Exercises for Introduction to Interpolation 1. Consider P(x)=-0.02x^3+0.1x^2-0.2x+1.66 which passes through the four points (1,1.54),(2,1.5),(3,1.42) and (5,0.66) (a) Find P(4) (b) Find P'(4) (c) Find the definite integral of P(x) taken over [1,4] (d) Find the extrapolated value P(5.5) (e) Show how to find the coefficients of P(x)

Question

Answer 1

(a) To find

, we substitute

into the polynomial

:

(b) To find

, we first need to find the derivative of

:

Now, we substitute

into

:

(c) To find the definite integral of

taken over

, we integrate

with respect to

from 1 to 4:

To evaluate this integral, we can use the Fundamental Theorem of Calculus. First, we find the antiderivative of

:

Now, we evaluate this antiderivative at the upper and lower limits of integration:

(d) To find the extrapolated value

, we substitute

into the polynomial

:

(e) To find the coefficients of

, we can use the given points to set up a system of equations. Since

passes through the points

,

, and

, we can

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