4. Example-4 `x=root(3)(48)`
Find `root(3)(48)` using Bisection method
Solution:
Let `x=48^(1/3)`
`:.x^3=48`
`:.x^3-48=0`
i.e. `f(x)=x^3-48`
Here
| `x` | 0 | 1 | 2 | 3 | 4 |
|---|
| `f(x)` | -48 | -47 | -40 | -21 | 16 |
|---|
`1^(st)` iteration :
Here `f(3) = -21 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `3` and `4`
`x_0=(3+4)/2=3.5`
`f(x_0)=f(3.5)=3.5^3-48=-5.125 < 0`
`2^(nd)` iteration :
Here `f(3.5) = -5.125 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `3.5` and `4`
`x_1=(3.5+4)/2=3.75`
`f(x_1)=f(3.75)=3.75^3-48=4.7344 > 0`
`3^(rd)` iteration :
Here `f(3.5) = -5.125 < 0` and `f(3.75) = 4.7344 > 0`
`:.` Now, Root lies between `3.5` and `3.75`
`x_2=(3.5+3.75)/2=3.625`
`f(x_2)=f(3.625)=3.625^3-48=-0.3652 < 0`
`4^(th)` iteration :
Here `f(3.625) = -0.3652 < 0` and `f(3.75) = 4.7344 > 0`
`:.` Now, Root lies between `3.625` and `3.75`
`x_3=(3.625+3.75)/2=3.6875`
`f(x_3)=f(3.6875)=3.6875^3-48=2.1414 > 0`
`5^(th)` iteration :
Here `f(3.625) = -0.3652 < 0` and `f(3.6875) = 2.1414 > 0`
`:.` Now, Root lies between `3.625` and `3.6875`
`x_4=(3.625+3.6875)/2=3.6562`
`f(x_4)=f(3.6562)=3.6562^3-48=0.8773 > 0`
`6^(th)` iteration :
Here `f(3.625) = -0.3652 < 0` and `f(3.6562) = 0.8773 > 0`
`:.` Now, Root lies between `3.625` and `3.6562`
`x_5=(3.625+3.6562)/2=3.6406`
`f(x_5)=f(3.6406)=3.6406^3-48=0.2534 > 0`
`7^(th)` iteration :
Here `f(3.625) = -0.3652 < 0` and `f(3.6406) = 0.2534 > 0`
`:.` Now, Root lies between `3.625` and `3.6406`
`x_6=(3.625+3.6406)/2=3.6328`
`f(x_6)=f(3.6328)=3.6328^3-48=-0.0566 < 0`
`8^(th)` iteration :
Here `f(3.6328) = -0.0566 < 0` and `f(3.6406) = 0.2534 > 0`
`:.` Now, Root lies between `3.6328` and `3.6406`
`x_7=(3.6328+3.6406)/2=3.6367`
`f(x_7)=f(3.6367)=3.6367^3-48=0.0982 > 0`
`9^(th)` iteration :
Here `f(3.6328) = -0.0566 < 0` and `f(3.6367) = 0.0982 > 0`
`:.` Now, Root lies between `3.6328` and `3.6367`
`x_8=(3.6328+3.6367)/2=3.6348`
`f(x_8)=f(3.6348)=3.6348^3-48=0.0208 > 0`
`10^(th)` iteration :
Here `f(3.6328) = -0.0566 < 0` and `f(3.6348) = 0.0208 > 0`
`:.` Now, Root lies between `3.6328` and `3.6348`
`x_9=(3.6328+3.6348)/2=3.6338`
`f(x_9)=f(3.6338)=3.6338^3-48=-0.0179 < 0`
`11^(th)` iteration :
Here `f(3.6338) = -0.0179 < 0` and `f(3.6348) = 0.0208 > 0`
`:.` Now, Root lies between `3.6338` and `3.6348`
`x_10=(3.6338+3.6348)/2=3.6343`
`f(x_10)=f(3.6343)=3.6343^3-48=0.0014 > 0`
`12^(th)` iteration :
Here `f(3.6338) = -0.0179 < 0` and `f(3.6343) = 0.0014 > 0`
`:.` Now, Root lies between `3.6338` and `3.6343`
`x_11=(3.6338+3.6343)/2=3.634`
`f(x_11)=f(3.634)=3.634^3-48=-0.0082 < 0`
`13^(th)` iteration :
Here `f(3.634) = -0.0082 < 0` and `f(3.6343) = 0.0014 > 0`
`:.` Now, Root lies between `3.634` and `3.6343`
`x_12=(3.634+3.6343)/2=3.6342`
`f(x_12)=f(3.6342)=3.6342^3-48=-0.0034 < 0`
`14^(th)` iteration :
Here `f(3.6342) = -0.0034 < 0` and `f(3.6343) = 0.0014 > 0`
`:.` Now, Root lies between `3.6342` and `3.6343`
`x_13=(3.6342+3.6343)/2=3.6342`
`f(x_13)=f(3.6342)=3.6342^3-48=-0.001 < 0`
`15^(th)` iteration :
Here `f(3.6342) = -0.001 < 0` and `f(3.6343) = 0.0014 > 0`
`:.` Now, Root lies between `3.6342` and `3.6343`
`x_14=(3.6342+3.6343)/2=3.6342`
`f(x_14)=f(3.6342)=3.6342^3-48=0.0002 > 0`
Approximate root of the equation `x^3-48=0` using Bisection method is `3.6342` (After 15 iterations)
| `n` | `a` | `f(a)` | `b` | `f(b)` | `c=(a + b)/2` | `f(c)` | Update | | 1 | 3 | -21 | 4 | 16 | 3.5 | -5.125 | `a = c` | | 2 | 3.5 | -5.125 | 4 | 16 | 3.75 | 4.7344 | `b = c` | | 3 | 3.5 | -5.125 | 3.75 | 4.7344 | 3.625 | -0.3652 | `a = c` | | 4 | 3.625 | -0.3652 | 3.75 | 4.7344 | 3.6875 | 2.1414 | `b = c` | | 5 | 3.625 | -0.3652 | 3.6875 | 2.1414 | 3.6562 | 0.8773 | `b = c` | | 6 | 3.625 | -0.3652 | 3.6562 | 0.8773 | 3.6406 | 0.2534 | `b = c` | | 7 | 3.625 | -0.3652 | 3.6406 | 0.2534 | 3.6328 | -0.0566 | `a = c` | | 8 | 3.6328 | -0.0566 | 3.6406 | 0.2534 | 3.6367 | 0.0982 | `b = c` | | 9 | 3.6328 | -0.0566 | 3.6367 | 0.0982 | 3.6348 | 0.0208 | `b = c` | | 10 | 3.6328 | -0.0566 | 3.6348 | 0.0208 | 3.6338 | -0.0179 | `a = c` | | 11 | 3.6338 | -0.0179 | 3.6348 | 0.0208 | 3.6343 | 0.0014 | `b = c` | | 12 | 3.6338 | -0.0179 | 3.6343 | 0.0014 | 3.634 | -0.0082 | `a = c` | | 13 | 3.634 | -0.0082 | 3.6343 | 0.0014 | 3.6342 | -0.0034 | `a = c` | | 14 | 3.6342 | -0.0034 | 3.6343 | 0.0014 | 3.6342 | -0.001 | `a = c` | | 15 | 3.6342 | -0.001 | 3.6343 | 0.0014 | 3.6342 | 0.0002 | `b = c` |
This material is intended as a summary. Use your textbook for detail explanation.
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