IB›IB Maths: Analysis and Approaches SL›Mind maps5.1 Limits and the derivativeIB Maths: Analysis and Approaches SL: Mind mapStudy pack PDFAlso for this subtopic:Revision notesFlashcardsSubtopic testCover factsLimitsValue an expression approacheslimh→0(4+h)=4\lim_{h\to0}(4+h)=4limh→0(4+h)=4Can exist even if undefined at the pointSL needs only an informal ideaEstimatingCheck values from both sidesOutputs settling on one number: that is the limitValues closest to the point are most reliableQuote to about 3 s.f.Chord to tangentChord gradient: f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)As h→0h\to0h→0 the chord becomes the tangentCurve's gradient is the limit of chord gradientsx3x^{3}x3 at x=2x=2x=2: 12+6h+h2→12+6h+h^{2}\to12+6h+h2→ 12Derivativelimits and gradientf′(x)f'(x)f′(x)h→0h\to0h→0The derivativeAlso called the gradient functionWritten dydx\frac{\mathrm{d}y}{\mathrm{d}x}dxdy or f′(x)f'(x)f′(x)dsdt\frac{\mathrm{d}s}{\mathrm{d}t}dtds is velocityUnits: top per bottom quantityNegative derivative means decreasingRates of changeAverage: gradient of a chordInstantaneous: gradient of the tangentInstantaneous is the limit of shrinking intervalss=4.9t2s=4.9t^{2}s=4.9t2 at t=2t=2t=2: 19.6 m/sExam tipsUndefined at the point does not mean no limitDivide by hhh for a gradientGive the rate at an instant, not an averageSay it in words with units