Continuity of a Function
* A function ƒ(x) is continuous at x = c, if ƒ(x) is defined at x = c and lim_{x \rightarrow } c ƒ ( x)
=n ƒ(c).
* If ƒ is not continuous at x = c, then we say ƒ is discontinuous at c, and c is called a point of discontinuity of ƒ.
* A real function ƒ is said to be continuous, if it is continuous at every point in the domain ƒ.
Concept of Infinity
+ ∞ is a number larger than any given real number.
– ∞ is a number smaller than any given real number.
Algebra of Continues Functions
* The sum, difference, product and quotient of two real functions ƒ(x) and ɡ(x) continuous at x = c are also continuous.
* The product of a real valued constant and continuous function at some real number is also continuous.
* The negative of any continuous function is also continuous.
* A real valued number divided by any continuous function at some real number is also continuous.
Differentiability Of A Function
* A function ƒ is said to be differentiable at point c in its domain, if:
5 lim_{h \rightarrow } 0^{-} \frac{ƒ ( c + h) - ƒ (c)}{h} = lim_{h \rightarrow } 0^{+} \frac{ƒ ( c + h) - ƒ (c)}{h}
* A function is said to be differentiable in interval [a, b], if it is differentiable at every point of [a, b].
* A function is said to be differentiable in (a, b), if it is differentiable at every point of (a, b).
* If function ƒ is differentiable at point c, then it is also continuous at that point.
* A continuous function need not be differentiable.
* Derivatives of Composite, Implicit and Inverse Trigonometric Functions
ƒ is a real valued function, which is a composite of two functions, u and v; i.e.
ƒ = u ◦ v. suppose t = v(x) and if \frac{dt}{dx} and \frac{du}{dt} exists,
then : \frac{dƒ}{dx} = \frac{du}{dt}. \frac{dt}{dx}
ƒ is a real valued function, which is a composite of three functions, u, v and w; i.e.
ƒ = (u ◦ v) ◦ w. If s = w(x) and t = v(s), then:
\frac{dƒ}{dx} = \frac{d((u º v )}{ds} . \frac{ds}{dx} = \frac{du}{dt} . \frac{dt}{ds} . \frac{ds}{dx}
| ƒ ( x) | sin^{-1} x | cos^{-1} x | tan^{-1} x | cot^{-1} x | sec^{-1} x | cosec^{-1} x |
| ƒ’ ( x) | \frac{1}{ \sqrt{1 - x²} } | \frac{1}{ \sqrt{1 - x²} } | \frac{1}{ 1 + x² } | \frac{- 1}{ 1 + x² } | \frac{1}{ x \sqrt{x² - 1 } } | \frac{- 1}{ x \sqrt{x² - 1 } } |
| Domain of ƒ’ | ( -1, 1 ) | ( -1, 1 ) | R | R | ( -∞,-1) U (1 ,∞) | ( -∞,-1 ) U ( 1,∞) |
Exponential and Logarithmic Functions
* An exponential function with a positive base b > 1, is a function denoted by y = ƒ(x) = b^{x}
* Exponential functions with base 10 are called common exponential functions.
* Exponential functions with base e are called natural exponential functions.
log_{b} a = x if b^{x} = y, ∀ b > 1 and ∀ b ∈ R.
Change of base rule: = log_{a} þ = \frac{ log_{a} p}{ log_{b} a}
log_{b} pq = log_{b} p + log_{b} q
log_{b} \frac{ p}{ q} = log_{b} p – log_{b} q
Derivatives of Exponential and Logarithmic Functions
\frac{ d}{ dx} ( e^{x} ) = e^{x}
\frac{ d}{ dx} ( log x ) = \frac{ 1}{ x}
Logarithmic differentiation is a process by which a complex function can be differentiated by taking logarithm to base e of the function and then differentiating
both the sides.
Derivatives of Functions in Parametric Form
Expressing a relation between two variables x and y in the form x = (t) and y = ɡ(t) is said to be its parametric form, with t as a parameter.
If x = (t) and y = ɡ(t), then \frac{ dy}{ dx} = \frac{ g' (t)}{ f' (t)} , provided ƒ’ (t) ≠ 0.
Second Order Derivatives
If x = ƒ(x), then \frac{ d}{ dx} ( \frac{ dy}{ dx} ) is called the second order derivative of y w.r.t.x, provided
ƒ’ (x) is differentiable.
\frac{ d}{ dx} ( \frac{ dy}{ dx} ) is denoted by \frac{ d² y}{ dx²} or D²y or ‘ y ” or y_{2}
Mean Value Theorem
Rolle’s theorem: if ƒ: [a, b] → R is continuous on [a, b] and differentiable on (a, b), such that (a) = ƒ(b), where a and b are some real numbers, then their exists
some c (a, b), such that ƒ’(c) = 0.
Mean value theorem: if ƒ: [a, b] → R is continuous on [a, b] and differentiable on (a, b), then their exists some c (a, b), such that ƒ’(c) =\frac{ƒ (b) - ƒ ( a)}{b - a} .