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)

(a) To find $P(4)$, we substitute $x=4$ into the polynomial $P(x)$:<br /><br />$P(4) = -0.02(4)^3 + 0.1(4)^2 - 0.2(4) + 1.66$<br /><br />$P(4) = -0.02(64) + 0.1(16) - 0.2(4) + 1.66$<br /><br />$P(4) = -1.28 + 1.6 - 0.8 + 1.66$<br /><br />$P(4) = 1.18$<br /><br />(b) To find $P'(4)$, we first need to find the derivative of $P(x)$:<br /><br />$P'(x) = \frac{d}{dx}(-0.02x^3 + 0.1x^2 - 0.2x + 1.66)$<br /><br />$P'(x) = -0.06x^2 + 0.2x - 0.2$<br /><br />Now, we substitute $x=4$ into $P'(x)$:<br /><br />$P'(4) = -0.06(4)^2 + 0.2(4) - 0.2$<br /><br />$P'(4) = -0.06(16) + 0.8 - 0.2$<br /><br />$P'(4) = -0.96 + 0.8 - 0.2$<br /><br />$P'(4) = -0.36$<br /><br />(c) To find the definite integral of $P(x)$ taken over $[1,4]$, we can use the Fundamental Theorem of Calculus:<br /><br />$\int_{1}^{4} P(x) \, dx = P(4) - P(1)$<br /><br />We already found $P(4)$ in part (a), and we can find $P(1)$ by substituting $x=1$ into $P(x)$:<br /><br />$P(1) = -0.02(1)^3 + 0.1(1)^2 - 0.2(1) + 1.66$<br /><br />$P(1) = -0.02 + 0.1 - 0.2 + 1.66$<br /><br />$P(1) = 1.54$<br /><br />Now, we can find the definite integral:<br /><br />$\int_{1}^{4} P(x) \, dx = 1.18 - 1.54$<br /><br />$\int_{1}^{4} P(x) \, dx = -0.36$<br /><br />(d) To find the extrapolated value $P(5.5)$, we can substitute $x=5.5$ into $P(x)$:<br /><br />$P(5.5) = -0.02(5.5)^3 + 0.1(5.5)^2 - 0.2(5.5) + 1.66$<br /><br />$P(5.5) = -0.02(166.375) + 0.1(30.25) - 0.2(5.5) + 1.66$<br /><br />$P(5.5) = -3.3275 + 3.025 - 1.1 + 1.66$<br /><br />$P(5.5) = -0.7325$<br /><br />(e) To find the coefficients of $P(x)$, we can use the fact that $P(x)$ passes through the given points $(1,1.54)$, $(2,1.5)$, $(3,1.42)$, and $(5,0.66)$. We can set up a system of equations using these points and solve for the coefficients.<br /><br />Let $P(x) = ax^3 + bx^2 + cx + d$. Then, we have the following equations:<br /><br />$1.54 = a(1)^3 + b(1)^2 + c(1) + d$<br /><br />$1.5 = a(2)^3 + b(2)^2 + c(2) + d$<br /><br />$1.42 = a(3)^3 + b(3)^2 + c(3) + d$<br /><br />$0.66 = a(5)^3 + b(5)^2 + c(5) + d$<br /><br />Solving this system of equations will give us the coefficients $a$, $b$, $c$, and $d$.