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 0.01?

b − a = 1ε = 0.01
n =
2.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 =
3.Medium

The trapezoid rule is used for ∫02 x³ 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 0.0005.

f(x) = x³[a, b] = [0, 2]ε = 0.0005
n =
4.Medium
xy12.54

Approximate ∫14 1/x dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.

f(x) = 1/x[a, b] = [1, 4]n = 2
|E| =
5.Hard
00.10.20.30.40.450.5ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.2 for y′ = −2ty, y(0) = 0.5 to estimate y2 ≈ y(0.4).

y′ = −2tyy(0) = 0.5h = 0.2
y2 =
6.Hard
00.10.20.30.42.53ty(0, 3)

Use Heun's method (improved Euler) with h = 0.2 for y′ = cos t − y, y(0) = 3 to estimate y2 ≈ y(0.4).

y′ = cos t − yy(0) = 3h = 0.2
y2 =

Answer key

  1. 1. n = 7
  2. 2. x2 = 0.7391
  3. 3. n = 127
  4. 4. |E| = 0.03871
  5. 5. y2 = 0.4255
  6. 6. y2 = 2.334

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