4Y5 + 3Y 4 - Y+ 9 = 0 How can you solve y from the above polynomials using an ap

Question

4Y5 + 3Y4 - Y+ 9 = 0

How can you solve y from the above polynomials using an approximation method such as Midpoint method or Newton method?

How do midpoint method and Newton method work, and how we shall apply it? Please solve me the above polynomials using both methods. Please write me the steps to follow for both methods. When shall we use the Midpoint method and Newton method, and which one is easier?

Can you please mention me other approximation methods and write a few descriptions of each?

Thanks

Explanation / Answer

Given

3Y2 - 2Y3 -0.95=0 ........(1)

4Y5 +3Y4 -Y +9 =0 .........(2)

3Y2 = 2Y3+0.95 .........(3)

substitute in(1)

4Y5 + 2Y5 +0.95Y2 -Y +9 =0

6 Y5 + 0.95Y2 -Y +9 =0

f(y) = 6Y5+ 0.95Y2 -Y +9 =0

Using midpoint approximation

y=0

f(0) = 9

f(0.1) = 6(0.1)5 + 0.95 (0.1)2 - 0.1+9

= 8.909

f(0.2) = 6(0.2)5 +0.95(0.2)2-0.2+9

= 8.84

f(0.3) = 6(0.3)5 + 0.95(0.3)2 -o.3 +9

= 8.0008

f(0.4) = 6(0.4)5 +0.95(0.4)2 -0.4 +9

= 8.81344

f(0.5) = 6(0.5)5 +0.95(0.5)5-0.5 +9

= 8.925

midpoint method =f(0)+f(0.1) +f(0.2)+f(0.3)+f(0.4)+f(0.5)

= 8.925+8.008+8.8134+9+8.84+8.909

= 52.496

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