Newton's Method for Approximating Zeros Worksheet

**Page 1**

1.

Use Newton's method to find the value of $\sqrt[3]{-6}$ to the nearest thousendth.

a. | - 2.182 | ||

b. | - 1.818 | ||

c. | 1.833 | ||

d. | 1.818 | ||

e. | - 1.833 |

Let,

Draw the graph of the function.

From the graph, the zero of the function is in between - 2 and - 1.

The Newton's approximating sequence is generated by the formula:

[Zero of the function lies between - 2 and - 1]

Similarly,

So, the zero of the function is - 1.818.

Therefore,

Correct answer : (2)

2.

Find the approximate value of $\sqrt{\frac{3}{2}}$ to the nearest thousendth by using Newton's method.

a. | - 1.235 | ||

b. | 1.235 | ||

c. | 0.775 | ||

d. | 0.765 | ||

e. | 1.225 |

Let,

Draw the graph of the function.

From the graph, the approximate value of

The Newton's approximating sequence is generated by the formula:

Consider

[Zero of the function lies between 1 and 2. ]

Similarly,

One zero of the function is 1.225.

So, the approximate value of

Correct answer : (5)

3.

Use Newton's method to find a zero of the function between - 1 and 0 to three places of decimals.

$f$($x$) = $x$^{3} + $x$ + 1

a. | - 0.683 | ||

b. | 1.317 | ||

c. | 0.683 | ||

d. | - 1.317 | ||

e. | - 0.686 |

The Newton's approximating sequence is generated by the formula:

[Starting point.]

Similarly,

Since

So, the zero of function

Correct answer : (1)

4.

Use Newton's method to find a zero of the function in [2, 3] to nearest four decimals.

$f$($x$) = ${x}^{3}$ - 2$x$ - 5 = 0

a. | 2.1982 | ||

b. | 2.0945 | ||

c. | 2.1569 | ||

d. | 2.0016 | ||

e. | 2.8032 |

The Newton's approximating sequence is generated by the formula:

[Find

[Starting point.]

= 2 -

Similarly,

So, the zero of the function in [2, 3] to nearest four decimals is 2.0945

Correct answer : (2)

5.

Use Newton's method to find the zero of the function between 0 and 1, to the four places of decimals.

$f$($x$) = 2$x$ - ln(2 + $x$^{2})

a. | 0.3465 | ||

b. | 0.3716 | ||

c. | - 0.3465 | ||

d. | - 0.3817 | ||

e. | 0.3817 |

[Write the function.]

The Newton's approximating sequence is generated by the formula:

[Starting point.]

Similarly,

Therefore, the zero of function

Correct answer : (5)

6.

Use Newton's method to find the roots of the equation to four decimal places.

$x$^{2} - 8 = 0

a. | - 1.1716 and 1.1716 | ||

b. | - 2.8334 and 2.8334 | ||

c. | - 2.8284 and 2.8284 | ||

d. | - 2.8284 and 1.1716 | ||

e. | - 1.1716 and 2.8284 |

[Write the function.]

Draw the graph of the function.

From the graph, the zeros of the function are in between - 2 and - 3, 2 and 3.

The Newton's approximating sequence is generated by the formula,

Let us consider - 2 as the starting point to find a root in between - 2 and - 3.

[Find

[Starting point.]

Similarly,

The zero of the function is - 2.8284.

Let us consider 2 as the starting point to find a root in between 2 and 3.

= 2 -

Similarly,

The zero of the function is 2.8284.

Therefore, the roots of the equation are - 2.8284 and 2.8284, to four decimal places.

Correct answer : (3)

7.

Find the zero of the function to the nearest four decimals by using Newton's method.

$f$($x$) = 3 - $e$^{2$x$}

a. | 0.3964 | ||

b. | 0.5518 | ||

c. | - 0.4507 | ||

d. | 0.4481 | ||

e. | 0.5493 |

[Write the function.]

Draw the graph of the function.

From the graph, the zero of the function is in between 0 and 1.

The Newton's approximating sequence is generated by the formula,

[Find

[Starting point.]

= 0.5 -

Similarly,

Therefore the zero of the function

Correct answer : (5)

8.

Use Newton's method to find the solution for the function in [- 1, 0] to the nearest four decimals $f$($x$) = $e$^{$x$} + 2$x$.

a. | 0.3333 | ||

b. | - 0.3333 | ||

c. | - 0.3517 | ||

d. | 0.3829 | ||

e. | 0.3517 |

The Newton's approximating sequence is generated by the formula,

[Find

[Starting point.]

= 0 -

Similarly,

So, the solution of

Correct answer : (3)

9.

Find the zero of the function in [- 1, 0] to the nearest four decimals $f$($x$) = $e$^{cos $x$} + 2$x$ by using the Newton's method.

a. | - 1 | ||

b. | - 0.9177 | ||

c. | - 1.0823 | ||

d. | - 0.9179 | ||

e. | - 1.1506 |

The Newton's approximating sequence is generated by the formula,

[Find

= - 1 - [

Similarly,

So, the zero of the function

Correct answer : (4)

10.

Use Newton's method to find the zero of the function between - 1 and 0 to nearest four decimals.

$f$($x$) = 2$x$ + $\sqrt{3-{x}^{2}}$

a. | - 0.5469 | ||

b. | - 1 | ||

c. | - 0.7837 | ||

d. | - 0.8 | ||

e. | - 0.7746 |

The Newton's approximating sequence is generated by the formula,

[Find

[Starting point.]

= - 1 - [

Similarly,

So, the zero of the function

Correct answer : (5)