Exercises Flashcards
2.2
Let f : R -> R be uniformly continuous. Show that there exists a, b > 0 such that |f(x)| <= a|x| + b for every x \in R.
2.3
Example of f_n -> f differentiable, but f’_n -/-> f’
2.4
Example f_n -> f cts. but not diff.
3.1
f_n(x) = x/(1+nx^2), determine pointwise and uniform convergence of f_n and f’_n
3.2
f_n cts with f_n ->-> f. for every convergent sequence x_n -> x, we have f_n(x_n) -> f(x).
counter example if we only assume pointwise.
3.3a
study uniform and pointwise convergence of f_n(x) = x^2/(n^2 + x^2) on R and [0,1].
3.3b
Study uniform and pointwise convergence of g_n(x) = x^n/(n+x^n)
3.4(a)
if f_n -> on compact K, is it true that f_n ->-> f on K.
3.4(e)
If f_n -> f on interval, and f_n increasing, is it true that f also increasing.
4.2(a)
Show that g(x) = \sum_{n=1}^{\infty} \cos{2^{n}x}/2^n is cts on R.
4.3(b)
If h(x) = \sum_{n=1}^{\infty} 1/(x^{2} + n^{2}), show that h’ is cts.
4.4
f_n ->-> f and g_n ->-> g on A. does f_n x g_n ->-> fg on A?
4.5
f_n cts. diff. on [a,b], f’->->. show that if there exists x_0 \in [a,b] st f_n(x_0) converges, then f_n ->->
4.6
f_n \in C^1([a,b]) ->-> f, is f diff.
5.3(a)
f:U -> R^k cts at p. Prove that if f(p) \neq 0, then there exists d > 0 such that |f(x)| > \frac{1}{2} |f(p)| for every x \in U s.t. |x-p| < d.
(b)
suppose that k = 1 and that f(x) \neq 0 for every x \in U. prove that 1/f is cts at p.
5.4(b)
g(x,y) = x^3y/(x^2 + y^2)^2. prove that lim_(x,y)->(0,0) g(x,y) does not exist.
5.5
f(x,y) = sin(x-y)/(x-y), f(t,t) = 1, prove that f is cts on R.
6.4
Prove that U = {(a,b,c,d) \in R^4 : |ad - bc| > 1} is open.
7.1(a)
A \in R^n,n , B_j = \sum_{r=0}^{j} A^r/r!.
Prove that B_j is Cauchy wrt operator norm.
(b)
B_j -> e^A, compute e^A when A = (0 t // -t 0)
8.1
for v \in R^n, let Ax = v . x. Prove that ||A||op = ||v||
8.2
Show that if A \in GL(n,R) and ||A^-1||op = u, then ||Ax|| >= 1/u||x|| for every x \in R^n
8.3
Prove that Df(x) is unique.
8.4(d)
verify that f_X(x,y) = 2xy(y^2 - 2x^6)/(x^6 + y^2)^2 is unbounded on B(0, d) for every d>0.
8.5
f:R^n -> R, r:(ab) -> R^n cts. diff. and r solves r’(t) = -\nabla f(r(t)). Prove that if [\alpha, beta] \subset (a,b), then f(r(\alpha)) >= f(r(\beta)), with equality if and only if \nablda f(r(\alpha)) = 0 and r()t) = r(\alpha) for every t \in [\alpha,. \beta].
