Math by Tathagata
Tathagata The number e is an irrational number and its values lies between 2 and 3. In this article, let us learn how to prove that e is irrational. Question Prove that e is irrational. Answer: If possible, suppose that e is an irrational number. So $e=\dfrac{p}{q}$ where p and q are positive integers prime to … Read more
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) + $\dfrac{(x-a)^2}{2!}f”(a)$ + $\dfrac{(x-a)^3}{3!}f”'(a) + \cdots$)f'(a) + $\dfrac{(x-a)^2}{2!}f”(a)$ + $\dfrac{(x-a)^3}{3!}f”'(a) + \cdots$ Maclaurin Series Formula Taylor series … 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
Engineering Mathematics Assignments. This page contains a list of assignments for Engineering Mathematics. Unit I and IV: Differential Calculus & Vector Algebra Q1: Show that the following limits do not exist. Q2: If $y=(ax+b)^m$, then find $y_n$. For this, you study the article nth Derivative. Q3: If $y=e^{m \sin^{-1}x}$, then show that $(1-x^2)y_{n+2} -(2n+1)xy_{n+1}$ $- … Read more
Special functions practice problems. A list of practice problems on Special Functions are given here. Problems on Beta Gamma Functions Q1: For what values of m and n, the integral $\displaystyle \int_0^1 x^{m+1}(1-x)^{n-3}\,dx \quad$ converges. Q2: Find the Value of $B\left(\dfrac{1}{2}, \dfrac{1}{2} \right)\quad$ [Solution] Q3: What are the values of $\Gamma(1)$ and $\Gamma(5)$? Q4: Find … Read more
Engineering Mathematics Syllabus: In this page, the syllabus of Engineering Mathematics I is provided. Study Material Books: Engineering Mathematics I: Best Books, Practice Problems Engineering Mathematics I Syllabus Unit I: Differential Calculus Introduction to limit, continuity, derivative for function of one variable; Successivedifferentiation, Leibnitz’s theorem; Rolle’s theorem, Lagrange’s mean value theorem, Taylor’and Maclaurin’s theorems with … Read more
Answer: The Fourier series of x in (-π, π) is given as follows: f(x) = x = $\sum_{n=1}^{\infty} \frac{2}{n} (-1)^{n+1} \sin(nx)$. Fourier Series of x Question: Find the Fourier series of x in the interval (-π, π). Answer: We know that the Fourier series of a function ( f(x) ) in the interval (-π, π) … 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