Limits And Continuity

CALCULUS AM 1001

Prepared By: Deptt. Of Mathematics Chitkara University

Lecture 1

Limits And Continuity

Introduction to Limits • The concept of limit is to distinguish calculus from algebra and trigonometry • Here we discuss the behavior of functions with the help of theory of limits. • Its concept is also used to define tangent lines to the graph of functions

Definition of Limit • Def: Let f (x) be defined on an open interval about x0 , except possibly at x0 itself. Then f(x) approaches the limit ‘L’ as x approaches x0 i.e. lim f ( x) = L x→ x • In other words, for every number ε > 0 , there exists a corresponding number δ > 0 such that, f ( x) − L < ε , 0 < | x − x0 | < δ , ∀ x 0

Rules of Limit 1. Sum/Difference Rule: lim[ f ( x) ± g ( x)] = L ± M x →c

2. Product Rule:

lim[ f ( x) ⋅ g ( x)] = L ⋅ M x →c

3. Constant Multiple Rule: lim kf ( x) = kL x →c

f ( x) L = • Quotient Rule: lim x →c g ( x ) M • Power Rule If m and n are integers, then

lim[ f ( x)] x →c

m

n

= [ L]

m

n

Left Hand Limit A function f (x) is said to have a left hand f ( x) = L limit L at x0 , written as xlim →x If for every number ε > 0 there exists a corresponding numberδ > 0 such that for all − 0

x,

x0 − δ < x < x0 ⇒ f ( x) − L < ε

Right Hand Limit A function f (x) is said to have a right hand f ( x) = L limit L at x0 , written as xlim →x If for every number ε > 0 there exists a corresponding numberδ > 0 such that for all x, x0 < x < x0 + δ ⇒ f ( x) − L < ε + 0

Introduction to Continuity • It is a function whose outputs vary continuously with the inputs and do not jump from one value to another without taking on the values in between. • Eg. Variation in time on Earth.

Definition of Continuity •

Def: A function f is said to be continuous at a point x = a if the following three conditions are satisfied: 2. f (a) is defined f ( x) exists (i.e.lim f ( x) = lim f ( x) ). 3. lim x→ a x→a x→a f ( x) = f (a) 4. lim x→ a +

−

Introduction to Derivatives • These are used widely in science, economics, medicine and in computer science to calculate velocity and acceleration which helps in explaining: • the behavior of machinery and • to define the consequences of making errors in measurements.

Introduction to Derivatives •Method of Computation. •Change in Variable with reference to other variable. Eg. 1. Speed. 2. Useful in Business Eg. Demand versus price.

Differentiation Def: The derivative of function f with respect to the variable x is f ( x + h) − f ( x ) f ( x) = lim h →0 h '

provided the limit exists. dy Denoted by: dx which represents the rate of change of y w.r.t. change in x

y = f ( x) The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.

y = f ′( x) →

y = x −3 2

y′ = lim

( x + h)

2

− 3 − ( x − 3) 2

h

h →0

x + 2 xh + h − x y′ = lim h →0 h 0 y  lim 2 x  h 2

2

h 0

y′ = 2 x

2

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

Steps of Differentation 1. Write expression for f (x) and f ( x + h) 2. Expand and simplify the difference quotient f ( x + h) − f ( x ) h ' f 4. Using the simplified quotient find ( x)

by evaluating the limit

f ( x + h) − f ( x ) f ( x) = lim h →0 h '

Differentiable on an Interval • A function y = f (x)is differentiable on an open interval if it is differentiable at each point of interval. • It is differentiable on a closed interval [ a, b] if it is differentiable on the interior (a, b)and the limits lim f (a + h) − f (a) right hand derivative h →0 h at a and similarly for left hand derivative at b exists at the end pionts f (b + h) − f (b) +

lim−

h →0

h

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at:

f  x  x

f  x  x

corner

2 3

cusp

 1, x  0 f  x    1, x  0

f  x  3 x

vertical tangent

discontinuity

→

Sum Rule • The derivative of sum of finite number of functions is the sum of their derivatives. Eg. y=u+v+w+…… Then, its derivative is dy du dv dw = + + + ....... dx dx dx dx

Product Rule • The derivative of product of two functions is equal to the product of the first and the derivative of the second plus the product of the second and the derivative of the first. • Eg. If y=uv dy dv du • Then =u +v dx dx dx

Quotient Rule • The derivative of the quotient of two functions is equal to the product of denominator and derivative of numerator minus product of numerator and derivative of denominator, whole divided by square of denominator. • Eg. If y=u/v du dv v −u • Then dy dx dx dx

=

( v)

2

Derivative of function of function • Chain Rule:If y is function of u say y=f(u), where u itself is a function of x say u= φ (x) then ‘y’ is called a function of a function. • Its derivative is given by:

dy dy du = ∗ dx du dx

Power chain rule • If u (x) is a differentiable function and n is an integer , then u n is differentiable and d n n −1 du u = nu dx dx

Derivative of Trigonometric funs. • • • • • •

Derivative of : sin x = cos x Derivative of : cos x = − sin x 2 Derivative of : tan x = sec x 2 Derivative of : cot x = csc x Derivative of : sec x = sec x tan x Derivative of : csc x = − csc x cot x

Differentiation by Substitution 1. Pre requisite: • Trigonometry • Algebra 2. It reduces the given expression to be differentiated in simple form by suitable substitution.

Leibnitz’s Theorem(Successive Differentiation) • Successive Differentiation: If function then its derivative is

y = f (x)

dy = f ' ( x) (first derivative) dx further its derivative is d2y '' = f ( x) (second derivative of y) 2 dx and so on, its n th derivative is dny n = f ( x) n dx

is any

Leibnitz’s theorem •If u, v are two functions of x possessing derivative of nth order then

( uv ) n = un v + nc1un −1v1 + nc2un −2v2 + ............ + n cr u n − r vr + ....+ n cnuvn

Absolute Extreme Values • Let f be a function with domain D. Then f has an absolute maximum values on D at point ‘a’ if f ( x) ≤ f (a) for all x in D. and an absolute minimum value on D at ‘a’ if f ( x) ≥ f (a) for all x in D.

Local Extremes Values • A function f has a local maximum value at an interior point ‘a’ of its domain if f ( x) ≤ f (a) for all x in some open interval containing ‘a’ •A function f has a local minimum value at an interior point ‘a’ of its domain if f ( x) ≥ f (a) for allx in some open interval containing ‘a’

Finding Extrema • First Derivative Theorem (Local Extreme Values): If f has a local maximum or minimum value at an interior point ‘a’ of its domain and if f ' is defined at ‘a’ then f ' (a) = 0 • Critical Points: An interior point of the domain of a function f where f ' is zero or undefined is a critical point of f

Taylor’s Series Expansion • Definition: Let f be a function with derivatives of all orders through out some interval containing ‘a’ as an interior point. Then the Taylor series generated by f at x = a is '' f (a ) f ( x) = f (a) + f (a )( x − a) + ( x − a) 2 + ..... 2! f n (a) .. + ( x − a ) n + ....... n! '

Maclaurin Series • Maclaurin series generated by f is ''

f (0) 2 f ( x) = f (0) + f (0)( x) + ( x) + ..... 2! f n (0) n .. + ( x) + ....... n! '

• This is Taylor series generated by x=0

f

at