Determine the Maclaurin polynomial of minimal degree needed

Determine the Maclaurin polynomial of minimal degree needed to calculate sin(1) to an accuracy of 3 decimal places. 2. Let f(x) = (x - 1)^7 e^x. Calculate f^(50)(1).

Solution

There are mutiple question s given. Only one can be answered here at a time. It is #1 therefore, as given below:

#1 To calculate sin(1) using Maclaurin polynomial

Let f(x) = sin x, so that

f\'(x)= cosx ; f\" (x)= -sin x ; f\'\'\' (x) = -cos x and so on.

Hence f(0)=0; f\'(0)=1; f\"(0)=0; f\'\'\'(0)= -1 ; f4(0)=0; f5(0)= 1...

In general, nth Maclaurin Polynomial for a function f(x) is given by

Pn(x) = f(0) + f\'(0) x + f\" (0) x²/2! + f\'\'\'(0) x³/3! +....

In the present case it would thus be Sin x = 0 + x +0- x³/3! +0 +x⁵/5! -....

Thus P₁= x ; P₃= x- x³/3!; P₅ = x-x³/3! +x⁵/5!: P₇ = x-x³/3! +x⁵/5! - x⁷/7!..

(P₂, P₄... will all be 0)

P₁ gives sin1 =1; P₃ gives sin1= 1- 1/6=5/6=0.833333.., P₅ would give sin 1= 1-1/6 +1/120=0.841666 and P₇ would give sin1 = 1-1/6 +1/120 -1/5040 = 0.841468

The value of sin 1 is given by0.841471(approx) , where only the last digit has been rounded. It would thus be seen that P₅ is the minimal degree polynomial which gives sin(1) to the accuraccy of 3 decimal places.

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