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 = 2

Apply Newton–Raphson to f(x) = ln x + x − 3 with the starting value x₀ = 2. Find x2.

f(x) = ln x + x − 3f′(x) = 1/x + 1x₀ = 2
x2 =
2.Easy
xya = 0b = 1

Use the bisection method on f(x) = cos x − x with the starting interval [0, 1]. Find the midpoint c4 of step 4.

f(x) = cos x − x[a, b] = [0, 1]n = 4
c4 =
3.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 √x·e−x dx with Simpson's rule using n = 6 strips. Give S.

f(x) = √x·e−x[a, b] = [0, 1]n = 6
S =
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
00.511.5123ty(0, 1)

Use Euler's method with h = 0.5 for y′ = y − t², y(0) = 1 to estimate y3 ≈ y(1.5).

y′ = y − t²y(0) = 1h = 0.5
y3 =
6.Hard
00.20.411.52ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = t² + y², y(0) = 1. Find y₁ ≈ y(0.5).

y′ = t² + y²y(0) = 1h = 0.5
y1 =

Answer key

  1. 1. x2 = 2.208
  2. 2. c4 = 0.6875
  3. 3. S = 0.3732
  4. 4. n = 13
  5. 5. y3 = 2.688
  6. 6. y1 = 2.055

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