x), image, preimage, rule, domain and co-domain; evaluate a function at a stated exact input and solve a simple equation of the form f(x)=cCH 6 RoundingCH 3 Apply the distributive law repeatedly, preserve signs and collect like powers to write a polynomial in descending-power formCH 2 Multiply and divide fractionsCH 2 Substitute known values into a formula and evaluate it correctlyCH 1 Remove the greatest common numerical and algebraic factor from every term before considering any further factorisationCH 2 Form the coefficient and exponent data of a binomial power, including alternating signs for subtraction, and write the terms in descending powers of the first algebraic factorCH 2 State a polynomial's degree, leading coefficient, constant term, monic status and coefficient set after arranging its terms by descending powerCH 4 Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powersCH 4 Factor numerators and denominators, cancel common factors rather than terms, retain the original excluded values, and present the remaining numerator and denominator in a consistent formCH 2Rewrite 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 requiredFOUNDATIONSRearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutionsCH 4 Write a line's equation from its y-intercept and slopeCH 6 Find the equation of the line through two pointsCH 1 Combine ordered roots, multiplicities, vertical intercept, sign intervals and end behaviour into a consistent qualitative graph of a factorised polynomial of degree at least fourCH 5

Average and instantaneous rates

Derivative from first principles

Polynomial differentiation

Calculate the average rate of change of a function over a non-degenerate interval as the exact change in output divided by the exact change in input, from a rule, table, two points or a factual context, keeping the subtraction order matched between the output change and the input change

Not started yet

Practise this

RESTS ON

Compute a slope from two coordinate pairs with the slope formulaNot startedEvaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variableNot started

UNLOCKS

Attach the signed rate, its dependent-unit-per-independent-unit form and its increase/decrease meaning to the variables in a stated contextDistinguish an instantaneous rate at one input from an average rate over an interval and connect it to the gradient of the tangent at the pointForm and evaluate the exact gradient of the secant through two points of a function, including a neighbouring point whose input differs by an exact incrementCalculate and distinguish average velocity, average speed and average acceleration from exact endpoint and journey data CH 13