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Unit-I: Differential Calculus
Contents
Rolle’s Theorem, Mean Value Theorems, Taylor's Series and Maclaurin's Series, Expansion of functions using standard expansions, Indeterminate Forms, L' Hospital's Rule, Evaluation of Limits and Applications.
Rolle's Theorem
Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative is zero. The theorem is named after Michel Rolle.
Lagrange's Mean Value Theorem
Lagrange’s mean value theorem (MVT) states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one point x=c on this interval, such that
f(b)−f(a)=f′(c)(b−a).
This theorem (also known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment.
Cauchy's Mean Value Theorem
Lagrange’s mean value theorem (MVT) states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one point x=c on this interval, such that
f(b)−f(a)=f′(c)(b−a).
This theorem (also known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment.
Taylor's Series
The Taylor Series of a function is an infinite sum of terms that are expressed in terms of the functions derivatives at a single point.The English Mathematician, Brook Taylor first wrote about expansion and was published in 1715.The Taylor's Theorem can be derived from generalized mean value theorem.
Maclaurin's Series
The Taylor's series expansion of a function about 'zero' is known as Maclaurin's Series.Named after Scottish Mathematician Colin Maclaurin who formulated this in the 18th century.
Indeterminate Forms
L'Hospital's Rule is named after a French Mathematician Guillaume de L'Hospital.The rule provides a technique to evaluate limits of indeterminate forms.Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. Although the rule is often attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician John Bernoulli.
Guillame de l'hospital
Applications
Mean value theorems are applies to prove some trigonometric equalities, to find extreme values of function.Taylor's Series is used to find the sum of a series.It is also used in approximating functions. The concept of approximating function using a series of function is a fundamental tool in modern science and used in data analytics,differential equations,etc.
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