Question

a fixed point of a function f is a number c in its domain such
that f(c)=c. use the intermediate value theorem to prove that any
continious function with domain [0,1] and range a subset of [0,1]
must have a fixed point.[hint: consider the function f(x)-x]

“Recall the intermediate value theorem:suppose that f is
countinous function with domain[a,b]and let N be any number between
f(a)and f(b), where f(a)not equal to f(b). Then there exist at
least one number c in [a,b] such that f(c)=N

Answer #1

(i) Use the Intermediate Value Theorem to prove that there is a
number c such that 0 < c < 1 and cos (sqrt c) = e^c- 2.
(ii) Let f be any continuous function with domain [0; 1] such
that 0smaller than and equal to f(x) smaller than and equal to 1
for all x in the domain. Use the Intermediate Value Theorem to
explain why there must be a number c in [0; 1] such that f(c)
=c

4. Let f be a function with domain R. Is each of the following
claims true or false? If it is false, show it with a
counterexample. If it is true, prove it directly from the FORMAL
DEFINITION of a limit.
(a) IF limx→∞ f(x) = ∞ THEN limx→∞ sin (f(x)) does
not exist.
(b) IF f(−1) = 0 and f(1) = 2 THEN limx→∞ f(sin(x)) does not
exist.

4) For each of the following functions, establish on
which subset of its domain it is C1. (You do not need to provide
formal proofs in your answers: an explanation or brief graphical
argument is sufficient.) Compute partial derivatives for these
subsets.
(a) f(x) = ⌈x⌉ (recall ⌈x⌉ is the lowest integer above
or equal to x).
(b) f(x, y) = min{x, y}.
(c) f(x, y) = xαyβ where α, β ∈ (0,1).

4. Given the function f(x)=x^5+x-1, which of the following is
true?
The Intermediate Value Theorem implies that f'(x)=1 at some
point in the interval (0,1).
The Mean Value Theorem implies that f(x) has a root in the
interval (0,1).
The Mean Value Theorem implies that there is a horizontal
tangent line to the graph of f(x) at some point in the interval
(0,1).
The Intermediate Value Theorem does not apply to f(x) on the
interval [0,1].
The Intermediate Value Theorem...

Consider the function f(x) = c/x , where c is a nonzero real
number. What is the vertical asymptote, the horizontal asymptote,
the domain and range?

1a.
Find the domain and range of the function. (Enter your answer
using interval notation.)
f(x) = −|x + 8|
domain=
range=
1b.
Consider the following function. Find the composite
functions
f ∘ g
and
g ∘ f.
Find the domain of each composite function. (Enter your domains
using interval notation.)
f(x) =
x − 3
g(x) = x2
(f ∘ g)(x)=
domain =
(g ∘ f)(x) =
domain
are the two functions equal?
y
n
1c.
Convert the radian...

Let C [0,1] be the set of all continuous functions from [0,1] to
R. For any f,g ∈ C[0,1] define dsup(f,g) =
maxxE[0,1] |f(x)−g(x)| and d1(f,g)
= ∫10 |f(x)−g(x)| dx. a) Prove that for any
n≥1, one can find n points in C[0,1] such that, in dsup
metric, the distance between any two points is equal to 1. b) Can
one find 100 points in C[0,1] such that, in d1 metric,
the distance between any two points is equal to...

a) Find the domain and range of the following function:
f(x,y)=sin(ln(x+y))
b) sketch the domain.
c) on seperate graph, sketch three level curves

A function f is said to be Borel measurable provided its domain
E is a Borel set and for each c, the set {x in E l f(x) > c} is
a Borel set. Prove that if f and g are Borel measurable functions
that are defined on E and are finite almost everywhere on E, then
for any real numbers a and b, af+bg is measurable on E and fg is
measurable on E.

Let f:[0,1]——>R be define by f(x)= x if x belong to rational
number and 0 if x belong to irrational number and let g(x)=x
(a) prove that for all partitions P of [0,1],we have
U(f,P)=U(g,P).what does mean about U(f) and U(g)?
(b)prove that U(g) greater than or equal 0.25
(c) prove that L(f)=0
(d) what does this tell us about the integrability of f ?

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