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MISCELLANEOUS QUIZ ON DIFFERENTIAL CALCULUS

MISCELLANEOUS QUIZ ON DIFFERENTIAL CALCULUS

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Question 1 of 10

1. Question
Let it possible $\displaystyle\alpha=\lim_{(x,y) \to (0,0)} \dfrac{{\sin ( {x^2} + {y^2})}} {{x^2} + {y^2}}$
Hence, $\alpha$ is equal to _______.
- @@wpProQuizCloze-0@@
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Question 2 of 10

2. Question
The value of $\displaystyle{\lim_{n \to \infty }} \sum\limits_{r = 1}^n {\dfrac{1}{{\sqrt {2rn - {r^2}} }}}$ is ________.
- @@wpProQuizCloze-0@@
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Question 3 of 10

3. Question
If z = x^myⁿ, then
- $\dfrac{{dz}}{z} = m\dfrac{{dx}}{x} - n\dfrac{{dy}}{y}$
- $\dfrac{{dz}}{z} = m\dfrac{{dx}}{y} - n\dfrac{{dy}}{x}$
- $\dfrac{{dz}}{z} = m\dfrac{{dx}}{x} - n\dfrac{{dy}}{y}$
- None
Correct

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Question 4 of 10

4. Question
If z = f(u, v), where u = x² – 2xy – y² , v = y then

$(x + y) \dfrac{{\partial z}}{{\partial x}} + (x - y)\dfrac{{\partial z}}{{\partial y}} = 0$ is equivalent to
- $\dfrac{{\partial z}}{{\partial u}} = 0$
- $\dfrac{{\partial z}}{{\partial v}} = 0$
- $\dfrac{{\partial z}}{{\partial x}} = 0$
- $\dfrac{{\partial z}}{{\partial y}} = 0$
Correct

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Question 5 of 10

5. Question
If z = x² + y² and y = z² + x then $\dfrac{{dz}}{{dx}} + \dfrac{{dy}}{{dx}}$ equal to
- $2\left( {\dfrac{{1 + x + 2xz}}{{1 - 2z}}} \right)$
- $2\left( {\dfrac{{1 - x - 2xz}}{{1 - 2z}}} \right)$
- $2\left( {\dfrac{{1 - x + 2xz}}{{1 - 2z}}} \right)$
- $2\left( {\dfrac{{x + 2xz}}{{1 - 2z}}} \right)$
Correct

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Question 6 of 10

6. Question
If $z = {\tan ^{ - 1}}\dfrac{y}{x}$ then dz = _____________.
- $\dfrac{{xdy + ydx}}{{{x^2} + {y^2}}}$
- $\dfrac{{xdy - ydx}}{{{x^2} + {y^2}}}$
- $\dfrac{{2xdy + 2ydx}}{{{x^2} + {y^2}}}$
- $\dfrac{{ydx - xdy}}{{{x^2} + {y^2}}}$
Correct

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Question 7 of 10

7. Question
For the function f (x, y) = x³ + y³ – 3x – 12y + 20 has
- minima at (1, 2)
- maxima at (–1, –2)
- Neither maxima nor minima at (1, –2) and (–1, 2)
- All of the above
Correct

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Question 8 of 10

8. Question
If f(x, y) = (x³ + y³)^1/3 at what point(s) does f_y(x, y) fails to exist
- No such point
- does not exists if $y \ne 0$
- does not exist if $y \ne 0$ and $x \ne y$
- does not exist if y = 0 and x = –y
Correct

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Question 9 of 10

9. Question
The accompanying figure shows a contour plot for unspecified function f(x, y), make conjecture about the signs of partial derivatives f_x(x₀,y₀) and f_y(x₀, y₀)
- $f_x(x_0, y_0) > 0$
- $f_x(x_0, y_0) = 0$
- $f_y(x_0, y_0) > 0$
- $f_y(x_0, y_0)$ < 0
Correct

Incorrect
Question 10 of 10

10. Question
Let $f\left( {x,\;y} \right)$ = Then
- $f (x, y)$ is continuous at origin
- $f (x, y)$ is discontinuous at origin
- $f (x, y)$ has removable discontinuity at origin
- $f (x, y)$ is not differentiable at origin
Correct

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