Numerical methods worksheet with answers

Free printable Numerical methods worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xyx₀ = 0.5

Apply Newton–Raphson to f(x) = cos x − x with the starting value x₀ = 0.5. Find x2.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 0.5
x2 =
2.Easy

The bisection method starts on an interval of length 0.5. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.0001?

b − a = 0.5ε = 0.0001
n =
3.Medium

The trapezoid rule is used for ∫13 1/x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [1, 3]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = 1/x[a, b] = [1, 3]ε = 0.001
n =
4.Medium
xy00.511.5

Approximate ∫01.5 √(1 + x³) dx with the trapezoid rule using n = 3 strips. Give T.

f(x) = √(1 + x³)[a, b] = [0, 1.5]n = 3
T =
5.Hard
00.10.20.30.411.5ty(0, 1)

Solve y′ = t + y, y(0) = 1 with Euler's method and step h = 0.1 up to t = 0.4. The exact solution is y = −t − 1 + (y₀ + 1)eᵗ. Find the global error |E| = |y(0.4) − y4|.

y′ = t + yy(0) = 1h = 0.1
|E| =
6.Hard
00.51−3−2−1ty(0, −1)

Solve y′ = y − t², y(0) = −1 with Euler's method and step h = 0.25 up to t = 1. The exact solution is y = t² + 2t + 2 + (y₀ − 2)eᵗ. Find the global error |E| = |y(1) − y4|.

y′ = y − t²y(0) = −1h = 0.25
|E| =

Answer key

  1. 1. x2 = 0.7391
  2. 2. n = 13
  3. 3. n = 37
  4. 4. T = 2.01
  5. 5. |E| = 0.05545
  6. 6. |E| = 0.4703

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