IB›IB Maths: Analysis and Approaches HL›Mind maps5.1 Limits and the derivativeIB Maths: Analysis and Approaches HL: Mind mapStudy pack PDFAlso for this subtopic:Revision notesFlashcardsSubtopic testCover factsWhat is a limit?Value an expression approaches as input nears a pointlimh→0(4+h)=4\lim_{h\to0}(4+h)=4limh→0(4+h)=4Can exist even where the expression is undefined4h+h2h=4+h\frac{4h+h^2}{h}=4+hh4h+h2=4+h for every h≠0h\ne0h=0EstimatingCheck values from both sidesLook for the number the outputs settle towardsValues closest to the point are most reliableQuote to about 3 s.f.Curve gradientChord 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 tangentx2−2xx^2-2xx2−2x at x=3x=3x=3: 4+h→4+h\to4+h→ 4x3x^3x3 at x=2x=2x=2: 12+6h+h2→12+6h+h^2\to12+6h+h2→ 12Limitsand the derivativeh→0h\to0h→0f′(x)f'(x)f′(x)The derivativeAlso called the gradient functionWritten dydx\frac{\mathrm{d}y}{\mathrm{d}x}dxdy or f′(x)f'(x)f′(x)Units: top quantity per bottom quantityNegative derivative means decreasingAverage vs instantAverage rate: gradient of a chordInstantaneous rate: gradient of the tangents=4.9t2s=4.9t^2s=4.9t2 at t=2t=2t=2: 19.6 m/sExam tipsDivide by hhh: change in yyy is not the gradientSay the rate in words with unitsUse the rate at the instant, not an average