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

The bisection method starts on an interval of length 1. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.05?

b − a = 1ε = 0.05
n =
2.Easy
xyx₀ = 0

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0
x1 =
3.Medium

The trapezoid rule is used for ∫02 sin x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 2]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = sin x[a, b] = [0, 2]ε = 0.001
n =
4.Medium

The trapezoid rule is used for ∫12 1/x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [1, 2]. Find the smallest number of strips n that guarantees an error below 0.0005.

f(x) = 1/x[a, b] = [1, 2]ε = 0.0005
n =
5.Hard
00.10.20.311.21.4ty(0, 1)

Use Heun's method (improved Euler) with h = 0.1 for y′ = 1 + sin(ty), y(0) = 1 to estimate y3 ≈ y(0.3).

y′ = 1 + sin(ty)y(0) = 1h = 0.1
y3 =
6.Hard
00.10.20.30.40.60.8ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.2 for y′ = e−ty, y(0) = 0.5 to estimate y2 ≈ y(0.4).

y′ = e−tyy(0) = 0.5h = 0.2
y2 =

Answer key

  1. 1. n = 5
  2. 2. x1 = 0.5
  3. 3. n = 26
  4. 4. n = 19
  5. 5. y3 = 1.355
  6. 6. y2 = 0.845

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