Engineering Mathematics
Calculus
Practice questions from Calculus.
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IncorrectConsider the following series:
(i)
(ii)
(iii)
Choose the correct option.
(i) (where )
By -test,
series is divergent.
(ii)
series is convergent.
(iii)
By ratio test,
series is convergent.
Hence, option (b) is correct.
Consider the function , defined as
Which of the following statements is/are correct?
(Here, is the set of real numbers.)
has no global maximizer
has no global minimizer
is a local minimizer of
is a local maximizer of
Stationary points:
Local Maxima
So, function has neither global maxima nor global minima.
Hence, option (a) and (b) are correct.
Consider a non-negative function which is continuous and bounded over the interval . Let and denote, respectively, the maximum and the minimum values of over the interval.
Among the combinations of and given below, choose the one(s) for which the inequality is guaranteed to hold.
Given: for
Thus, m , also
And, , also
Given that
Hence, option (A) is correct.
Consider the Earth to be a perfect sphere of radius R. Then the surface area of the region, enclosed by the N latitude circle, that contains the north pole in its interior is__________.
In spherical coordinate system for 60° latitude,
As
Let and represent density and velocity, respectively, at a point and time t. Assume is continuous. Let be an arbitrary volume in space enclosed by the closed surface and be the outward unit normal of Which of the following equations is/are equivalent to ?
Let be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true?
For every closed path T, we have
There exists a differentiable scalar function F(x,y,z) such that
is independent of C
is conservative.
Line Integral of conservative field over a closed curve is 0
is correct
For conservative field the curl is zero
Hence the conservative field can be expressed as gradient of a scalar field
We can write
is correct.
Also,
is correct
The rate of increase, of a scalar field in the direction at a point is
2
4
Given, scalar field,
finding the gradient of this Scalarfield.
By the formula.
Gradient of scalar field,
At point
Now finding the direction derivative,
Directional derivative,
The value of the line integral along the straight line joining the points and is
Given line integral, along the line joining the points and is
Putting the values into the integration then integrating to the point. so
The value of the integral over the region , given in the figure, is __________(rounded off to the nearest integer).
Given, integral,
Here we Break the integration limits into 2 parts.
Part-A varies from and
Part-B varies from x=y-2 to 2-y and y=1 to 2
