Engineering Maths · all questions

Calculus
practice.

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252

Questions

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Q221

Open question

The value of the definite integral

\int_{1}^{e} x ln (x) dx

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Q222

Open question

In the Taylor series expansion of e^x about x = 2, the coefficient of (x - 2)⁴ is

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Q223

Open question

The value of the integral

\int_{0}^{π} x cos^{2} xdx

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Q224

Open question

The value of the line integral

c \int (2 x y^{2} dx + 2 x^{2} ydy + dz)

along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is

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Q225

Open question

Consider a vector field

A (r) .

The closed loop line integral

\oint A \cdot d l

can be expressed as

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Q226

Open question

The area between the parabolas y² = 4ax and x² = 4ay is

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Q227

Open question

x \to \infty lim x^{\frac{1}{x}}

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Q228

Open question

For a position vector

r = x i^+ y j^+ z k^

the norm of the vector can be defined as

r = x^{2} + y^{2} + z^{2} .

Given a function

ϕ = ln r,

its gradient

\nabla ϕ

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Q229

Open question

According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value

ξ

in this interval such that

a \int b f (x) dx =

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Q230

Open question

Let

\nabla \cdot (f v) = x^{2} y + y^{2} z + z^{2} x,

where f and v are scalar and vector fields respectively. If

v = y i + z j + x k,

then

v \cdot \nabla f

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Q231

Open question

Let

f (x, y) = \frac{a x ^{2} + b y ^{2}}{xy},

where a and b are constants. If

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}

at x = 1 and y = 2, then the relation between a and b is

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Q232

Open question

The curve y = f(x) is such that the tangent to the curve at every point (x, y) has a Y-axis intercept c, given by c = -y. Then f(x) is proportional to

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Q233

Open question

If f(x) is an even function and a is a positive real number, then

\int_{- a}^{a} f (x) dx

equals

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Q234

Open question

Curl of vector

F = x^{2} z^{2} i^- 2 x y^{2} z j^+ 2 y^{2} z^{3} k^

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Q235

Open question

The directional derivative of the scalar function f(x, y, z) = x² + 2y² + z at the point P = (1, 1, 2) in the direction of the vector

a = 3 i^- 4 j^

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Q236

Open question

The curl of the gradient of the scalar field defined by V = 2x²y + 3y²z + 4z²x is

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Q237

Open question

The value of

(x, y) \to (0, 0) lim \frac{x ^{2} - xy}{x - y}

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Q238

Open question

e^{y} = x^{\frac{1}{x}},

then y has a

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Q239

Open question

x \to 0 lim \frac{sin ^{2} x}{x}

is equal to

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Q240

Open question

Equation of the line normal to function

f (x) = (x - 8)^{\frac{2}{3}} + 1

at P(0, 5) is

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Calculus
practice.