Anti-differentiation and integrals
Chapter 140 of 18 strong
RESTS ONConnect f'(x), dy/dx, a derivative with respect to a named independent variable, and evaluation notation at a stated input, while treating each as notation for a function or value rather than ordinary fraction arithmeticCH 12Simplify a polynomial to an explicit sum of powers where necessary and produce its complete derivative in descending-power form, including zero coefficients where the response table labels themCH 12Multiply and divide fractionsCH 2Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powersCH 4Apply the distributive law repeatedly, preserve signs and collect like powers to write a polynomial in descending-power formCH 2Factor 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 2Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variableCH 4Solve a one-step linear equationFOUNDATIONSRelate an original polynomial graph to its derivative graph through tangent-gradient sign, zeros of the derivative, degree reduction and relative steepness, in either direction where the supplied information is sufficientCH 12Identify stationary inputs from derivative zeros and use an exact local derivative-sign pattern to distinguish an increasing-to-decreasing turn, a decreasing-to-increasing turn and a stationary crossing, without claiming that every derivative zero is a turning pointCH 12Attach the signed rate, its dependent-unit-per-independent-unit form and its increase/decrease meaning to the variables in a stated contextCH 12State and respect a linear model's domain of interpretationCH 1Substitute an exact value into a derivative and compute the exact rational gradient, distinguishing f'(a) from f(a) and from the unsimplified derivative ruleCH 12Rearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutionsCH 4Signed numbersCH 2Interpret 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 6Determine the sign of a factorised polynomial on intervals separated by its ordered real roots, changing sign exactly at odd-multiplicity rootsCH 5Select and apply the area formula matching the named shape and measurementsCH 8
Anti-derivatives and power rules
Gradient functions and motion
Definite integrals and accumulation