5. This question concerns the function given by Fin,n)-in-sin(n-n). given by F (

Question

5. This question concerns the function given by Fin,n)-in-sin(n-n). given by F (i,2)12-sin (1- x2). (e) Explain why the solution to the equation F (x1,2)0 may be written in the form a2 f(xi) near the point (0,0). Hint: As part of your explanation you will need to evaluate one of the partial derivatives of F at the appropriate point. point (0, 0). {(yi, ya) E R2 : yl + y (bFind the degree two Taylor polynomial T2 (1.2) for the function F centred at the (c) Show that the error in approximating F by T2 on the unit disk S1 is bounded by 11 marks

Explanation / Answer

concerns the function given by Fin,n)-in-sin(n-n). given by F (i,2)12-sin (1- x2)

F (x1,2)0 may be written in the form a2 f(xi) near the point (0,0)

point (0, 0). {(yi, ya) E R2 : yl + y

This is a linear homogeneous recurrence so solution is of the form

f(n)=r^n

r^2=2r-1

r=1,repeated roots

f(n)=(1)^n(A+Bn) is general solution

f(1)=A+B=1

f(2)=A+2B=5

So, A=-3,B=4

f(n)=-3+4n

Proof

Base case n=3

f(3)=2f(2)-f(1)=2*5-1=9=-3+3*4

So base case is true

Now assume true for all k<=n for some n>=3

We show it is true for n+1

f(n+1)=2f(n)-f(n-1)=2(-3+4n)-(-3+4(n-1))=-6+8n+3-4n+4=1+4n=-3+4(n+1)

HEnce true for all n

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