Solve your IGNOU Doubts

Question:

Find $\frac{dz}{dt}$ for $z=x^{2}y+4y^{2}$ where x = cost and y = sin t using the chain rule.

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Question:

Compute $f_{xy}$ and $f_{yx}$ for the function $f (x,y )=e^{x+y}sin x+9x^{2}+2xy \: \: \: at\: \: (1,2)$

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Question:

Let the production function by $\varrho$ AL^a K^b . Find the elasticity of production with respect to labour (L).

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Question:

Explain the concept of maximum value function.

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Question:

Find $f_{x}( 0,0)$ and $f_{x}( x,y)$ , where $( x,y ) \neq (0,0)$ for the function R² → R defined by

$f (x,y)=\left\{\begin{matrix} \frac{xy^{3}}{x^{2}+y^{2}}&(x,y) \neq (0,0)\\ 0 & \: \: \: \: \: (x,y)= ( 0,0) \end{matrix}\right.$

Is x f₂ continuous at (0,0) ? Justify your answer.

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Question:

Evaluate $\int (7x-2)\sqrt{3x+2}\; \; dx$

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Question:

Singular matrix

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Question:

Decomposable matrix

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Question:

Adjugate of a matrix

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Question:

If Z = f(x,y) = xy

Find the maximum value for f(x,y) if x and y are constrained to sum to 1 (That is, x +y = 1). Solve the problem in two ways: by substitution and by using the Lagrangian multiplier method.

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Question:

Solve the following differential equation

$\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+10y=0,$

given Y(0) = 4

$\frac{dy}{dx}(0)=1$

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Question:

Let Z = f(x,y) = 3x³-5y² -225x + 70y + 23.

(i) Find the stationary points of z.

(ii) Determine if at these points the function is at a relative maximum, relative minimum, infixion point, or saddle point.

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Question:

Check whether the function $f ( x,y) =\left\{\begin{matrix} \frac{4x^{2}y}{4x^{4}+y^{2}}\; \; \; (x,y)\neq (0,0)& \\ 0,\; \; \; \; \; \; \; \: \: \: \: \: (x,y)=(0,0)& \end{matrix}\right.$

is continuous at ( 0,0 ).

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Question:

An individual consumer consumes two commodities X₁ & X₂. The utility function is

$\cup =X_{1}^{0.4}\; X_{2}^{0.6}$

The price of commodity one is P₁ = Rs.3.00, the price of commodity two is P₂ = Rs.4.00, the individual’s income per period is Rs.108. Determine the utility maximizing level of X₁ & X₂ and derive the demand curves for the two commodities.

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Question:

1. Consider the following two matrices

$A=\begin{pmatrix} -1 &1 &2 \\ 1& -1 &-2 \\ -2&2 &4 \end{pmatrix}$

$B=\begin{pmatrix} 1 &0 &-1 \\ -1& 1 &1 \\ 1&-1 &-1 \end{pmatrix}$

(i) Find the rank of ‘A’ and ‘B’

(ii) Show that (AB)^-1 = B^-1 A^-1

(iii) Show that (A^-1 )^-1 = A

(iv) Show that (B^-1 )^-1 = B

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Question:

Check whether the limit of the function $f ( x,y) =\frac{3x^{3}y}{x^{6} +2y^{2}}$ exists as $( x,y)\rightarrow ( 0,0)$

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Question:

Show that the closed sphere with centre (7,3,2) and radius 10 in R³ is contained in the open cube $P = \left \{ (x,y,z ):\mid -2\mid < 11,\mid \mid z-7\mid < 11) \right \}$

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Question:

Find the cylindrical coordinates of the points where the Cartesian coordinates are

i) (8,6,6)

ii) $\left ( \sqrt{2,1,1} \right )$

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Question:

Express the following surfaces in spherical coordinates

i) $xz=3$

ii) $x^{2}+y^{2}-z^{2}=1$

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Question:

a) The function $f : R^{3}\rightarrow R$ given byv $f ( x,y, z)= \mid x\mid +\mid y\mid +\mid z\mid$ is differentiable at , $( 2,3 ,-1)$ b) The function $f (x,y) = max \left \{ \frac{y}{x} ,x\right \}$ . is a homogeneous function on R²