A mathematically acceptable statement is a sentence which is either true or false.
Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p.
Compound statements and their related component statements: A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement.
The role of “And”, “Or”, “There exists” and “For every” in compound statements.
The meaning of implications “If ”, “only if ”, “ if and only if ”. A sentence with if p, then q can be written in the following ways.
p implies q (denoted by p ⇒ q)
p is a sufficient condition for q
q is a necessary condition for p
p only if q
∼ q implies ∼ p
The contrapositive of a statement p ⇒ q is the statement ∼q ⇒ ~p . The converse of a statement p ⇒ q is the statement q ⇒ p. p ⇒ q together with its converse, gives p if and only if q.
The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit.
Limit of a function at a point is the common value of the left and right hand limits, if they coincide.
For a function f and a real number a, $\lim_{a \rightarrow b}$ f(x) and f (a) may not be same (In fact, one may be defined and not the other one).
For functions $f$ and $g$ the following holds:
$ \lim_{a \rightarrow b} f(x) \pm g(x) = \lim_{a \rightarrow b} f(x) \pm \lim_{a \rightarrow b} g(x) $
$ \lim_{a \rightarrow b} \left[ f(x) \cdot g(x) \right]= \lim_{a \rightarrow b} f(x) \cdot \lim_{a \rightarrow b} g(x) $
$ \lim_{a \rightarrow b} \left[ \frac{f(x)}{g(x)} \right]= \frac{\lim_{a \rightarrow b} f(x)}{ \lim_{a \rightarrow b} g(x)}$
Following are some of the standard limits:
$ \lim_{a \rightarrow b} \frac{x^n - a^n} {x-a}= na^{n-1}$
$ \lim_{x \rightarrow 0} \frac{\sin x} {x}= 1$
$ \lim_{x \rightarrow 0} \frac{1 - \cos x} {x}= 0$
The derivative of a function f at a is defined by :
f'(a) = $ \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)} {h}$
Derivative of a function f at any point x is defined by :
f'(x) = $ \frac{\text{d}f(x)}{\text{d}x}= \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)} {h}$
For functions u and v the following holds:
$(u \pm v)' = u' \pm v'$
(uv)' = u' v + uv'
$\left( \frac{u}{v}\right )' = \left( \frac{u'v - uv'}{v^2}\right )$ ; provided all are defined.
Following are some of the standard derivatives.
$\frac{\text{d}\left(x^n\right)}{\text{d}x} = nx^{n-1}$
$\frac{\text{d}\left( \sin x \right)}{\text{d}x} = \cos x$
$\frac{\text{d}\left( \cos x \right)}{\text{d}x} = - \sin x$