Test: Functions Of One,Two Or Three Real Variables -1

# Test: Functions Of One,Two Or Three Real Variables -1

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Test: Functions Of One,Two Or Three Real Variables -1 - Question 1

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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 1

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.

Test: Functions Of One,Two Or Three Real Variables -1 - Question 2

### The value of dydx by changing the order of integration is

Test: Functions Of One,Two Or Three Real Variables -1 - 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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 3

Choose x> a such that k(x1 - a) < 1,
let α = sup | f(t) | on a < t < x1 then by Lagrange’s mean value theorem

where a < t< x ≤ x1
implies f(x) - f(a) = (x - a) f '(t)
[since f(a) = 0]

Test: Functions Of One,Two Or Three Real Variables -1 - Question 4

The volume of ellipsoide  is

Test: Functions Of One,Two Or Three Real Variables -1 - 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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 5

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

Test: Functions Of One,Two Or Three Real Variables -1 - 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

Test: Functions Of One,Two Or Three Real Variables -1 - Question 7

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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 7

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

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

Test: Functions Of One,Two Or Three Real Variables -1 - Question 8

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

Test: Functions Of One,Two Or Three Real Variables -1 - Question 9

dx dy is equal to

Test: Functions Of One,Two Or Three Real Variables -1 - Question 10

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

Test: Functions Of One,Two Or Three Real Variables -1 - Question 11

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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 11

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

Test: Functions Of One,Two Or Three Real Variables -1 - 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?

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 12

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.

Test: Functions Of One,Two Or Three Real Variables -1 - Question 13

For (x, y) ∈ R2, let

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 13

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 fx and fy exists at origin.

Test: Functions Of One,Two Or Three Real Variables -1 - Question 14

Let

Then,

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 14

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 fy exists at origin.

Test: Functions Of One,Two Or Three Real Variables -1 - Question 15

Let f : R2 —> R be defined by

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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 15

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

Test: Functions Of One,Two Or Three Real Variables -1 - Question 16

L et f : R2 —> R be defined by

Then,

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 16

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).

Test: Functions Of One,Two Or Three Real Variables -1 - Question 17

The inverse of function f(x) = x3 + 2 is ____________

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 17

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.

Test: Functions Of One,Two Or Three Real Variables -1 - Question 18

Let f : R2 —> R be such that and exist at all points. Then,

Test: Functions Of One,Two Or Three Real Variables -1 - Question 19

Let if possible,

Test: Functions Of One,Two Or Three Real Variables -1 - Question 20

Let f: R2 → R be defined by f(x, y)

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

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables -1 - Question 20

Here, f(x, y)

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