Problem 1 For this problem, let the inner product of f and g be defined as < f,g>= S f(x)g(x) dx. (1) Show that the functions 1 and x – are orthogonal with respect to this inner product. (2) Find a polynomial of degree two that is orthogonal to both 1 and x – . Hint: find numbers a, b, and c for so that a + bx + cx² is orthogonal to both 1 and x . (3) Write x? as a linear combination of 1, x – , and your degree-2 polynomial from part 2.

Q: ) Task 6: Given a scalar-valued function of multiple variables, f(x) = x₁sin(x₂) + x²x3 + x1x2x3 +…

A: The given scalar-valued function is fx=x1sinx2+x22x3+x1x2x3+x3 where x=x1x2x3. The quadratic…

Q: Form the interpolating polynomial of degree k that satisfies pG) (xo) = f(G) (xo) for 0<j< k.

A: To form: The interpolating polynomial of degree k that satisfies pjx0=fjx0 for 0≤j≤k

Q: nsider the following. -2x5 + 9x4-7x³ - 12x = 0, [3,4] ) Explain how we know that the given equation…

A: In this question, we will find the solution of the equation.

Q: Verify that the third Maclaurin polynomial for f (x) = ex sin x is equal to the product of the third…

A:

Q: Show that f (−1) = f (1), but that there is no point in (−1, 1) where f (0) is zero. Why does this…

A: Given: fx=1-x2/3 So, f-1=1--12/3=1-1=0 Also, f1=1-12/3=0 Thus, f-1=f1

Q: Show that f(x) = x4 + 4.x + c has at most two real roots for any umber c.

A: Consider the given equation. fx=x4+4x+c Case 1: When c&gt;0. No sign change, by Descartes rule of…

Q: Obtain the Lagrange interpolating polynomial for a certain function f of which we know that: f (-1)…

A: Lagrange interpolating polynomial: The Lagrange interpolating polynomial is a degree polynomial that…

Q: The following data is given: 2.5 3 4 5 1 7 8 1 a) Write the polynomial in Lagrange form that passes…

A:

Q: Let P0. (x) denote the quadratic polynomial that interpolates the data {(xo, yo), (x1, y1), (x2,…

A:

Q: of A, B, and C.

A: We are given that the polynomial function is of the form f(x) = Ax2+Bx+C. We need to find the values…

Q: Suppose we want to form an interpolant at equally spaced points xo = 0, x1 = h, and x2 = 2h of the…

A:

Q: points that correspond to the functions s(t), v(t), and a(t) to any point on the curves that match…

A: We will use relation between a function its 1st derivative and 2nd derivative to identify which…

Q: The point (3, 7) is a

A: Follow the procedure given below.

Q: 2. (а) Find the function P(x) = a +b cos(n x) +c sin(nx), which interpolates the data 0.5 1 y 4 (b)…

A:

Q: Use a Lagrange interpolating polynomial to approximate the derivative of the function tabulated…

A: Consider the given information: To use Lagrange interpolating polynomial to approximate the…

Q: Find a polynomial f with the following properties: i) f is concave up on (-, –2) U (0, ∞) and…

A: Function and derivative

Q: For the polynomial below, 1 is a zero. h(x) =x- 7x + 10x- 4 x=(x)4 Express h (x) as a product of…

A:

Q: i. 0 (the zero vector) is in H ii. It is closed under addition. i.e. if u, v are in H, then u + v…

A: We have the set of all polynomials of at most second degree ,…

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.