1. Limits
Let us consider a function f(x) defined in an interval l. if we see the behavior of f(x) become closer and closer to a number l as x a then l is said to be limit of f(x)at x=a.
1.1 Left Hand Limit –
Let function f(x) is said to approach l as x →a from left if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that
1.2 Right Hand Limit
Let function f(x) is said to approach l as x → a from right if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that
1.3. L- Hospital Rule:
When
must not be zero, where f(n) and g(n) are nth derivative of f(x) and g(x).
1.3.1. L- Hospital Rule for the form (∞ – ∞, 0 × ∞):
1.3.2. L-Hospital Rule for the form (0°, 1∞, ∞°):
2. CONTINUITY
A function y = f(x) is said to be continuous if the graph of the function is a continuous curve. On the other hand, if a curve is broken at some point say x = a, we say that the function is not continuous or discontinuous.
2.1. Definition:
A function f(x) is said to be continuous at x = a if and only if the following three conditions are satisfied:
2.2. Properties of continuous functions:
(i) A function which is continuous in a closed interval is also bounded in that interval.
(ii) A continuous function which has opposite signs at two points vanishes at least once between these points and vanishing point is called root of the function.
(iii) A continuous function f(x) in the closed interval [a, b] assumes at least once every value between f(a) and f(b), it being assumed that
f(a) ≠ f(b).
3. DIFFERENTIABILITY
Note.
A Necessary condition for the Existence of a Finite Derivative
Continuity is a necessary but not the Sufficient for the existence of a finite derivatives.
4. Fundamental Theorem:
4.1. Rolle’s Theorem:
If
(i) f(x) is continuous is the closed interval [a, b],
(ii) f’(x) exists for every value of x in the open interval (a, b) and
(iii) f (a) = f(b), then there is at least one value c of x in (a, b) such that f’ (c) = 0.
4.2 Lagrange’s Mean-Value Theorem:
If
(i) f(x) is continuous in the closed interval [a, b], and
(ii) f’(x) exists in the open interval (a, b),
then there is at least there is at one value c of x (a, b),
4.3. Cauchy’s Mean-value theorem:
If (i) f(x) and g(x) be continuous in [a, b]
(ii) f’(x) and g’(x) exist in (a, b) and
(iii) g’(x) ≠ 0 for any value of x in (a, b),
Basic Differentiation Formulas.
Suppose f and g are differentiable functions, c is any real number, then
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