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Oxbridge Assessment Prep513588020447000Week 5 – Differentiation, Integration and SeriesPhysics/Engineering at Oxford: PATEngineering at Cambridge: ENGAANatural Sciences at Cambridge: NSAAAssessment date: 4th November 2020Week 1 – Mechanics Week 2 – Electricity Week 3 – Area/volume of shapesWeek 4 – WavesWeek 5 – Differentiation/Integration/SeriesWeek 6 – General physics Week 7 – General maths (logs, binomial, trig). Differentiation factsCommon FunctionsFunctionDerivativeConstantc0Linex1?axaSquarex22xSquare Root√x(?)x-?Exponentialexex?axln(a) axLogarithmsln(x)1/x?loga(x)1 / (x ln(a))Trigonometry (x is in radians)sin(x)cos(x)?cos(x)?sin(x)?tan(x)sec2(x)Inverse Trigonometrysin-1(x)1/√(1?x2)?cos-1(x)?1/√(1?x2)?tan-1(x)1/(1+x2)???RulesFunctionDerivativeMultiplication by constantcfcf’Power Rulexnnxn?1Sum Rulef + gf’ + g’Difference Rulef - gf’ ? g’Product Rulefgf g’ + f’ gQuotient Rulef/g(f’ g ? g’ f )/g2Reciprocal Rule1/f?f’/f2Chain Rule (using ’ )f(g(x))f’(g(x))g’(x)Chain Rule (using?ddx?)dydx?=?dydududxIntegration factsCommon FunctionsFunctionIntegralConstant∫a dxax + CVariable∫x dxx2/2 + CSquare∫x2?dxx3/3 + CReciprocal∫(1/x) dxln|x| + CExponential∫ex?dxex?+ C?∫ax?dxax/ln(a) + C?∫ln(x) dxx ln(x) ? x + CTrigonometry (x in radians)∫cos(x) dxsin(x) + C?∫sin(x) dx-cos(x) + C?∫sec2(x) dxtan(x) + C???RulesFunctionIntegralMultiplication by constant∫cf(x) dxc∫f(x) dxPower Rule (n≠-1)∫xn?dxxn+1n+1+ CSum Rule∫(f + g) dx∫f dx +?∫g dxDifference Rule∫(f - g) dx∫f dx -?∫g dx228600028638500Integration by parts: ∫u v dx = u∫v dx ?∫u' (∫v dx) dxIntegration by substitution: Geometric Series {a, ar, ar2, ar3, ... } Arithmetic Series {a, a+d, a+2d, a+3d, ... }Sn=na1+an2Q1 PAT 2018Q5 PAT 2017 SpecimenQ5 PAT 2018Q3 PAT 2017Q15 PAT 2017 SpecimenQ17 Section 1 NSAA 2018(no calculator)Q37 Section 1 ENGAA 2018(no calculator)Q39 Section 1 ENGAA 2018(no calculator)Q1 PAT 2017Q3 PAT 2017 SpecimenQ8 PAT 2018Q23 section 1 ENGAA 2019(no calculator)Q5 PAT 2017Q49 ENGAA 2018(no calculator)Q17 PAT 2017 SpecimenQ21 PAT 2017Q33 Section 1 ENGAA 2019 ................
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