Newton's Method Worksheet

**Page 1**

1.

Using Newton's approximation method, find the zero of the function $x$^{2} - 55.

a. | ± 7.4161 | ||

b. | ± 7.4928 | ||

c. | ± 7.2137 | ||

d. | ± 7.3973 |

[Differentiate.]

From the graph, the zeros of the function are in between - 7 & - 8 and + 7 & + 8.

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

Let

[Starting point.]

[Substitute

Similarly,

Since the values of

Let

[Starting point.]

[Substitute

Similarly,

Since the values of

Hence, the approximate roots of

Correct answer : (1)

2.

Use Newton's method for approximating the zero(s) of the function $f$($x$) = 2$x$^{2} - 2$x$ - 4.

a. | 2.1000 | ||

b. | - 2.0000 | ||

c. | - 1.0000 | ||

d. | Both - 2.0000 & - 1.0000 |

[Differentiate.]

From the graph, the zeros of the function are - 1 & 2.

Though we are getting the roots from the graph directly, Let us check the zeros of the function by using " NewtonÃ¢â‚¬â„¢s method".

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

Let

[Starting point.]

Similarly,

[Substitute

Since, the values of

Let

[Starting point.]

[Substitute

Similarly,

Since, the values of

Hence, the approximate zeros of

Correct answer : (4)

3.

Use Newton's method for approximating the zeros of the function $f$($x$) = 3$x$^{2} - 28.

a. | ± 3.113 | ||

b. | ± 3.055 | ||

c. | ± 9.235 | ||

d. | ± 9.332 |

[Differentiate.]

From the graph, the zeros of the function lie between - 3 & - 4 and 3 & 4.

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

Let,

[Starting point.]

[Substitute

Similarly

Since the values of

Let

[Starting point.]

[Substitute

Similarly

Since the values of

Hence, the approximate roots of

Correct answer : (2)

4.

Use Newton's method for approximating the roots of the function $f$($x$) = $x$^{4} + 3$x$^{3} + 5.

a. | -1.00438 & -3.6611 | ||

b. | -1.2234 & -3.7861 | ||

c. | -1.49074 & -2.7629 | ||

d. | -1.36827 & -2.7629 |

[Differentiate.]

From the graph, the zeros of function are in between - 1 & - 2 and - 2 & - 3.

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

Let

[Starting point.]

[Substitute

Similarly,

Since the values of

Let

[Starting point.]

[Substitute

Similarly,

Since the values of

Hence, the approximate roots of

Correct answer : (3)

5.

Use Newton's method for approximating the root(s) of the function $f$($x$) = 12$x$^{2} - 5$x$ - 19.

a. | -1.06710 and 1.48376 | ||

b. | 1.48376 | ||

c. | - 1.06710 | ||

d. | 1.06710 |

[Differentiate.]

From the graph, the zeroÃ¢â‚¬â„¢s of function are in between 0 & - 1 and 1 & 2.

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

Let

[Starting point.]

[Substitute

Similarly

Since, the values of

Let

[Starting point.]

[Substitute

Similarly,

Since, the values of

Hence, the approximate roots of

Correct answer : (1)

6.

Identify the Newton's iterative formula that helps to approximate the 5^{th} root of the positive number $k$.

a. | ${x}_{n+1}=4{x}_{n}$ | ||

b. | ${x}_{n+1}=-\frac{1}{5}(4{x}_{n}+\frac{k}{{{x}_{n}}^{4}})$ | ||

c. | ${x}_{n+1}=4{x}_{n}+\frac{k}{{{x}_{n}}^{4}}$ | ||

d. | ${x}_{n+1}=\frac{1}{5}(4{x}_{n}+\frac{k}{{{x}_{n}}^{4}})$ |

So,

[Evaluate

[Write the Newton's iterative formula.]

=

=

So,

Correct answer : (4)

7.

Find the roots of $f$($x$) = 2$x$^{2} - 17.

a. | ± 2.7865 | ||

b. | ± 2.5569 | ||

c. | ± 2.9154 | ||

d. | ± 2.6781 |

[Differentiate.]

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

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

Let

[Starting point.]

[Substitute

Similarly

Since the values of

Let

[Starting point.]

[Substitute

Similarly

Since the values of

Hence, the approximate roots of

Correct answer : (3)