List of top Multivariable Calculus Questions on Functions of Two or Three Real Variables

For a > b > 0, consider
\(D=\left\{(x,y,z) \in \R^3 :x^2+y^2+z^2 \le a^2\ \text{and } x^2+y^2 \ge b^2\right\}.\)
Then, the surface area of the boundary of the solid D is

Define the function \(f: \R^2 → \R\) by
f(x, y) = 12xy e^-(2x+3y-2).
If (a, b) is the point of local maximum of f, then f(a, b) equals

Let
\(𝑆 = x\left\{(𝑥, 𝑦) ∈ ℝ^2 : 𝑥 > 0, 𝑦 > 0\right\} ,\)
and f: S → ℝ be given by
\(f(x,y)=2x^2+3y^2-\log x-\frac{1}{6}\log y.\)
Then, which of the following statements is/are TRUE ?

Let f : \(\R^2 → \R\) be the function defined as follows :
\(f(x,y)=\begin{cases} (x^2-1)^2\cos^2(\frac{y^2}{(x^2-1)^2}) & \text{if }x \ne ±1 \\ 0 & \text{if } x=±1\end{cases}\)
The number of points of discontinuity of f(x, y) is equal to _________.

Let f(x, y) = \(e^{x^2}+y^2\) for (x, y) ∈ \(\R^2\) , and a_n be the determinant of the matrix
\(\begin{pmatrix} \frac{∂^2f}{∂x^2} & \frac{∂^2f}{∂x∂y} \\ \frac{∂^2f}{∂y∂x} & \frac{∂^2f}{∂y^2} \end{pmatrix}\)
evaluated at the point (cos(n),sin(n)). Then the limit \(\lim\limits_{n \rightarrow \infin}\) a_n is

For each t ∈ (0, 1), the surface P_t in \(\R^3\) is defined by
\(P_t = \left\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 ≤ x^2 + y^2 ≤ 1\right\}.\)
Let a_t ∈ R be the surface area of P_t. Then

The value of
\(\int^1_0\int_0^{1-x}\cos(x^3+y^2)dy\ dx-\int^1_0\int_0^{1-x}\cos(x^3+y^2)dx\ dy\)
is

Let f : \(\R^2 → \R^2\) be defined by f(x, y) = (e^x cos(y), e^x sin(y)). Then the number of points in \(\R^2\) that do NOT lie in the range of f is

Let f : \(\R^2 → \R\) be defined as follows :
\(f(x,y)=\begin{cases} \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\ 0 & \text{if } (x,y)=(0,0) \end{cases}\)
Then

A subset S ⊆ \(\R^2\) is said to be bounded if there is an M > 0 such that |x| ≤ M and |y| ≤ M for all (x, y) ∈ S. Which of the following subsets of \(\R^2\) is/are bounded ?

Let \( c \) be a positive real number and let \( u: \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ u(x,t) = \frac{1}{2c} \int_{x-ct}^{x+ct} e^{s^2} \, ds \quad \text{for} \ (x,t) \in \mathbb{R}^2. \] Then which one of the following is true?

Let M be a positive real number and let u, v: \(\R^2\rightarrow\R\) be continuous functions satisfying \(\sqrt{u(x,y)^2+v(x,y)^2}\ge M\sqrt{x^2+y^2}\ for\ all\ (x,y)\isin\R^2.\)
Let F: \(\R^2\rightarrow\R^2\) be given by
F(x, y) = (u(x, y), v(x, y)) for (x, y)\(\isin\R^2\).
Then which of the following is/are true?

Let \( f(x, y) = e^x \sin y, \, x = t^3 + 1, \, y = t^4 + t. \) Then \( \dfrac{df}{dt} \) at \( t = 0 \) is ............. (rounded off to two decimal places).

Let \( f(x, y, z) = x^3 + y^3 + z^3 - 3xyz. \) A point at which the gradient of \( f \) is equal to zero is

Let \( f(x, y) = \begin{cases} x^2 \sin \dfrac{1}{x} + y^2 \sin \dfrac{1}{y}, & xy \ne 0 \\ x^2 \sin \dfrac{1}{x}, & x \ne 0, y = 0 \\ \\ y^2 \sin \dfrac{1}{y}, & y \ne 0, x = 0 \\ 0, & x = y = 0 \end{cases} \).

Which of the following is true at \( (0, 0)? \)

If the directional derivative of the function \( z = y^2 e^{2x} \) at \( (2, -1) \) along the unit vector \( \vec{b} = \alpha \hat{i} + \beta \hat{j} \) is zero, then \( |\alpha + \beta| \) equals

If the equation of the tangent plane to the surface \( z = 16 - x^2 - y^2 \) at the point \( P(1, 3, 6) \) is \( ax + by + cz + d = 0 \), then the value of \( |d| \) is

Let \( g: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. If \( f(x, y) = g(y) + x g'(y) \), then

The minimum value of the function \( f(x, y) = x^2 + xy + y^2 - 3x - 6y + 11 \) is ............

Let \( D = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \le 1\} \) and \( f : D \to \mathbb{R} \) be a non-constant continuous function. Which of the following is TRUE?

Let \( a, b \in \mathbb{R} \) and let \( f: \mathbb{R} \to \mathbb{R} \) be a thrice differentiable function. If \( z = e^u f(v) \), where \( u = ax + by \) and \( v = ax - by \), then which one of the following is TRUE?

Let \( f(x, y) = \begin{cases} \dfrac{xy}{(x^2 + y^2)^{\alpha}}, & (x, y) \neq (0,0) \\ 0, & (x, y) = (0,0) \end{cases} \) Then which one of the following is TRUE for \( f \) at the point \( (0, 0) \)?

In \( \mathbb{R}^3 \), the cosine of the acute angle between the surfaces \( x^2 + y^2 + z^2 - 9 = 0 \) and \( z - x^2 - y^2 + 3 = 0 \) at the point (2, 1, 2) is

The tangent plane to the surface \( z = \sqrt{x^2 + 3y^2} \) at (1, 1, 2) is given by

Let \( \phi(x, y, z) = 3y^2 + 3yz \) for \( (x, y, z) \in \mathbb{R}^3 \). Then the absolute value of the directional derivative of \( \phi \) in the direction of the line \[ \frac{x-1}{2} = \frac{y-2}{-1} = \frac{z}{-2} \] at the point \( (1, -2, 1) \) is .............