Calculus MCQs Question and Answer in Engineering Maths

Let f be a real-valued function of a real variable defined as f(x) = x² for x ≥ 0, and f(x) = -x² for x < 0. Which one of the following statements is true?

The value of \[\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + \frac{1}{{\text{x}}}} \right)^{\text{x}}}\]   is

The divergence of the vector field \[\overrightarrow {\rm{U}} = {{\rm{e}}^{\rm{x}}}\left( {\cos \,{\rm{y\hat i}} + \sin {\rm{y\hat j}}} \right)\]     is

Given a function f(x, y) = 4x² + 6y² - 8x - 4y + 8. The optimal value of f(x, y)

To evaluate the double integral \[\int_0^8 {\left( {\int_{\frac{{\text{y}}}{2}}^{\frac{{\text{y}}}{2} + 1} {\left( {\frac{{2{\text{x}} - {\text{y}}}}{2}} \right){\text{dx}}} } \right){\text{dy,}}} \]      we make the substitution \[{\text{u}} = \frac{{2{\text{x}} - {\text{y}}}}{2}\]   and \[{\text{v}} = \frac{{\text{y}}}{2}.\]  The integral will reduce to

\[\iint {\left( {\nabla \times {\text{P}}} \right) \cdot {\text{ds,}}}\]    where P is a vector, is equal to

\[\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{\sin {\text{x}}}}{{\text{x}}}\]   is

At t = 0, the function \[{\text{f}}\left( {\text{t}} \right) = \frac{{\sin {\text{t}}}}{{\text{t}}}\]   has

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x⁴ - 16x³ - 24x² + 37 is

If P, Q and R are three points having coordinates (3, -2, -1), (1, 3, 4), (2, 1, -2) in XYZ space, then the distance from point P to plane OQR (O being the origin of the coordinate system) is given by