Solve your IGNOU Doubts

Question:

L'Hôpital's rule.

See Answer →

Question:

Homogeneous and Homothetic functions.

See Answer →

Question:

For the function $f(x) = \cos(x)$ :

i) Find the linear and quadratic approximations.

ii) Derive the Maclaurin series expansion.
b) Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be defined as

$f(x, y) = \begin{pmatrix} e^{2xy} \\ 2x^2 + 3y^2 \end{pmatrix}.$

Find the Jacobian $J_f$ at the point $(2, 1)$ .

See Answer →

Question:

Consider the Lagrange problem:

$\text{Maximize } f(u, v) = u^3 + v, \quad \text{subject to } g(u, v) = u^2 + v^2 = a^2.$

Use the envelope theorem to estimate the maximum value $f^<em>(a)$ when $a = 1.01$ , and verify this by computing the optimal value function $f^</em>(a)$ .
b) Maximize the function

$f(a, u) = 4u^2 + 2u + a^2 - 2a,$

with respect to $a > 0$ .

See Answer →

Question:

In a random sample of 400 university students, 300 failed the examination. In another sample of 500 affiliated college students, 300 failed the same examination. Find the standard error (S.E.) of the difference between the proportion of failures:

i) In university departments.

ii) Between university departments and affiliated colleges.

See Answer →

Question:

What is a standard error, and why is it important?

See Answer →

Question:

Let $J(y)$ be the functional defined by

$J(y) = \int_0^1 \left( y'^2 - y^2 + 2ty \right) \, dt,$

with boundary conditions $y(0) = 0$ and $y(1) = 1$ . Find the extremals in the interval $[0, 1]$ .
b) Find the extremals for

$\int_0^1 \left( [x'(t)]^2 + 10x(t)t \right) \, dt,$

subject to $x(0) = 2$ and $x(1) = 3$ .

See Answer →

Question:

Consider the fishing optimal control problem defined by

$\dot{P}_t = aP_t - bP_t x_t,$

where $P_t$ is the fish population at time $t$ , and $x_t$ is the fishing intensity or catch ( $a$ and $b$ are constants). If $r$ is the discount rate, the objective function is given by

$V(x) = \int_0^\infty e^{-rt} u(x_t) \, dt,$

where the utility of consumption is $c_t = x_t$ .
a) State the transversality condition.

b) Find the optimal consumption $x_t$ , if $u(x_t) = \ln(x_t)$ .

See Answer →

Question:

Consider an investor who, at time $t = 0$ , is endowed with an initial capital of $x(0) = x_0 > 0$ . At any time $t \in [0, T]$ , where $T > 0$ is given, the investor decides their rate of consumption $c(t) \in [0, \bar{c}]$ , where $\bar{c} > 0$ is the maximum allowable rate of consumption. The capital stock evolves according to

$\dot{x} = \alpha x - c(t),$

where $\alpha > 0$ is the rate of return. The investor's time- $t$ utility for consuming at rate $c(t)$ is $u(c(t))$ , where $u: \mathbb{P}_+ \to \mathbb{P}$ is an increasing, strictly concave utility function. The investor's problem is to find a consumption plan $c(t), t \in [0, T]$ , that maximizes the discounted utility