🔬 Tutorial problems zeta \zeta

🔬 Tutorial problems zeta \(\zeta\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\zeta\).1#

Using L’Hôpital’s rule compute the following limit

\[ \lim _{x \rightarrow \infty} \frac{x^{2}}{e^{x}} \]

\(\zeta\).2#

Consider function \(f \colon X \to \mathbb{R}\) defined by \(f(x) = \frac{1}{x} e^x\).

Find the minimizer(s) and the maximizer(s) of this function on \(X = (0, 2]\).

Follow all the required steps and explain your reasoning.

Review the algorithm for univariate optimization in the lecture notes

\(\zeta\).3#

Find an example of a nonlinear univariate function \(f \colon D \subset \mathbb{R} \to \mathbb{R}\) that:

(a) has exactly one maximizer and one minimizer (b) has has neither a maximizer nor a minimizer (c) has an infinite number of maximizers and minimizers (d) has exactly finite number \(n\) of maximizers and \(n\) minimizers

Remember to define both the function \(f(x)\) and its domain \(D\) for each case.

First, review the relevant definitions. Then, try to draft some ideas on a piece of paper. Think of how they can be expressed in mathematical terms.

\(\zeta\).4#

A square tin plate whose edges are 18 cm long is to be made into an open square box of depth \(x\) cm by cutting out equally sized squares of width \(x\) in each corner and then folding over the edges. Draw a figure, and show that the volume of the box is, for \(x \in [0, 9]\):

\[ V(x) = x(18 - 2x)^2 = 4x^3 - 72x^2 + 324x \]

Also find maximum point of V in \([0, 9]\) and show it is indeed the maximum using second order conditions.

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 8.3, Question 3

\(\zeta\).5#

The portion of families whose income is no more than \(x\), and who have a home computer, is given by

\[ p(x) = a + k (1-e^{-cx}), \]

where \(a\), \(k\), and \(c\) are positive constants. Determine \(p'(x)\) and \(p''(x)\). Does \(p(x)\) have a maximum? Sketch the graph of \(p(x)\) for \(x \geqslant 0\).

\(\zeta\).6#

Find the own price elasticity of demand for each of the following demand or inverse demand functions. If possible, find the price or prices for which the demand curves will: (i) be inelastic, (ii) have unitary elasticity and (iii) be elastic.

(a) \(P=100-2 Q\);

(b) \(Q=200-0.8 P\);

(c) \(P=100 Q^{-1}\);

(d) \(Q=200 P^{-0.8}\);

(e) \(P=50 e^{-0.7 Q}\); and

(f) \(Q=\frac{150}{\ln (P)}\).

[Shannon, 1995], p. 404

Contents