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 can use the Fundamental Theorem of Calculus:

We already found

in part (a), and we can find

by substituting

into

:

Now, we can find the definite integral:

(d) To find the extrapolated value

, we can substitute

into

:

(e) To find the coefficients of

, we can use the fact that

passes through the given points

,

, and

. We can set up a system of equations using these points and solve for the coefficients.Let

. Then, we have the following equations:

Solving this system of equations will give us the coefficients

,

, and

.

Soru