Question

Consider the following functions.

f_{1}(x) = x, f_{2}(x) =
x^{2}, f_{3}(x) = 6x −
4x^{2}

g(x) = c_{1}f_{1}(x) +
c_{2}f_{2}(x) + c_{3}f_{3}(x)

Solve for

c_{1}, c_{2},

and

c_{3}

so that

g(x) = 0

on the interval

(−∞, ∞).

If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.)

{c_{1}, c_{2}, c_{3}} =

Answer #1

Consider the following functions.
f1(x) = x, f2(x) = x-1, f3(x) = x+4
g(x) = c1f1(x) + c2f2(x) + c3f3(x)
Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =?
Determine whether f1, f2, f3 are linearly independent on the
interval (−∞, ∞).
linearly dependent or linearly independent?

The functions
f1(x) = x
and
f2(x) = x6
are orthogonal on
[−4, 4].
Find constants
C1
and
C2
such that
f3(x) = x + C1x2 +
C2x3
is orthogonal to both
f1
and
f2
on the same interval.

Determine if the set of functions is linearly independent:
1. f1(x)=cos2x, f2(x)=1, f3(x)=cos^2 x
2. f1(x)=e^ x, f2(x)=e^-x, f3(x)=senhx

Consider the following predicate formulas.
F1: ∀x ( P(x) → Q(x) )
F2: ∀x P(x) → Q(x)
F3: ∃x ( P(x) → Q(x) )
F4: ∃x P(x) → Q(x)
For each of the following questions, answer Yes or No &
Justify briefly
. (a) Does F1 logically imply F2?
(b) Does F1 logically imply F3?
(c) Does F1 logically imply F4?
(d) Does F2 logically imply F1?

Write a Matlab script that plots the following functions over 0
≤ x ≤ 5π:
f1(x) = sin2 x − cos x,
f2(x) = −0.1 x 3 + 2 x 2 + 10,
f3(x) = e −x/π ,
f4(x) = sin(x) ln(x + 1).
The plots should be in four separate frames, but all four frames
should be in one figure window. To do this you can use the subplot
command to create 2 × 2 subfigures.

QUESTION 2
Consider the differential equation:
x2 y'' - 4 x y' + 6 y = 4 x3
If yc= c1 x2 + c2
x3, then yp(1) equals
(enter only a number; yp(1) is the particular
solution for the differential equation, evaluated at 1)

1. For the following, make sure you explain which basic
functions in F1-F3 you are using, and how exactly you are applying
the operations O1-O3.
(a) Show that every constant function g : N → N is recursive.
You may need to use induction.
(b) Show that the function f : N → N such that f(x) = x 3 is
primitive recursive.

For each of the following functions fi(x), (i) verify that they
are legitimate probability density functions (pdfs), and (ii) find
the corresponding cumulative distribution functions (cdfs) Fi(t),
for all t ? R.
f1(x) = |x|, ? 1 ? x ? 1
f2(x) = 4xe ?2x , x > 0
f3(x) = 3e?3x , x > 0
f4(x) = 1 2? ? 4 ? x 2, ? 2 ? x ? 2.

Find the solution of each of the following initial-value
problems.
x'=y+f1(t), x(0)=0
y'=-x+f2(t), y(0)=0

f(x)=x2+9 and g(x)=x2-8, find the
following functions. a. (f * g) (x); b. (g * f)(x) c. (f *
g)(4);
d. (g * f)(4)

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