Question

For functions f and g, where f(x) = 1 + x 2 , g(x) = x − 1 in C[0, 1] with the inner product defined by the integral: hf , gi = Z 1 0 f(x)g(x)dx, (a) find the norm of f and g. (b) find unit vectors in the directions of f and g. (c) find the cosine of the angle θ between f and g. (d) find the orthogonal projection of f along g

Answer #1

Let f and g be measurable unsigned functions on R^d . Assume
f(x) ≤ g(x) for almost every x. Prove that the integral of f dx ≤
Integral of g dx.

Let V={f∈C1([−1,1])|f(1) =f(−1)}, then〈f, g〉=∫(f g+f'g′)dx (from
-1 to 1) is an inner product on V (you do not have to verify
this).
Under this inner product, prove that:
{x2}⊥={f∈V | ∫(x^2−2)f(x)dx= 0} (from -1 to 1)

1. V is a subspace of inner-product space R3,
generated by vector
u =[2 2 1]T and v
=[ 3 2 2]T.
(a) Find its orthogonal complement space V┴ ;
(b) Find the dimension of space W = V+ V┴;
(c) Find the angle θ between u and
v and also the angle β between u
and normalized x with respect to its 2-norm.
(d) Considering v’ =
av, a is a scaler, show the
angle θ’ between u and...

a). Find dy/dx for the following integral.
y=Integral from 0 to cosine(x) dt/√1+ t^2 ,
0<x<pi
b). Find dy/dx for tthe following integral
y=Integral from 0 to sine^-1 (x) cosine t dt

1. a True or False? If ∫ [ f ( x ) ⋅ g ( x ) ] d x = [ ∫ f ( x )
d x ] ⋅ [ ∫ g ( x ) d x ]. Justify your answer.
B. Find ∫ 0 π 4 sec 2 θ tan 2 θ + 1 d θ
C. Show that ∫ 0 π 2 sin 2 x d x = ∫ 0 π 2 cos...

suppose and are functions that are differentiable at x=0 and
that f(1)=2, f'(1)=-1, g(1)=-2, and g'(1)=3. Find the value of
h'(1).
1) h(x)=f(x) g(x)
2) h(x)=xf(x) / x+g(x)

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 f, g : X −→ C denote continuous functions from the open
subset X of C. Use the properties of limits given in section 16 to
verify the following:
(a) The sum f+g is a continuous function. (b) The product fg is
a continuous function.
(c) The quotient f/g is a continuous function, provided g(z) !=
0 holds for all z ∈ X.

The functions f and g are defined as ?f(x)=10x+33 and
?g(x)=11?55x.
?a) Find the domain of? f, g, fplus+?g, f??g, ?fg, ff, f/ g, and
g/f.
?b) Find ?(f+?g)(x), (f??g)(x), ?(fg)(x), (ff)(x),(f/g)(x), and
(g/f)(x).

Consider the functions f(x) and g(x), for which f(0)=2, g(0)=4,
f′(0)=12 and g′(0)= -2
find h'(0) for the function h(x) = f(x)/g(x)

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