Idea of Derivatives
* Limit: lim x→a ƒ(x) is the expected value of ƒ(x) at x = a is given the values of ƒ(x) when approaches a.
* Left hand limit: lim x→a – ƒ(x) is the expected value of ƒ(x) at x = a given the values of ƒ(x) near x to the left of a.
* Right hand limit: lim x→a + ƒ(x) is the expected value of ƒ(x) at x = a given the values of ƒ(x) near x to the right of a.
Introduction to Limits
* Limit: lim x→a ƒ(x) is the expected value of ƒ(x) at x = a given the values of ƒ(x) when x approaches a.
* Left hand limit: lim x→a – ƒ(x) is the expected value of ƒ(x) at x = a given the values of ƒ(x) near to x to the left of a.
* Right hand limit: lim x→a + ƒ(x) is the expected value of ƒ at x = a given the values of ƒ(x) near x to the right of a.
Illustrations of Limits of Functions
The limit of a function exists only if the hand limit and the right hand limit coincide with each other.
Algebra of Limits
* The limit of the sum of two functions is equal to the sum of the limits of the function.
lim x→a [ƒ(x) + g(x)] = lim x→a ƒ(x) + lim x→a g(x)
* The limit of the difference of two functions is equal to difference of the limits of the functions.
lim x→a [ƒ(x) – g(x)] = lim x→a ƒ(x) – lim x→a g(x)
* The limit of the product of two functions is equal to the product of the limits of the functions.
lim x→a [ƒ(x) – g(x)] = lim x→a ƒ(x) . lim x→a g(x)
lim x→a [ k . g(x)] = k . lim x→a g(x), k is a constant.
* The limit of quotient of two functions is equal to the quotient of the limits of the functions.
\lim_{x \longrightarrow } a \frac{f(x)}{g (x)} = \frac{ \lim_{x \rightarrow} a \thinspace f(x)}{ \lim_{x \rightarrow} a \thinspace g(x) } g(x) ≠ 0
If ƒ(x) is a well defined polynomial function, then lim x→a ƒ(x) = ƒ(a).
Limits of Trigonometric Functions
Theorem: Let g and h be two real valued functions with the same domain, such
that g(x) ≤ h(x) for all x in the domain of definition.
* For some a, if both
lim x→a g(x) and lim x→a h(x) exists, then lim x→a g(x) lim x→a h(x).
Sandwich Theorem: let ƒ, g and h be real functions, such that ƒ(x) ≤ g(x) ≤ h(x)
for all x in the common domain of definition.
* For some real number a, if lim x→a ƒ(x)= l = lim x→a h(x), then lim x→a g(x) = l.
lim x→ 0 \frac{sin \thinspace x}{x} = 1
log x→ 0 \frac{1 - cos \thinspace x}{x} = 0
Introduction to Derivatives
* Suppose f is a real valued function and a is a point in its dooming of definition.
* The derivative of f at a is defined by (provided the limit exists):
f’ (a) = lim h→ 0 \frac{f (a + h) - f (a)}{h}
* Derivative of f(x) at a is denoted by f’ (a).
Suppose f is a real valued function. The function defined by f’ (x) = lim h→ 0
\frac{f ( x + h) - f (x)}{h}
Wherever the limit exists,is defined to be the derivative of f at x and is denoted by f’(x).
Algebra of Derivatives
* The derivative of the sum of two functions is equal to the sum of the derivatives of
the functions.
\frac{d}{dx} [ f (x) + g (x) ] = \frac{d}{dx} [ f (x) ] + \frac{d}{dx} [ g (x) ]
* The derivative of the difference of two functions is equal to the difference of the derivatives of the functions.
\frac{d}{dx} [ f (x) - g (x) ] = \frac{d}{dx} [ f (x) ] - \frac{d}{dx} [ g (x) ]
* The derivative of the product of two functions is:
\frac{d}{dx} [ f (x) . g (x) ] = f (x) \frac{d}{dx} [ g (x) ] + g (x) \frac{d}{dx} [ f (x) ]
* The derivative of the quotient of two functions is:
\frac{d}{dx} [ \frac{f (x)}{g (x)} ] = \frac{g (x) \frac{d}{dx}[ f(x) ] -f (x) \frac{d}{dx} [ g (x)]}{(g (x) )²}
g(x) ≠ 0
Derivatives of Polynomial Functions
\frac{d}{dx} (xn)= nxn-1, where n is a real number
Polynomial functions:
f(x)=anxn + an-1xn-1 + … + a2x2 + a1x +a0,
ai= Real number, ∀ i ∈ Z and an ≠ 0
Derivatives of polynomial function:
\frac{d}{dx} f(x) = nanxn-1 + (n-1)an-1xn-2 + … +2a2x +a1
Derivatives of Trigonometric Functions
Derivative of sin x: \frac{d}{dx} (sin x) = cox x
Derivatives of cos x: \frac{d}{dx} (cos x) = -sin x
Derivative of tan x: \frac{d}{dx} (tan x) = sec2x