Question

The functions

f_{1}(x) = x

and

f_{2}(x) = x^{6}

are orthogonal on

[−4, 4].

Find constants

C_{1}

and

C_{2}

such that

f_{3}(x) = x + C_{1}x^{2} +
C_{2}x^{3}

is orthogonal to both

f_{1}

and

f_{2}

on the same interval.

Answer #1

Hey mate!

I hope you understand all the steps.

If you still have any questions please let me know in comments.

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?

Consider the following functions.
f1(x) = x, f2(x) =
x2, f3(x) = 6x −
4x2
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} =

Let f1, f2, f3: [a,b] -->R be nonnegative concave functions
such that f1(a) = f2(a) = f3(a) = f1(b) = f2(b) = f3(b) = 0.
Suppose that max(f1) <= max(f2) <= max(f3).
Prove that: max(f1) + max(f2) <= max(f1+f2+f3)

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?

Find the derivative of each function. (a) F1(x) = 3(x4 + 5)5 2
F1'(x) = (b) F2(x) = 3 2(x4 + 5)5 F2'(x) = (c) F3(x) = (3x4 + 5)5 2
F3'(x) = (d) F4(x) = 3 (2x4 + 5)5

Determine whether the given functions are linearly dependent or
linearly independent.
f1(t) =
4t − 7,
f2(t) =
t2 + 1,
f3(t) =
6t2 − t,
f4(t) =
t2 + t + 1
linearly dependentlinearly independent
If they are linearly dependent, find a linear relation among them.
(Use f1 for f1(t),
f2 for f2(t),
f3 for f3(t), and
f4 for f4(t).
Enter your answer in terms of f1,
f2, f3, and
f4. If the system is independent, enter
INDEPENDENT.)

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.

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.

In addition to NAND and NOR, find four more two-input boolean
functions that are each individually universal. Give a logic
expression for each of your functions, using AND, OR, and NOT, and
prove that each of these functions is individually universal, by
showing for example that you can implement AND, OR, and NOT with
your function. You may use the constant values zero and one as
inputs to your universal functions to implement other functions.
Name your functions f1, f2,...

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