Math by Tathagata
In this page, we will learn Taylor series expansion formula along with its remainder form and solved examples. Taylor Series Formula The Taylor series expansion formula of f(x) about x=a is given by f(x) = f(a) + (x-a)f'(a) + $\dfrac{(x-a)^2}{2!}f”(a)$ + $\dfrac{(x-a)^3}{3!}f”'(a) + \cdots$ Maclaurin Series Formula Taylor series expansion of f(x) about x=0 is … Read more
Indeterminate Form and L’Hospital Rule: L’Hôpital’s rule in Calculus is a powerful tool for solving indeterminate limits of the forms like 0/0 or ∞/∞ by taking the derivatives of the top and bottom simultaneously and then re-evaluating the limit of the resulting quotient. This rule is repeatedly applied till the limit exists, and this value … Read more
Answer: The derivative of f(x) defined by f(x) = x2sin(1/x) if x≠0 and 0 if x=0 is equal to 0. That if, if f(x) = $\begin{cases} x^2 \sin \left(\dfrac{1}{x} \right) & \text{ if } x \neq 0 \\ 0 & \text{ if } x=0 \end{cases}$, then $f'(0)=0$. Differentiate x^2sin(1/x) Question: Find $f'(0)$ where f(x) is … Read more
Uniform Continuity is a special kind of continuity. In this page, we will learn about uniformly continuous functions with examples and related theorems. Notation: I= An interval.ℝ= The set of all real numbers. MOTIVATION: Let f: I → ℝ be a continuous function and c ∈ I. As f is continuous at c, for every … Read more
Continuous functions is an important topic in Real Analysis. On this page, we will study about continuous functions along with its several properties and examples. Definition Definition of a Continuous Function: Let f: D → ℝ be a function and let c ∈ D. We say f is continuous at x=c, if limx→c f(x) = … Read more
Rolle’s theorem is very useful in Calculus for continuous functions. In this post, we will learn Rolle’s theorem with its geometrical interpretation along with some solved examples. Statement of Rolle’s Theorem Let f(x) be a real-valued function defined on the closed interval [a, b] satisfying the following: Then there is a point c ∈ (a, … Read more
