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Calculus
practice.

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Questions

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Q201

Open question

θ \to 0 lim \frac{sin \frac{θ}{2}}{θ}

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Q202

Open question

At x = 0, the function f(x) = x³ + 1 has

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Q203

Open question

If T(x, y, z) = x² + y² + 2z² defines the temperatures at any location (x, y, z) then magnitude of temperature gradient at P(1, 1, 1) is

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Q204

Open question

The value of

x \to 0 lim \frac{x ^{3} - sin ( x )}{x}

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Q205

Open question

The divergence of the vector field

(x - y) i^+ (y - x) j^+ (x + y + z) k^

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Q206

Open question

The solution of

1 \int a 1 \int b \frac{dxdy}{xy}

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Q207

Open question

The line integral

\int V . d r

of the vector

V . (r) = 2 xyz i^+ x^{2} z j^+ x^{2} y k^

from the origin to the point P(1, 1, 1)

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Q208

Open question

Given a vector

u = \frac{1}{3} (- y^{3} i^+ x^{3} j^+ z^{3} k^)

and

n^

as the unit normal vector to the surface of the hemisphere (x² + y² + z² = 1; z ≥ 0), the value of integral

\int (\nabla \times u) \cdot n^dS

evaluated on the curved surface of the hemisphere S is

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Q209

Open question

The minimum value of function y = x² in the interval [1, 5] is

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Q210

Open question

Let f(x) =

x^{- \frac{1}{3}}

and A denote the area of the region bounded by f(x) and the X-axis, when x varies from -1 to 1. Which of the following statements is/are True?
1. f is continuous in [-1, 1]
2. f is not bounded in [-1, 1]
3. A is nonzero and finite

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Q211

Open question

The parabolic arc y = √x, 1 ≤ x ≤ 2 is revolved around the x-axis. The volume of the solid of revolution is

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Q212

Open question

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

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Q213

Open question

A function f(x) = 1 - x² + x³ is defined in the closed interval [-1, 1]. The value of x in the open interval (-1, 1) for which the mean value theorem is satisfied, is

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Q214

Open question

For 0 ≤ t <

\infty

, the maximum value of the function f(t) = e^-t - 2e^-2t occurs at

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Q215

Open question

Let x be a continuous variable defined over the interval

(- \infty, \infty)

, and f(x) = e^{-x-e^-x} . The integral

g (x) = \int f (x) dx

is equal to

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Q216

Open question

Divergence of vector field

V (x, y, z) = - (x cos xy + y) i^+ (y cos xy) j^+ [(sin z^{2}) + x^{2} + y^{2}] k^

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Q217

Open question

One of the roots of the equation x³ = j, where j is the positive square root of -1, is

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Q218

Open question

The value of the integral

- \infty \int \infty \frac{dx}{1 + x ^{2}}

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Q219

Open question

f (x) = \frac{2 x ^{2} - 7 x + 3}{5 x ^{2} - 12 x - 9},

then

x \to 3 lim f (x)

will be

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Q220

Open question

For two non-zero vectors

A

and

B

, if

A

B

is perpendicular to

A

B

then,

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Calculus
practice.