Reference no: EM132318403
Mathematical Analysis Assignment Questions -
Question 1 -
a) Let f : [a, b] → R be continuous. Prove that f is bounded.
b) State (but do not prove) the Mean Value Theorem. Deduce that if f is differentiable on R with derivative 0 then f is constant.
c) Let c: R → R be a twice differentiable function satisfying
c''(x) = -c(x), for all x
and c(0) = 1, c'(0) = 0.
Let f (x) = c'(x) cos x + c(x) sin x for x ∈ R.
Show that f(x) = 0 for all real x.
d) With c as in the previous part consider the function u given by
u(x) = c(x)/cos x, for - π/2 < x < π/2
and show that c(x) = cos x for - π/2 < x < π/2.
Question 2 -
a) Show that the maximum value of (1 - x)ex on the real line is 1.
b) State (but do not prove) 1'Hopital's rule for computing the limit of a ratio of two differentiable functions
limx→c f(x)/g(x).
c) Consider the function f defined by
![1444_figure.png](https://secure.expertsmind.com/CMSImages/1444_figure.png)
(i) Prove that f is differentiable at 0 and find its derivative there.
(ii) Show that f' is non-negative at all points other than 0.
Question 3 -
a) Assuming that log t ≤ t - 1 for all t > 0 show that
log t ≥ 1 - 1/t
for all t > 0.
b) Deduce that for all t > 0
t log t ≥ t - 1.
c) Let p1, p2, . . . , pm be positive numbers with 1∑mpi =1.
Show that
1∑mpilog(mpi) ≥ 0.
d) With p1, p2, . . . , pm as before show that
1∑mpilogpi ≥ -logm.
e) Find positive numbers p1, p2, . . . , pm adding up to 1 for which
1∑mpilogpi = -log m.
Question 4 -
a) Taylor's Theorem with Lagrange remainder states that if f is n-times differentiable on an open interval containing a and x then
![2080_figure1.png](https://secure.expertsmind.com/CMSImages/2080_figure1.png)
for some t between a and x.
Use this with n = 3 and a = 0 to approximate ex by its quadratic Taylor approximation arid find an expression for the error.
b) Show that if x ≥ 0 then ex ≥ 1 + x + x2/2 but if x < 0 then ex < 1 + x + x2/2.
c) Show that for any x
ex ≤ 1 + x + x2/2 + x3ex/6.
d) Deduce that if x < 61/3
ex ≤ (1 + x + x2/2)/(1 - x3/6).
e) Expand
1/(1 - x3/6)
as a power series centred at 0 and use this to calculate the first 5 terms of the power series for
(1+ x + x2/2)/(1 - x3/6).
Attachment:- Assignment File.rar