Let f(x) = x - x^3/6 - sin(x). (a) What is the 5th order...

Let f(x) = x - x^3/6 - sin(x). (a) What is the 5th order Taylor polynomial P_5(x) of f(x) about a = 0. (b) Find a "reasonable" upper-bound on the error in approximating f(x) by P_5(x) valid for every value of x such that |x| <= 0.1.

Transcript

00:01 Hi, given fx is equal to x minus x cubed by 6 minus sine x.

00:07 Now in part a, we have to write what is the fifth order polynomial, that is p5x for this fx about a is equal to 0.

00:17 So now fx by taylor series expansion is equal to f of a plus f dash a by 1 factorial into x minus a plus f double derivative at a by 2 factorial.

00:30 Into x minus a whole square and so on so in this case at a is equal to zero f x is equal to now putting a to x to be zero this becomes zero plus d by d x of x minus x cube by six minus sine x at this point zero divided by one factorial into x plus d by d x d square x d square by d x of double derivative of x minus x cubed by 6 minus sine x at this point 0 by 2 factorial into x square and so on so now on expanding this we see that first four terms will be zero and then fifth term will be minus x to the power 5 by 120 so therefore we had we had to write the fifth degree fifth order polynomial so fifth order polynomial will be equal to minus x to the power 5 by 120 now in part b we have to find the error bound error so now error is given by error is given as r n x magnitude is less than m by n plus 1 factorial into x minus to the power n plus 1 and f n plus 1 x derivative is less than equal to m for mod x minus a less than d so now we have to find r5 of x so r5 of x from this formula must be less than equal to m by n plus 1 that is 6 factorial into x minus a to the power 6 and mod of of f six derivative at x must be less than equal to m for all mod x less than 0 .1 now this is given to us so now we have now mod f's sixth derivative of f must be is equal to sine x so now mod on both sides must be equal to mod of sine x and this is always less than equal to one so now if we put m is equal to 1 so we have r5 of x less than equal to 1 by 6 factorial into x minus a.

03:09 So a is 0.

03:11 So this becomes x to the par 6, mod x to the par 6.

03:14 And now which is further less than 0 .1...

Question

Let f(x) = x - x^3/6 - sin(x). (a) What is the 5th order Taylor polynomial P_5(x) of f(x) about a = 0. (b) Find a "reasonable" upper-bound on the error in approximating f(x) by P_5(x) valid for every value of x such that |x| <= 0.1.

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