Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) =
sin(?? + ?? + ?2 ).
1. Find a bound for the directional derivative of ? in any
direction, i.e. find a constant ? such that |???(?, ?)|
≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ 2 with
|?| = 1.
4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 =
?2}. Prove that ? is an equivalence relation, and
concisely characterize how its equivalence classes are different
from simple real-number equality.
Define a function ?∶ ℝ→ℝ by
?(?)={?+1,[?] ?? ??? ?−1,[?]?? ????
where [x] is the integer part function. Is ? injective?
(b) Verify if the following function is
bijective. If it is bijective, determine its inverse.
?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)
Topology question:
Show that a function f : ℝ → ℝ is continuous in the ε − δ
definition of continuity if and only if, for every x ∈ ℝ and every
open set U containing f(x), there exists a neighborhood V of x such
that f(V) ⊂ U.
Consider the user-defined MATLAB function below,
ht_mp_ch().
The function ht_mp_ch(), outputs the sampled version of the impulse
response hmp(t) of the multipath fading channel as a
vector.
function
impulse_response=ht_mp_ch(max_delay,L,decay_base,t_step)
t_vector=0:t_step:max_delay;
mp_tmp=0*(t_vector);
path_delays=[0 sort(rand(1,L-1)*max_delay)];
impulse_positions=floor(path_delays/t_step);
mp_tmp(impulse_positions+1)=exp(j*2*pi*rand(1,L));
mp_tmp=mp_tmp.*(decay_base.^(t_vector/max_delay));
impulse_response=mp_tmp/sqrt(sum(abs(mp_tmp).^2));
Explain what the variable on the left-hand side represents and
justify how the right-hand side expression is formulated by adding
comments to every line.
Consider a firm using a technology defined by the linear
production function = q = f(K, L) = K + L
Which of the following statements are correct? There might be
multiple
a) The elasticity of output with respect to capital is equal to
the inverse of the average product of capital
b) The elasticity of output with respect to labor is
constant
c) The elasticity of long-run cost with respect to output is
equal to 1
d) The elasticity...
Consider the Basic growth model from your text and lecture as
defined below: Production function: GDP = f(At, Kt, Lt); where
capital (K) and labor (L) are substitutes and technology (A) aids
production Demographic Behavior: Lt+1= Lt(1+n) = Nt+1; where n is
the population growth rate, N is the total population and there is
no unemployment Capital and Savings/Investment Dynamics: Kt+1 = It
+ Kt(1-?); It = St = s*Y; where ? is the depreciation rate of
capital, s is...
Consider the function f(x)= 7 - 7x^2/3 defined on the interval
[-1, 1]. State which of the three hypotheses of Rolle’s Theorem
fail(s) for f(x) on the given interval.
Prove that the function defined to be 1 on the Cantor
set and 0 on the complement of the Cantor set is discontinuous at
each point of the Cantor set and continuous at every point of the
complement of the Cantor set.