Worksheets · Calculus

Newton's method worksheet

Guess a root, draw the tangent there, and see where it crosses the axis. That crossing is a better guess, and repeating it closes in fast.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Newton's Method

Date Period

Answer each question in exact form unless it says to round.

  1. Apply Newton's method

    For f(x)=x2−3x−20, start at x0=6 and find x2 by Newton's method, exactly.
  2. For f(x)=x2+x−10, start at x0=3 and find x2 by Newton's method, exactly.
  3. For f(x)=x2+2x−13, start at x0=3 and find x2 by Newton's method, exactly.
  4. For f(x)=x2−x−25, start at x0=6 and find x2 by Newton's method, exactly.
  5. For f(x)=x2−22, start at x0=5 and find x2 by Newton's method, exactly.
  6. For f(x)=x2+3x−21, start at x0=3 and find x2 by Newton's method, exactly.
  7. For f(x)=x2−28, start at x0=5 and find x2 by Newton's method, exactly.
  8. For f(x)=x2−2x−30, start at x0=7 and find x2 by Newton's method, exactly.
  9. For f(x)=x2−x−13, start at x0=4 and find x2 by Newton's method, exactly.
  10. For f(x)=x2+x−26, start at x0=5 and find x2 by Newton's method, exactly.

  1. Approximate with Newton's method

    Approximate the solution of ex=5x starting from x0=0. Use two steps of Newton's method and give x2 to four decimal places.
  2. Approximate the solution of cos⁡x=x starting from x0=1. Use two steps of Newton's method and give x2 to four decimal places.
  3. Approximate 2 starting from x0=2. Use two steps of Newton's method and give x2 to four decimal places.
  4. Approximate 854 starting from x0=3. Use two steps of Newton's method and give x2 to four decimal places.
  5. Approximate a root of x3−2x2+8x+4=0 starting from x0=−1. Use two steps of Newton's method and give x2 to four decimal places.
  6. Approximate the solution of cos⁡x=2x starting from x0=1. Use two steps of Newton's method and give x2 to four decimal places.
  7. Approximate the solution of xex=2 starting from x0=1. Use two steps of Newton's method and give x2 to four decimal places.
  8. Approximate the solution of ln⁡x=2−x starting from x0=2. Use two steps of Newton's method and give x2 to four decimal places.
  9. Approximate the solution of sin⁡x=x3 starting from x0=2. Use two steps of Newton's method and give x2 to four decimal places.
  10. Approximate a root of x3−4x2+8x−2=0 starting from x0=0. Use two steps of Newton's method and give x2 to four decimal places.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Newton's MethodAnswer keyVersion 1

Date Period

Answer each question in exact form unless it says to round.

  1. Apply Newton's method

    For f(x)=x2−3x−20, start at x0=6 and find x2 by Newton's method, exactly.4756765
  2. For f(x)=x2+x−10, start at x0=3 and find x2 by Newton's method, exactly.851315
  3. For f(x)=x2+2x−13, start at x0=3 and find x2 by Newton's method, exactly.329120
  4. For f(x)=x2−x−25, start at x0=6 and find x2 by Newton's method, exactly.67461221
  5. For f(x)=x2−22, start at x0=5 and find x2 by Newton's method, exactly.4409940
  6. For f(x)=x2+3x−21, start at x0=3 and find x2 by Newton's method, exactly.28987
  7. For f(x)=x2−28, start at x0=5 and find x2 by Newton's method, exactly.56091060
  8. For f(x)=x2−2x−30, start at x0=7 and find x2 by Newton's method, exactly.105611608
  9. For f(x)=x2−x−13, start at x0=4 and find x2 by Newton's method, exactly.1478357
  10. For f(x)=x2+x−26, start at x0=5 and find x2 by Newton's method, exactly.57471243

  1. Approximate with Newton's method

    Approximate the solution of ex=5x starting from x0=0. Use two steps of Newton's method and give x2 to four decimal places.x2≈0.2592
  2. Approximate the solution of cos⁡x=x starting from x0=1. Use two steps of Newton's method and give x2 to four decimal places.x2≈0.7391
  3. Approximate 2 starting from x0=2. Use two steps of Newton's method and give x2 to four decimal places.x2≈1.4167
  4. Approximate 854 starting from x0=3. Use two steps of Newton's method and give x2 to four decimal places.x2≈3.0364
  5. Approximate a root of x3−2x2+8x+4=0 starting from x0=−1. Use two steps of Newton's method and give x2 to four decimal places.x2≈−0.4435
  6. Approximate the solution of cos⁡x=2x starting from x0=1. Use two steps of Newton's method and give x2 to four decimal places.x2≈0.4504
  7. Approximate the solution of xex=2 starting from x0=1. Use two steps of Newton's method and give x2 to four decimal places.x2≈0.8528
  8. Approximate the solution of ln⁡x=2−x starting from x0=2. Use two steps of Newton's method and give x2 to four decimal places.x2≈1.5571
  9. Approximate the solution of sin⁡x=x3 starting from x0=2. Use two steps of Newton's method and give x2 to four decimal places.x2≈2.2796
  10. Approximate a root of x3−4x2+8x−2=0 starting from x0=0. Use two steps of Newton's method and give x2 to four decimal places.x2≈0.2879

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Use Newton's method on f(x)=x2−2 from x0=1. Find x1.

  1. f of 1 is minus 1, and the derivative there is 2.
  2. So x1 is 1 minus minus one over two.
  3. That is three halves.

The answer is 32.

Where students go wrong

Subtracting the derivative instead of dividing by it. The formula is x minus f over f prime; the division is what turns a height into a horizontal step.

Also called Newton Raphson, root finding or iteration formula.

What you can put on this worksheet

Newton's method
For f(x)=x2−17, start at x0=4 and find x1 by Newton's method, exactly. → 338
Transcendental equations, intersections and roots
Approximate a root of x3+x2+4x−5=0 starting from x0=0. Use two steps of Newton's method and give x2 to four decimal places. → x2≈0.9358

Questions about these worksheets

Yes. Every newton's method sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the newton's method questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: newton's method and transcendental equations, intersections and roots. Tick as many as you want and set how many of each, or let it spread them evenly.

Subtracting the derivative instead of dividing by it. The formula is x minus f over f prime; the division is what turns a height into a horizontal step.