4. Verify whether S(x) is a cubic spline{−(x − 1)³ + 2(x − 1)² + 4(x − 1) + 1S(x)=x³ + x − 1If not, how can it be fixed?Is the original function or its revision a natural cubic spline?Is it a not-a-knot cubic spline?on [0, 1)on [1,2]

A function S(x) is called cubic spline in if S(x) satisfies the following condition.

Answered: 4. Verify whether S(x) is a cubic…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min
9. A polynomial function p(x) has exactly four roots: -2i, 2i,-5, and 1. Which of the following could be p(x)? p(z) = (z + 5) (z – 1) (z² + 4) B p(z) = (x - 5) (z + 1) (z² + 4) C2 p (x) = (x +5) (z - 1) (2 – 4) (p5)(+1)(+4)
Why is this not a cubic spline?
1. Let f(x) be the cubic function graphed below and g(x) be the quadratic function graphed below. Write an expression in terms of f(x) and g(x) that gives the area of shaded region between the two curves. You do not need to find the equations of the curves. f(x) N g(x)
A rectangle in the coordinate plane has its base along the x-axis and two vertices on the parabola defined by y = 9 - x². Which of the following equations represents the area of the rectangle A(x)? (0 ≤ x ≤ 3) O A(x) 2x(9-x²) O A(x) (9 x2)2 ○ A(x) = x² - O A(x) = x(9x2)
et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'? I would be very thankful if you could provide some explanation with the steps, thank you in advance.

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4. Verify whether S(x) is a cubic spline x³ x - 1 S(x) = { ²² + ² 17²³ + 2(x − 1)² + 4(x − 1) + 1 − If not, how can it be fixed? Is the original function or its revision a natural cubic spline? Is it a not-a-knot cubic spline? on [0, 1) on [1,2]