Antiderivatives
Chapter 70 of 31 strong
RESTS ONSimplify 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 5Interpret 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 zeroFOUNDATIONSAdd or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powersCH 1Multiplication and divisionCH 9Remove the greatest common numerical and algebraic factor from every term before considering any further factorisationCH 6Factor a simple quadratic with integer coefficients by finding factor data whose product and sum reproduce the constant and linear coefficientsCH 3Recognise and factor a difference of two squares, a perfect-square trinomial, and a sum or difference of two cubes, with the correct sign patternFOUNDATIONSExtract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical formCH 2Interpret (f∘g)(x) as f(g(x)), identify the inner and outer function and the order of application, and distinguish composition from multiplication and from the reversed compositionCH 3Extend the power rule to a rational exponent on the function's domain, including negative and fractional powers, and rewrite simple root and reciprocal forms as powers before differentiatingCH 6Reduce an angle to its first-quadrant reference angle measured from the horizontal axisCH 1Determine which of sine, cosine and tangent are positive, negative, zero or undefined in each quadrant or on a boundaryCH 1Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variableCH 1Relate 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 6Identify 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 6Attach the signed rate, its dependent-unit-per-independent-unit form and its increase/decrease meaning to the variables in a stated contextCH 6State and respect a linear model's domain of interpretationCH 2Substitute an exact value into a derivative and compute the exact rational gradient, distinguishing f'(a) from f(a) and from the unsimplified derivative ruleCH 5Substitute known values into a formula and evaluate it correctlyCH 4Solve two-step linear equations by inverse operations in reverse orderCH 1Unitary methodFOUNDATIONSConvert between unitsFOUNDATIONS
Antiderivatives and arbitrary constants
Power rule and integrand preparation
Integrating powers of linear functions
Exponential and circular antiderivatives
Antiderivative families and applications