List of top Calculus Questions

If $P(x,y,z)$ is the point nearest to the origin lying on the intersection of the surfaces $z=xy+5$ and $x+y+z=1$, then the distance (in integer) between the origin and $P$ is __________.

Which of the following real-valued functions is/are uniformly continuous on $[0, \infty)$?

Consider the following infinite series: \[ S_1 := \sum_{n=0}^{\infty} (-1)^n \frac{n}{n^2 + 4} \quad \text{and} \quad S_2 := \sum_{n=0}^{\infty} (-1)^n \sqrt{n^2 + 1 - n}. \] Which of the above series is/are conditionally convergent?

If the line $ y = \alpha x, \alpha \geq \sqrt{2} $, divides the area of the region \[ R := \{(x, y) \in \mathbb{R}^2 \mid 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 2\} \] into two equal parts, then the value of $ \alpha $ is equal to

Let $\{a_n\}_{n=1}^\infty$ be a sequence of positive real numbers satisfying \[ \frac{8}{a_{n+1}} = \frac{7}{a_n} + \frac{a_n^2}{343}, \quad n \geq 1, \] with $a_1 = 3$ and $a_n<7$ for all $n \geq 2$.
Consider the following statements: \begin{itemize} \item [(I)] $\{a_n\}$ is monotonically increasing. \item [(II)] $\{a_n\}$ converges to a value in the interval $[3, 7]$. \end{itemize} Then which of the above statements is/are true?

Let $ f : \mathbb{R}^2 \to \mathbb{R} $ be a function defined by \[ f(x, y) = \begin{cases} \frac{x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Find the value of $ \frac{\partial f}{\partial x} $ at $ (0, 0) $.

The maximum value of $ f(x, y) = 49 - x^2 - y^2 $ on the line $ x + 3y = 10 $ is __________.

If the function $ f(x, y) = x^2 + xy + y^2 + \frac{1}{x} + \frac{1}{y} $, $ x \neq 0, y \neq 0 $ attains its local minimum value at the point $ (a, b) $, then the value of $ a^3 + b^3 $ is __________ (round off to TWO decimal places).

The surface area of the paraboloid $ z = x^2 + y^2 $ between the planes $ z = 0 $ and $ z = 1 $ is __________ (round off to ONE decimal place).

The rate of change of $ f(x, y, z) = x + x \cos z - y \sin z + y $ at $ P_0(2, -1, 0) $ in the direction from $ P_0(2, -1, 0) $ to $ P_1(0, 1, 2) $ is __________.

The work done by the force $ \mathbf{F} = (x + y) \hat{i} - (x^2 + y^2) \hat{j} $,
where $ \hat{i} $ and $ \hat{j} $ are unit vectors in $ \mathbf{O X} $ and $ \mathbf{O Y} $ directions, respectively, along the upper half of the circle $ x^2 + y^2 = 1 $ from $ (1, 0) $ to $ (-1, 0) $ in the $ xy $-plane is

Let $ \Gamma $ denote the boundary of the square region $ R $ with vertices $ (0,0), (2,0), (2,2), (0,2) $, oriented in the counter-clockwise direction. Then $ \int_{\Gamma} (1 - y^2) dx + x \, dy = \underline{\hspace{1cm}}. $

Let $ f: \mathbb{R}^2 \to \mathbb{R} $ be given by \[ f(x, y) = \begin{cases} \sqrt{x^2 + y^2} \sin\left( \frac{y^2}{x} \right) & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases} \] Consider the following statements: P: $ f $ is continuous at $ (0, 0) $ but $ f $ is NOT differentiable at $ (0, 0) $.
Q: The directional derivative $ D_u f(0, 0) $ of $ f $ at $ (0, 0) $ exists in the direction of every unit vector $ u \in \mathbb{R}^2 $. Then:

Let $ f: \mathbb{R}^2 \to \mathbb{R} $ be given by \[ f(x, y) = 4xy - 2x^2 - y^4. \] Then $ f $ has

The equation \[ xy - z \log y + e^{xz} = 1 \] can be solved in a neighborhood of the point $ (0, 1, 1) $ as $ y = f(x, z) $ for some continuously differentiable function $ f $. Then

Let $ V $ be the solid region in $ \mathbb{R}^3 $ bounded by the paraboloid $ y = x^2 + z^2 $ and the plane $ y = 4 $. Then the value of \[ \iiint_V 15 \sqrt{x^2 + z^2} \, dV \] is

Let $ f: \mathbb{R}^2 \to \mathbb{R} $ be differentiable. Let $ D_u f(0,0) $ and $ D_v f(0,0) $ be the directional derivatives of $ f $ at $ (0,0) $ in the directions of the unit vectors $ u = \left( \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) $ and $ v = \left( \frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}} \right) $, respectively. If $ D_u f(0,0) = \sqrt{5} $ and $ D_v f(0,0) = \sqrt{5} $, then \[ \frac{\partial f}{\partial x} (0,0) + \frac{\partial f}{\partial y} (0,0) = $\underline{\hspace{1cm}}$. \]

Let $ f: \mathbb{R}^3 \to \mathbb{R} $ be a twice continuously differentiable scalar field such that $ \text{div}(\nabla f) = 6 $. Let $ S $ be the surface $ x^2 + y^2 + z^2 = 1 $ and $ \hat{n} $ be the unit outward normal to $ S $. Then the value of \[ \iint_S (\nabla f \cdot \hat{n}) \, dS \] is

The family of surfaces given by \[ u = xy + f(x^2 - y^2), \text{where} \, f : \mathbb{R} \to \mathbb{R} \, \text{is a differentiable function, satisfies:} \]

f(x,y)=$\left\{\begin{matrix} 1-cos(\frac{x^2}{y^2}) \\ \sqrt{x^2+y^2} \end{matrix}\right.$ ;if y≠0 & n∈R, then

Number of finite non-isomorphic groups with exactly 3 conjugacy classes_____.