GATE Questions & Answers of Calculus Computer Science and Information Technology

The value of $ \int_0^{\mathrm\pi/4}x\cos\left(x^2\right)dx $ correct to three decimal places (assuming that $ \mathrm\pi=3.14 $) is _________ .

The value of $\lim_\limits{x\rightarrow1}\frac{x^7-2x^5+1}{x^3-3x^2+2}$

If $f\left(x\right)=R\;\sin\;\left(\frac{\pi x}2\right)+S,\;f'\left(\frac12\right)=\sqrt2\;and\;\int_0^1f\left(x\right)dx=\frac{2R}\pi, $ then the constants R and S are respectively

Let the function


where θπ6,π3 and f'θ denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

(I) There exists θπ6,π3 such that f'θ=0.

(II) There exists θπ6,π3 such that f'θ≠0.

The function f(x)=x sin x Satisfies the following equation:f "(x)+f(x)+t cos x=0.the value of t is_________.

A funnction f(x) is continuous in the interval 0,2. It is known that f(0)=f(2)=-1 and f(1)=1. Which one of the following statements must be true?

If 02πxsinxdx=kπ,then the value of k is equal to ______ .

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified
pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. Suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1 + min(T(y), T(z)). Then the value of the product yz is _____.

The value of the integral given below is

$\int\limits_0^\mathrm\pi x^2\cos xdx$

Which one of the following functions is continuous at x = 3?

Consider the function f(x) = sin(x) in the interval x π/4,7π/4. The number and location(s) of the local minima of this function are

Given i=-1, what will be the evaluation of the definite integral $\int\limits_0^{\pi/2}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx$ ?

What is the value of limn1-1n2n?

$\int\limits_0^{\mathrm\pi/4}\left(1-\tan x\right)/\left(1+\tan x\right)\operatorname{d}x$

evaluates to

limxx-sinxx+cosx equals

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x4- 16x3 + 24x2 + 37 is

Consider the following two statements the function f(x) = |x|:

P. f(x) is continuous for all real values of x
Q. f(x) is differentiable for all real values of x

Which of the following is TRUE?

Subjects of Engineering Mathematics