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: If a and b are integers, then a/b is always a rational number. True False

A:

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: a. Find the linear approximating polynomial for the following function centered at the given point…

A:

Q: Given a function f(x). Suppose that Newton's interpolating polynomial P₂ (x) of f(x) at the points…

A:

Q: Given a function f(x). Suppose that Newton's interpolating polynomial P;(x) of f(x) at the points Xo…

A:

Q: c) Find the second degree polynomial approximation to f at (a, b) = (1, – 2) T2(x, y)

A: Given :- The second degree polynomial approximation to f at a,b = 1,-2 f x,y = -x2y2x2 +y2

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: 4. Verify Rolle's Theorem for f(x) = x² – 2x +1 on [-1,3]

A:

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: an initial investment of 1500 thousand dol ollars at the end of each of the next two qua The…

A: Given: Initial Investment=$1500000 Further investment=$250000 (Quarterly) Operating cost=$120000…

Q: 4. Let F(x) be a polynomial function such that F(4) =-1 and F'(4) = 0. If x 4, then F'(x) > 0, what…

A: Given that  F(x) be a polynomial function such that F(4) = -1 and F'(4) = 0. Also F'(x) &lt; 0 when…

Q: Let xj = xo + hj for j = 0, 1, 2. Use the quadratic polynomial interpolating f at those nodes to…

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: Let ro, r1,..., r10 be 11 distinct values. Let L10,i(x) (i = %3D 0,..., 10) be the polynomials for…

A: Lagrangian polynomial: If x0,x1,...,xn are any n+1 distinct numbers, then for i=0,1,...,n, the i-th…

Q: . Consider f(2)= (+ i)¹/3. Choose a set of branch cut (s) and sketch it in the complex plane. For…

A: As per the question we are given a complex function as : f(z) = (1/z + i)1/3 And we have to choose a…

Q: If f(x) is known at the following data: xi 1 2 3 4 5 fi 1 4 8 16 32 1. Show the Newton's Forward…

A:

Q: Junction f (x) given that f (2) = 12 f(0) = 2, f(1)= 3, and f(3) = 35. Hence find f(5).

A: f(5)=52 ....

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: 1: On the polynomial space P₁ John chooses the scalar product (flg)₁ = ₁f(x)g(x) dx, while Jim uses…

A:

Q: 2. Consider the Newton polynomial Pn(x) = ao+a1(xr-xo) + a₂(x-xo)(x − x₁) +... +an(x-xo)(x -…

A: Given that   that interpolate then we have Now  ,       putting

Q: 1. Identify and sketch the domain of f(x, y) = √y² + 4x + 2 In(1-x² - y²)

A:

Q: In linear space, consider all true polynomial functions with internal multiplication x(t)y(t)dt for…

A: We will use GRAM-SCHMIDT ORTHOGONALIZATION PROCESS to solve this problem For a basis y0,y1,y2 to…

Q: sketch h(x)=-x2-4 as a translation and reflection of f(x)=x2

A: h(x) =-x2 -4 is transformed and reflected graph of the parent function f(x)=x2 The graph h(x) is…

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.