Give the correct answer and solve the given equation Let p(x)=2+xandq(x)=x. Using the inner product ⟨ p, q⟩=∫−11pqdx find all polynomials r(x)=a+bx∈P1(R)P (R) such that {p(x), q(x), r(x)} is an orthogonal set.

Notice that {p,q}=∫−11(2+x)xdx=∫−112xdx+∫−11x2dx =x2|−11+x33|−11 =(12−(−1)2)+(133−(−1)33) =(1−1)+(13−−13) =2/3 Thus,{p,q}≠0, so the set {p(x), q(x), r(x)} cannot be orthogonal There exist none such r(x)

Answered: "Give the correct answer and solve the given..."

College
Calculus and Analysis
Integral Calculus
Integrals

UkusakazaL

Answered question

2020-11-01

Give the correct answer and solve the given equation
Let

p (x) = 2 + x and q (x) = x

. Using the inner product

⟨ p, q ⟩ = \int_{- 1}^{1} p q d x

find all polynomials

r (x) = a + b x \in P 1 (R) P

(R) such that {p(x), q(x), r(x)} is an orthogonal set.

Answer & Explanation

mhalmantus

Skilled2020-11-02Added 105 answers

Notice that

{p, q} = \int_{- 1}^{1} (2 + x) x d x = \int_{- 1}^{1} 2 x d x + \int_{- 1}^{1} x^{2} d x

= x^{2} {|_{- 1^{1} +} \frac{x^{3}}{3} |}_{- 1}^{1}

= (1^{2} - {(- 1)}^{2}) + (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})

= (1 - 1) + (\frac{1}{3} - \frac{- 1}{3})

=2/3 Thus,

{p, q} \neq 0

,
so the set {p(x), q(x), r(x)} cannot be orthogonal
There exist none such r(x)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Integral Calculus

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?
How to expand and simplify $2 (3 x + 4) - 3 (4 x - 5)$ ?
Find an equation equivalent to $x^{2} - y^{2} = 4$ in polar coordinates.
How to graph $r = 5 \sin θ$ ?
How to find the length of a curve in calculus?
When two straight lines are parallel their slopes are equal.
A)True;
B)False
Integration of 1/sinx-sin2x dx
Converting percentage into a decimal. $8.5 %$
Arrange the following in the correct order of increasing density.
Air
Oil
Water
Brick
What is the exact length of the spiraling polar curve $r = 5 e^{2 θ}$ from 0 to $2 π$ ?
What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?
What is the slope of the tangent line of $r = - 2 \sin (3 θ) - 12 \cos (\frac{θ}{2})$ at $θ = \frac{- π}{3}$ ?
How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?
Use the summation formulas to rewrite the expression $Σ \frac{2 i + 1}{n^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.
How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?

Didn't find what you were looking for?