Solution - Career Point

MATHSCALCULUSMathematics FundamentalsFunction, Limits, Continuity & Differentiability :If the domain of the function is in one quadrant thenthe trigonometrical functions are always one-one.If trigonometrical function changes its sign in twoconsecutive quadrants then it is one-one but if it doesnot change the sign then it is many one.In three consecutive quadrants tigonometricalfunctions are always many one.Any continuous function f(x), which has at least onelocal maximum, is many-one.Any polynomial function f : R → R is onto if degreeof f is odd and into if degree of f is even.An into function can be made onto by redefining thecordomain as the range of f is even.An into function can be made onto by redefining thecodomain as the range of the original function.1If f(x) is periodic with period T then is alsof (x)periodic with same period T.If f(x) is periodic with period T, f (x)is alsoperiodic with same period T.Period of x – [x] is 1. Period of algebraic functionsx , x 2 , x 3 + 5 etc. does't exist.If lim f (x)does not exist, then we can not removex→athis discontinuity. So this become a non-removablediscontinuity or essential discontinuity.If f is continuous at x = c and g is discontinuous at x= c, then(a) f + g and f – g are discontinuous(b) f.g may be continuousIf f and g are discontinuous at x = c, then f + g, f – gand fg may still be continuous.Point functions (domain and range consists one valueonly) is not a continuous function.If a function is differentiable at a point, then it iscontinuous also at that point.i.e., Differentiability ⇒ Continuity, but the converseneed not be true.If a function 'f' is not differentiable but is continuousat x = a, it geometrically implies a sharp corner orkink at x = a.If f(x) and g(x) both are not differentiable at x = athen the product function f(x).g(x) can still bedifferentiable at x = a.If f(x) is differentiable at x = a and g(x) is notdifferentiable at x = a then the sum function f(x) +g(x) is also not differentiable at x = a.If f(x) and g(x) both are not differentiable at x = a,then the sum function may be a differentiablefunction.Differentiation and Applications of Derivatives :dy d dis (y) in which is simply a symbol ofdx dx dxoperation and not 'd' divided by dx.If f´(x 0 ) = ∞, the function is said to have an infinitederivative at the point x 0 . In this case the line tangentto the curve of y = f(x) at the point x 0 isperpendicular to the x-axis.Of all rectangles of a given perimeter, the square hasthe largest area.All rectangles of a given area, the squares has theleast perimeter.A cone of maximum volume that can be inscribed in4 ra sphere of a given radius r, is of height . 3A right circular cylinder of maximum volume thatcan be inscribed in a square of radius r, is of height2 r .3If at any point P(x 1 , y 1 ) on the curve y = f(x), thetangent makes equal angle with the axes, then at theπ 3π dypoint P, ψ = or . Hence, at P tan ψ = = ±14 4dxIndefinite Integral :If F 1 (x) and F 2 (x) are two antiderivatives of afunction f(x) on an interval [a, b], then the differencebetween them is a constant.The signum function has an antiderivative on anyinterval which does not contain the point x = 0, anddoes not possess an anti=derivative on any intervalwhich contains the point.The antiderivative of every odd function is an evenfunction and vice-versa.XtraEdge for IIT-JEE 46 APRIL 2010