Calculus MCQs Question and Answer in Engineering Maths

If Z = e^{ax + by} F(ax - by); the value of \[{\text{b}} \cdot \frac{{\partial {\text{Z}}}}{{\partial {\text{x}}}} + {\text{a}} \cdot \frac{{\partial {\text{Z}}}}{{\partial {\text{y}}}}\]   is

Curl of vector V(x, y, z) = 2x²i + 3z²j + y³k at x = y = z = 1 is

For a scalar function f(x, y, z) = x² + 3y² + 2z², the directional derivative at the point P(1, 2, -1) in the direction of a vector \[\overrightarrow {\text{i}} - \overrightarrow {\text{j}} + 2\overrightarrow {\text{k}} \]   is

The directional derivative of the function f(x, y) = x² + y² along a line directed from (0, 0) to (1, 1), evaluated at point x = 1, y = 1 is

Value of the integral \[\oint\limits_{\text{c}} {\left( {{\text{xydy}} - {{\text{y}}^2}{\text{dx}}} \right)} \]   , where c is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (Use Green's theorem to change the line integral into double integral)

The tangent to the curve represented by y = x $$l$$nx is required to have 45° inclination with the x-axis. The coordinates of the tangent point would be

Consider the function f(x) = sin (x) in the interval \[{\text{x}} \in \left[ {\frac{\pi }{4},\,\frac{{7\pi }}{4}} \right].\]   The number and location(s) of the local minima of this function are

Euclidean norm (length) of the vector [4 -2 -6]^T is

∇ × ∇ × P where P is a vector is equal to

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is \[{\text{y}} = 4{\text{h}}\left( {\frac{{{{\text{x}}^2}}}{{{{\text{L}}^2}}}} \right)\]  , where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is