Recent questions in Calculus

Consider points $\mathrm{P}$ and $\mathrm{Q}$ in the $x\text{-}y$ plane, with $P=(1,0)$ and $Q=(0,1)$. The line integral $\displaystyle{}2 \int_{P}^{Q}(x d x+y d y)$ alon...

Simpson's rule for integration gives exact result when $f(x)$ is a polynomial of degree$1$$2$$3$$4$

Which of the following improper integrals is (are) convergent?$\int_{0}^{1} \frac{\sin x}{1-\cos x} d x$$\int_{0}^{\infty} \frac{\operatorname{cox} x}{1+x} d x$$\int_{0}^...

The function $f(x, y)=x^{2} y-3 x y+2 y+x$, hasno local extremumone local minimum but no local maximumone local maximum but no local minimumone local minimum and one loca...

The value of the double integral $\int_{0}^{1} \int_{x}^{1 / x} \frac{x}{1+y^{2}}$ $d x d y$ is

The Taylor series expansion of $\dfrac{\sin x}{x-\pi}$ at $x=\pi$ is given by$1+\frac{(x-\pi)^{2}}{3 !}+\ldots$$-1-\frac{(x-\pi)^{2}}{3 !}+\ldots$$1-\frac{(x-\pi)^{2}}{3 ...

If $e^{y}=x^{\frac{1}{x}}$, then $y$ has amaximum at $x=e$minimum at $x=e$maximum at $x=e^{-1}$minimum at $x=e^{-1}$

Given that $\mathrm{f(y})=|\mathrm{y}| / \mathrm{y},$ and $\mathrm{q}$ is any non-zero real number, the value of $|\mathrm{f}(\mathrm{q})-\mathrm{f}(-\mathrm{q})|$ is$0$$...

The function $f(x) = 8 \log_{e} x – x^{2} + 3$ attains its minimum over the interval $[1, e]$ at $x = $____________.(Here $\log_{e}x$ is the natural logarithm of $x$.)$2$...

Consider the following series:$$ \sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}}$$For which of the following combinations of $c, d$ values does this series converge?$ c= 1, d = –...

The partial derivative of the function$$f(x, y, z) = e^{1-x\cos y} + xze^{-1/(1+y^{2})}$$with respect to $x$ at the point $(1,0,e)$ is$-1$$0$$1 \\$$\dfrac{1}{e}$

For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.

The value of the contour integral$$\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$$evaluated over the unit circle $\mid z \mid=1$ is_______.

The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.

Consider the line integral$$\int_{c} (xdy-ydx)$$the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $...

The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______