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) = e−x − x with the starting value x₀ = 0.5. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x1 =
2.Easy
xya = 0b = 1

Use the bisection method on f(x) = cos x − x with the starting interval [0, 1]. Find the midpoint c3 of step 3.

f(x) = cos x − x[a, b] = [0, 1]n = 3
c3 =
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 1 × 10⁻⁵.

f(x) = 1/x[a, b] = [1, 3]ε = 1 × 10⁻⁵
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 0.001.

f(x) = 1/x[a, b] = [1, 2]ε = 0.001
n =
5.Hard
00.20.40.60.80.60.81ty(0, 1)

Use Euler's method with h = 0.2 for y′ = −2ty, y(0) = 1 to estimate y4 ≈ y(0.8).

y′ = −2tyy(0) = 1h = 0.2
y4 =
6.Hard
00.10.20.30.60.8ty(0, 0.5)

Use Euler's method with h = 0.1 for y′ = e−ty, y(0) = 0.5 to estimate y3 ≈ y(0.3).

y′ = e−tyy(0) = 0.5h = 0.1
y3 =

Answer key

  1. 1. x1 = 0.5663
  2. 2. c3 = 0.625
  3. 3. n = 366
  4. 4. n = 13
  5. 5. y4 = 0.5873
  6. 6. y3 = 0.7812

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