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
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 =
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
xy01.53

Approximate ∫03 √x·e−x dx with the trapezoid rule using n = 2 strips. Give T.

f(x) = √x·e−x[a, b] = [0, 3]n = 2
T =
4.Medium
xy00.40.81.21.62

Approximate ∫02 1/(1 + x³) dx with the trapezoid rule using n = 5 strips. Give T.

f(x) = 1/(1 + x³)[a, b] = [0, 2]n = 5
T =
5.Hard
00.20.40.60.60.81ty(0, 1)

Use Euler's method with h = 0.25 for y′ = t − y, y(0) = 1 to estimate y3 ≈ y(0.75).

y′ = t − yy(0) = 1h = 0.25
y3 =
6.Hard
00.20.400.20.4ty(0, 0)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = e−ty, y(0) = 0. Find y₁ ≈ y(0.5).

y′ = e−tyy(0) = 0h = 0.5
y1 =

Answer key

  1. 1. c4 = 0.6875
  2. 2. n = 13
  3. 3. T = 0.4746
  4. 4. T = 1.088
  5. 5. y3 = 0.5938
  6. 6. y1 = 0.4629

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