GATE EC Calculus questions

For a function $g(t),$ it is given that $\int_{- \infty}^{ + \infty} g(t)e^{-j\omega t}\:dt = \omega e^{-2\omega^{2}}$ for any real value $\omega.$ If $y(t) = \int_{- \infty}^{t}\:g(\tau)\:d\tau,$ then $\int_{- \infty}^{ + \infty}y(t)dt$ is

The volume under the surface $z(x,y) = x + y$ and above the triangle in the $x – y$ plane defined by $\{0 \leq y \leq x \: \text{and} \: 0 \leq x \leq 12\}$ is _______.

For $0 \leq t < \infty ,$ the maximum value of the function $f(t)= e^{-t}-2e^{-2t}$ occurs at

The value of $$\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$$ is

The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.

The maximum value of the function $f(x) = \text{ln } (1+x) – x$ (where $x >-1$) occurs at $x=$_______.

If $z= xy \text{ ln} (xy)$, then

The maximum value of $f(x)$= $2x^{3}$ – $9x^{2}$ + $12x – 3$ in the interval $0\leq x\leq 3$ is _______.

The series $\Sigma_{n=0}^{\infty} \frac{1}{n!}$ converges to

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the triangle, the angle between the hypotenuse and the side is

The maximum value of $\theta$ until which the approximation $\sin\theta \approx \theta$ holds to within $10\%$ error is

A polynomial $f(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x - a_{0}$ with all coefficients positive has

The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1,6]$ is

If $e^{y}=x^{\frac{1}{x}}$, then $y$ has a

The Taylor series expansion of $\dfrac{\sin x}{x-\pi}$ at $x=\pi$ is given by

For real values of $x$, the minimum value of the function $f(x)=\exp (x)+\exp (-x)$ is

Which of the following functions would have only odd powers of $x$ in its Taylor series expansion about the point $x=0?$

In the Taylor series expansion of $\exp (x)+\sin (x)$ about the point $x=\pi$, the coefficient of $(x-\pi)^{2}$ is

The following plot shows a function $y$ which varies linearly with $x$. The value of the integral $I=\displaystyle{}\int_1^2 y d x$ is

For $|x| \ll 1$, $\operatorname{coth}(x)$ can be approximated as