process in which a list of numbers $x_0, x_1, x_2, x_3 \text{…}$ is generated by starting with a number $x_0$ and defining $x_n = F (x_{n - 1})$ for $n \ge 1$

method for approximating roots of $f (x) = 0;$ using an initial guess $x_0;$ each subsequent approximation is defined by the equation $x_n = x_{n - 1} - \frac{f (x_{n - 1})}{f ' (x_{n - 1})}$

Calculus Volume 1 · 4.9 Newton’s Method

Newton’s Method: practice problems

30 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 407
For the following exercises, write Newton’s formula as $x_{n + 1} = F (x_{n})$ for solving $f (x) = 0.$
$f (x) = x^{3} + 2 x + 1$
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Exercises 409
For the following exercises, write Newton’s formula as $x_{n + 1} = F (x_{n})$ for solving $f (x) = 0.$
$f (x) = e^{x}$
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Exercises 411
For the following exercises, solve $f (x) = 0$ using the iteration $x_{n + 1} = x_{n} - c f (x_{n}),$ which differs slightly from Newton’s method. Find a that works and a that fails to converge, with the exception of
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Exercises 413
For the following exercises, solve $f (x) = 0$ using the iteration $x_{n + 1} = x_{n} - c f (x_{n}),$ which differs slightly from Newton’s method. Find a that works and a that fails to converge, with the exception of
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Exercises 415
For the following exercises, start at
a. $x_{0} = 0.6$ and
b. $x_{0} = 2.$
Compute $x_{1}$ and using the specified iterative method.
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Exercises 417
For the following exercises, start at
a. $x_{0} = 0.6$ and
b. $x_{0} = 2.$
Compute $x_{1}$ and using the specified iterative method.
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Exercises 419
For the following exercises, start at
a. $x_{0} = 0.6$ and
b. $x_{0} = 2.$
Compute $x_{1}$ and using the specified iterative method.
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Exercises 421
For the following exercises, start at
a. $x_{0} = 0.6$ and
b. $x_{0} = 2.$
Compute $x_{1}$ and using the specified iterative method.
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Exercises 423
For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess $x_{0}$ that is not the exact root.
$x^{4} - 100 = 0$
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Exercises 425
For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess $x_{0}$ that is not the exact root.
$x^{3} - x = 0$
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Exercises 427
For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess $x_{0}$ that is not the exact root.
$x + tan (x) = 0,$ choose $x_{0} \in (- \frac{π}{2}, \frac{π}{2})$
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Exercises 429
For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess $x_{0}$ that is not the exact root.
$1 + x + x^{2} + x^{3} + x^{4} = 2$
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Exercises 431
For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess $x_{0}$ that is not the exact root.
$x = sin^{2} (x)$
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Exercises 433
For the following exercises, use Newton’s method to find the fixed points of the function where $f (x) = x;$ round to three decimals.
$tan (x)$ on $x = (\frac{π}{2}, \frac{3 π}{2})$
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Exercises 435
For the following exercises, use Newton’s method to find the fixed points of the function where $f (x) = x;$ round to three decimals.
$ln (x) + 2$
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Exercises 437
Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton’s method to the derivative function $f^{'} (x)$ to find its roots, instead of the original function. For the following exercises, consider the formulation of the method.
What additional restrictions are necessary on the function $f ?$
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Exercises 439
For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.
Minimum of $f (x) = 3 x^{3} + 2 x^{2} - 16$
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Exercises 441
For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.
Maximum of $f (x) = x + \frac{1}{x}$
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Exercises 443
For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.
Maximum of $f (x) = \frac{x - 3 x}{x}$
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Exercises 445
For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.
Minimum of $f (x) = x^{4} + x^{3} + 3 x^{2} + 12 x + 6$
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Exercises 447
For the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work.
Newton’s method, $0 = e^{x}$
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Exercises 449
For the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work.
Solving $x_{n + 1} = - x_{n}^{3}$ starting at $x_{0} = - 1$
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Exercises 451
For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by
$x_{n} = x_{n - 1} - f (x_{n - 1}) \frac{x _{n - 1} - x _{n - 2}}{f ( x _{n - 1} ) - f ( x _{n - 2} )} .$
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Exercises 453
For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by
$x_{n} = x_{n - 1} - f (x_{n - 1}) \frac{x _{n - 1} - x _{n - 2}}{f ( x _{n - 1} ) - f ( x _{n - 2} )} .$
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Exercises 455
For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton’s method.
$f (x) = x^{2} + 2 x + 1, x_{0} = 1$
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Exercises 457
For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton’s method.
$f (x) = sin x, x_{0} = 1$
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Exercises 459
For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton’s method.
$f (x) = x^{3} + 2 x + 4, x_{0} = 0$
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Exercises 461
In the following exercises, consider Kepler’s equation regarding planetary orbits, $M = E - ε sin (E),$ where $M$ is the mean anomaly, $E$ is eccentric anomaly, and $ε$ measures eccentricity.
Use Newton’s method to solve for the eccentric anomaly $E$ when the mean anomaly $M = \frac{3 π}{2}$ and the eccentricity of the orbit round to three decimals.
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Exercises 463
The following two exercises consider a bank investment. The initial investment is $$ 10, 000.$ After $25$ years, the investment has tripled to $$ 30, 000.$
Use Newton’s method to determine the interest rate if the interest was compounded continuously.
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Review Exercises 543
Use Newton’s method to find the first two iterations, given the starting point.
$y = x^{3} + 1, x_{0} = 0.5$
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Key terms · 2

iterative process: process in which a list of numbers $x_0, x_1, x_2, x_3 \text{…}$ is generated by starting with a number $x_0$ and defining $x_n = F (x_{n - 1})$ for $n \ge 1$
Newton’s method: method for approximating roots of $f (x) = 0;$ using an initial guess $x_0;$ each subsequent approximation is defined by the equation $x_n = x_{n - 1} - \frac{f (x_{n - 1})}{f ' (x_{n - 1})}$

Learn 4.9 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 4.9 in the textbook · This section on openstax.org

Newton’s Method: practice problems

30 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 407

For the following exercises, write Newton’s formula as $x_{n + 1} = F (x_{n})$ for solving $f (x) = 0.$

$f (x) = x^{3} + 2 x + 1$

Exercises 409

For the following exercises, write Newton’s formula as $x_{n + 1} = F (x_{n})$ for solving $f (x) = 0.$

$f (x) = e^{x}$

Exercises 411

For the following exercises, solve $f (x) = 0$ using the iteration $x_{n + 1} = x_{n} - c f (x_{n}),$ which differs slightly from Newton’s method. Find a that works and a that fails to converge, with the exception of

Exercises 413

Exercises 415

For the following exercises, start at

a. $x_{0} = 0.6$ and

b. $x_{0} = 2.$

Compute $x_{1}$ and using the specified iterative method.

Exercises 417

For the following exercises, start at

a. $x_{0} = 0.6$ and

b. $x_{0} = 2.$

Compute $x_{1}$ and using the specified iterative method.

Exercises 419

For the following exercises, start at

a. $x_{0} = 0.6$ and

b. $x_{0} = 2.$

Compute $x_{1}$ and using the specified iterative method.

Exercises 421

For the following exercises, start at

a. $x_{0} = 0.6$ and

b. $x_{0} = 2.$

Compute $x_{1}$ and using the specified iterative method.

Exercises 423