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QUESTION: 1

If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) is

Solution:

For xy ∈ l, by Lagrange’s mean value theorem

where x < c < y

implies f(x)- f(y) = (x - y) f '(c )

implies |f(x) - f(y) | = x - y || f'(c)|

for a given such that |f(x) - f(y)| < ε, x, y ∈ l. Hence f(x) is uniformly continuous on l. We know that every uniformly continuous function is also continuous.

QUESTION: 2

The value of dydx by changing the order of integration is

Solution:

QUESTION: 3

If f(x) is differentiable on (a, b) and f(a) = 0 and there exists a real number k such that |f'(x) ≤ k |f (x)| on [a, b], then f(x) is

Solution:

Choose x_{1 }> a such that k(x_{1} - a) < 1,

let α = sup | f(t) | on a < t < x_{1} then by Lagrange’s mean value theorem

where a < t< x ≤ x_{1}

implies f(x) - f(a) = (x - a) f '(t)

[since f(a) = 0]

QUESTION: 4

The volume of ellipsoide is

Solution:

QUESTION: 5

If f(x+y) = f (x) . f(y) for all x and y. Suppose that f(3) = 3 and f'(0) = 11 then , f'(3) is equal to

Solution:

Given that f'(0) = 11

implies

implies

implies

[Since f(3 + 0) = f(3) - f(0)

=> f(3) = f(3) • (0)]

implies f(0) = 1 Now we have

QUESTION: 6

The area bounded by the curve y = ψ(x), x-axis and the lines x = l , x = m(l <m ) is given by

Solution:

QUESTION: 7

If f(x) and g(x) are real number function defined on and , , then which of following is correct?

Solution:

Here,

cos f(x) - sin (g(x))

using eq. (ii) in eq. (i), then we obtain cos (f (x)) - sin (g (x)) > 0

or

QUESTION: 8

The volume of an object expressed in spherical coordinates is given by

The value of the integral is

Solution:

QUESTION: 9

dx dy is equal to

Solution:

QUESTION: 10

Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integral is

Solution:

QUESTION: 11

Which of the following function is not called the Euler’s integral of the first kind?

Solution:

Euler’s integral of the first kind is nothing but Beta function. So, here only is not the definition of Beta function.

QUESTION: 12

Consider the following conditions

(i) f(x) is well defined at x = a

(ii) must exist.

(iii) f(x) is continuous.

(iv) f(x) ≠ 0 at x = a

Q. which of these conditions are necessary for a function f(x) to be derivable at a point x = a of its domain?

Solution:

The necessary condition for a function f(x) to be derivable at a point x = a of its domain.

(i) f(x) is well defined at x = a

(ii) must exist.

(iv) f (x) is continuous.

QUESTION: 13

For (x, y) ∈ R^{2}, let

Solution:

Let us approaches (0, 0) along the line y = mx which passes through origin. Put y = mx, we get

which depends on m. Hence, does not exists, Therefore f(x, y) is discontinuous at (0,0).

Now,

and

Hence, f(x,y) is discontinuous at (0, 0). But both the partial derivative f_{x} and fy exists at origin.

QUESTION: 14

Let

Then,

Solution:

Let us suppose (x, y) approaches (0, 0) along the line y = mx. Which is a line through the origin. Put y = mx and allows x —> 0, we get

which depends on m, therefore the limit of f(x, y) at (0, 0) does not exists. Hence, f(x, y) is discontinuous at origin.

Now,

since f_{y} exists at origin.

QUESTION: 15

Let f : R^{2 }—> R be defined by

Then, the directional derivative direction of f at (0 , 0) in the direction o f the vector is

Solution:

we are given that u = and a = (0,0). we have directional derivative at a in the direction u as

QUESTION: 16

L et f : R^{2} —> R be defined by

Then,

Solution:

We have,

But f(0 , 0) = 0

Hence,

implies f(x, y) is continuous at (0, 0) Now,

where A = 0, B = 0 which does not depends on h and k and

Hence, y(x, y) is continuous as well as differentiable at (0, 0).

QUESTION: 17

The inverse of function f(x) = x^{3 }+ 2 is ____________

Solution:

To find the inverse of the function equate f(x) then find the value of x in terms of y such that f ^{-1} (y) = x.

QUESTION: 18

Let f : R^{2} —> R be such that and exist at all points. Then,

Solution:

QUESTION: 19

Let if possible,

Solution:

QUESTION: 20

Let f: R^{2} → R be defined by f(x, y)

Then the value of at the point (0, 0) is

Solution:

Here, f(x, y)

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