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Differentiation
Functions
Modulus Functions
Differentiating trigonometry
Logarithms
The exponential functiony=exInverse is lnx which is log base e.y=ex increases at an ever increasing rate, described as exponential growthy=e-x exponential decay, approaches the x axis.
Making natural logarithm to solveN = 2000e 0.1tTaking natural logs;5=e 0.1tln5=ln(e 0.1t) = 0.1tt=10ln5 = 16.09
ln(ex)=xelnx = x
Making something subject using natural logs;ln(p) -ln(1-p) = tln(p/1-p) = te ln(p/1-p) = et(p/1-p) = etp + pet = etp(1+et) = etp - et/1+et
X numbers put in are the domain of the mappingthe y numbers out are the co-domain of the mapping
Mapping typesOne to oneMany to oneOne to manyMany to many
Inverse functions only work if the function is one to one mapping. Therefore domains have to be placed to make many to one mappings, one to one fora set of values.
Composite functions are functions within functions.Work from the inside out.gf(x) so f into g
Inverse functionsGive function f(x)interchange x and y valuesRearrange to make y the subject
Inverse trigonometry functionscosec0 = 1/sin0sec0 = 1/cos0cot0 = 1/tan0
Even functions have the y axis as a line of symmetryf(x) = f(-x)Odd functions have rotational symmetry of 180 about the originf(-x) = -f(x)
Periodic functions have a repeating pattern, repeats at regular intervalsf(x +k) = f(x)
Modulus of a function always takes the positive numerical value of x
Log ruleslogaN =x => N = a^xlogbM +logbN = logbMNlogbM - logbN = logbM/NnlogbM = logbM^nlogbB = 1logb1 = 0logbB^x = xlogbN = logaN/logaB
Functions are either many to one or one to one.
Z are all integer valuesR are all real numbersQ are all rational numbers
The Chain Ruledifferentiate dy/dxdy/dx= dy/du x du/dxy=u^2u= (3x +4)
Product ruledy/dx= uv' + vu'differentiate brackets x inside brackets differentiate.eg.(3x +1)^2= 6(3x +1)The factors found identify where the turning points are.
Related rates of changedA/dx is change in area ratedA/dx = dA/dr x dr/dt
Inverse functions are reflected in the mirror line of y = x.
Restrictions of the domainFor sin x it is /2For cos x it is 0For tan x it is /2
Quotient ruledy/dx = vu' - uv'/ v^2whenever it is y =
Inequalities involving modulus signs require you to take the positive and negative values to solve the equation.
Transformations of functionsf(x - t) + s translation of t/saf(x) = one way stretch in y axis, scale factor af(ax) = one way stretch in x axis, scale factor 1/a-f(x) = reflection in x axisf(-x) = reflection in y axis
Differentiating an inverse functiondy/dx is rate of change of y with xdx/dy is rate of change of x with ydy/dx = 1/dx/dy
Gradient of f-1x at (a,b) = 1/gradient of f(x) at (b,a)
dy/dx of y =lnx = 1/xy = e^x dy/dx = e^xy = e^ax => ae^ax where a is a numbery = ae^x => ae^xy = e^f(x) => f'(x)e^f(x)y = alnx => a/xy = ln(ax) => 1/xy = ln(f(x)) => f'x/fx
d/dx(sinx) = cos(x)d/dx(cosx) = -sin(x)d/dx(tanx) = 1/cos^2x or sec^2x
Chain rule on trigcos2xu = 2x y = cosuu' = 2 y'' = -sinu
Rules of dy/dx on trigy = cos kx => -ksinkxy= sinkx => kcoskxy = tankx => k/cos^2kx
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