MATHEMATICS FORMULAE by SUMITA BOSE

Author:SUMITA BOSE [BOSE, SUMITA]
Language: eng
Format: azw3
Publisher: V&S PUBLISHERS
Published: 2015-08-20T04:00:00+00:00

(i) decreasing on I if x1 < x2 in I => f (x1) ≥ f (x2) for all x1, x2 I. (ii) strictly decreasing on I if x1 < x2 in I => f (x1) > f (x2) for all

x1, x2 I. ∈

Increasing or Decreasing Function at a Point∈

Let xobe a point in the domain of definition of a real valued function f. Then f is said to be increasing, strictly increasing, decreasing or strictly decreasing at xo if there exists an open interval I containing xo such that f is increasing, strictly increasing, decreasing or strictly decreasing, respectively, in I.

An Important Theorem

Let the function f be continuous on [a, b] and differentiable on (a,b) then

(a) f is increasing in [a,b] if f ′(x) > 0 for each x (a, b) (b) f is strictly increasing in (a,b) if f ′(x) > 0 for each x (a, b) (c) f is decreasing in [a,b] if f ′(x) < 0 for each x∈ (a, b) (d) f is strictly decreasing in (a,b) if f ′(x) < 0 for each x∈ (a, b) (e) f is a constant function in [a,b] if f ′(x) = 0∈ for each x∈ (a, b) (f) A function will be increasing (decreasing) in R if it is increas∈ing(decreasing) in every interval of R.

Monotonic Function

If a function f in an interval I is either increasing or decreasing, then it is called a monotonic function.

Equation of Tangent and Normal

Slope of Tangent

(i) For a curve y = f (x) the slope of

the tangent at (x0 , y0) is given by

dy



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