Calculus MCQs Question and Answer in Engineering Maths

\[\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \frac{\theta }{2}}}{\theta }\]  is

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

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

The value of \[\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\text{x}}^3} - \sin \left( {\text{x}} \right)}}{{\text{x}}}\]   is

The divergence of the vector field \[\left( {{\rm{x}} - {\rm{y}}} \right){\rm{\hat i}} + \left( {{\rm{y}} - {\rm{x}}} \right){\rm{\hat j}} + \left( {{\rm{x}} + {\rm{y}} + {\rm{z}}} \right){\rm{\hat k}}\]       is

The solution of \[\int\limits_1^{\text{a}} {\int\limits_1^{\text{b}} {\frac{{{\text{dxdy}}}}{{{\text{xy}}}}} } \]  is

The line integral \[\int {\overrightarrow {\text{V}} .{\text{d}}\overrightarrow {\text{r}} } \]  of the vector \[\overrightarrow {\rm{V}} .\left( {\overrightarrow {\rm{r}} } \right) = 2{\rm{xyz\hat i}} + {{\rm{x}}^2}{\rm{z\hat j}} + {{\rm{x}}^2}{\rm{y\hat k}}\]      from the origin to the point P(1, 1, 1)

Given a vector \[\overline {\rm{u}} = \frac{1}{3}\left( { - {{\rm{y}}^3}{\rm{\hat i}} + {{\rm{x}}^3}{\rm{\hat j}} + {{\rm{z}}^3}{\rm{\hat k}}} \right)\]     and \[{{\rm{\hat n}}}\] as the unit normal vector to the surface of the hemisphere (x² + y² + z² = 1; z ≥ 0), the value of integral \[\int {\left( {\nabla \times \overline {\rm{u}} } \right) \cdot {\rm{\hat n}}} {\rm{dS}}\]    evaluated on the curved surface of the hemisphere S is

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

Let f(x) = \[{{\rm{x}}^{ - \frac{1}{3}}}\] 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