GATE Questions & Answers of Calculus Computer Science and Information Technology

Calculus 17 Question(s)

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

$f\left(\theta \right)=\left|\begin{array}{ccc}\mathrm{sin}\theta & \mathrm{cos}\theta & \mathrm{tan}\theta \\ \mathrm{sin}\left(\mathrm{\pi }}{6}\right)& \mathrm{cos}\left(\mathrm{\pi }}{6}\right)& \mathrm{tan}\left(\mathrm{\pi }}{6}\right)\\ \mathrm{sin}\left(\mathrm{\pi }}{3}\right)& \mathrm{cos}\left(\mathrm{\pi }}{3}\right)& \mathrm{tan}\left(\mathrm{\pi }}{3}\right)\end{array}\right|$

where $\theta \in \left[\begin{array}{cc}\frac{\pi }{6},& \frac{\pi }{3}\end{array}\right]$ and $f\text{'}\left(\theta \right)$ denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

(I) There exists $\theta \in \left[\begin{array}{cc}\frac{\pi }{6},& \frac{\pi }{3}\end{array}\right]$ such that $f\text{'}\left(\theta \right)$=0.

(II) There exists $\theta \in \left[\begin{array}{cc}\frac{\pi }{6},& \frac{\pi }{3}\end{array}\right]$ such that $f\text{'}\left(\theta \right)$≠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 ${\int }_{0}^{2\mathrm{\pi }}\left|x\mathrm{sin}x\right|dx=k\mathrm{\pi },$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 $\in$$\left[\mathrm{\pi }/4,7\mathrm{\pi }/4\right]$. The number and location(s) of the local minima of this function are

Given $i=\sqrt{-1}$, 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 $\underset{n\to \infty }{\mathrm{lim}}{\left(1-\frac{1}{n}\right)}^{2n}?$

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

evaluates to

$\underset{x\to \infty }{\mathrm{lim}}\frac{x-sinx}{x+\mathrm{cos}x}$ 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