For a root: check the sign change for bisection, or compute f and f′ for Newton, and keep six or more digits in a table of iterates.
For a table value: interpolate linearly between the two neighbouring points.
For an integral: find h = (b − a)/n, list f at every node, apply the weights (1, 2, …, 2, 1 for trapezoid; 1, 4, 2, …, 4, 1 for Simpson with even n) and multiply by h/2 or h/3.
For an ODE: step from t₀ with yₙ₊₁ = yₙ + h·f(tₙ, yₙ) (Euler) or the averaged slope (Heun), one row per step, until you reach the requested t.
Sequences & series
Arithmetic sequence
aₙ = a₁ + (n − 1)d, Sₙ = n(a₁ + aₙ)/2
Geometric sequence
aₙ = a₁rⁿ⁻¹, Sₙ = a₁(1 − rⁿ)/(1 − r)
Infinite geometric series
S = a₁/(1 − r), |r| < 1
p-series
Σ1/kᵖ converges only for p > 1
Ratio test
L = lim |ak+1/ak|: L < 1 converges, L > 1 diverges
Radius of convergence
R = lim |ck/ck+1|
Taylor polynomial
Pₙ(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ/k!
Maclaurin series
eˣ = Σxᵏ/k!, sin x = x − x³/3! + …, cos x = 1 − x²/2! + …, ln(1 + x) = x − x²/2 + …
Lagrange error bound
|Rₙ| ≤ M·|x − a|ⁿ⁺¹/(n + 1)!
Method
Identify the type: arithmetic (constant difference), geometric (constant ratio), a standard series, a power series or a Taylor polynomial.
For sums use Sₙ = n(a₁ + aₙ)/2, Sₙ = a₁(1 − rⁿ)/(1 − r), or S = a₁/(1 − r) when |r| < 1, with a₁ the first term of the sum.
For convergence: n-th term test first, then compare with Σ1/kᵖ or a geometric series, and use the ratio or root test for factorials and k-th powers.
For Taylor polynomials list f(a), f′(a), f″(a), …, divide the k-th derivative by k!, and bound the error with M·|x − a|ⁿ⁺¹/(n + 1)!.
Matrices & determinants
2×2 determinant
det = ad − bc
2×2 inverse
A⁻¹ = (1/det)·[d −b; −c a]
Product
(AB)ij = Σk AikBkj
Eigenvalues
det(A − λI) = 0
Cramer’s rule
xi = det(Ai)/det(A)
Differential equations
Exponential growth/decay
dy/dt = ky → y = y0ekt
Half-life
k = ln 2 / t½
Newton's law of cooling
T(t) = Ta + (T0 − Ta)e−kt
Integrating factor
y′ + p(x)y = q(x): μ = e∫p dx, (μy)′ = μq
Mixing tank
dm/dt = r·c − (r/V)·m
2Practice problems
1.Numerical methodsMedium
Approximate ∫03 sin x dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.
f(x) = sin x[a, b] = [0, 3]n = 2
|E| =
2.Numerical methodsHard
Solve y′ = −2ty, y(0) = −1 with Euler's method and step h = 0.2 up to t = 0.8. The exact solution is y = y₀e−t². Find the global error |E| = |y(0.8) − y4|.
y′ = −2tyy(0) = −1h = 0.2
|E| =
3.Sequences & seriesMedium
The power series Σk=1∞ k!·(x − 1)ᵏ/kᵏ converges on an interval around x₀ = 1. Find the left end x of that interval.
Σ k!·(x − 1)ᵏ/kᵏx₀ = 1
x =
4.Sequences & seriesHard
f(x) = cos x is approximated by its Maclaurin polynomial of degree 3 at x = 1. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₃|.
f(x) = cos xn = 3x = 1M = 1
|R₃| =
5.Matrices & determinantsMedium
Find the determinant det A of the 3×3 matrix A shown.
A = [−5 −5 −1; 0 4 2; 1 3 −4]
det A =
6.Matrices & determinantsHard
Find the larger eigenvalue λ of the matrix A shown.
A = [2 −1; −1 2]
λ =
7.Differential equationsMedium
A can of soda at 6 °C is taken out of the fridge into a room at 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 10 min.
T0 = 6 °CTa = 26 °Ck = 0.05 min⁻¹t = 10 min
T = °C
8.Differential equationsHard
A system obeys y″ + y′ + 9.25y = 92.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.
y″ + y′ + 9.25y = 92.5y(0) = 0y′(0) = 0
y(0.25) =
3Solutions
1.Numerical methods
Answer: |E| = 0.07556
Numerical integration: split [a, b] into n strips of width h = (b − a)/n with nodes x₀ … xₙ. Trapezoid rule: T = (h/2)[f₀ + 2f₁ + 2f₂ + … + 2fn−1 + fₙ]. Simpson's rule (n even): S = (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + … + 4fn−1 + fₙ]. The trapezoid error shrinks like h² (bound (b − a)h²·M/12 with M = max|f″|), Simpson's like h⁴, so Simpson is usually far more accurate for the same nodes.
Exact value
I = 1.989992
Simpson's rule
S = (1.5/3)·(0 + 4·0.997495 + 0.14112) = 2.06555
Error
|E| = |2.06555 − 1.989992| = 0.07556
Answer
|E| = 0.07556
Relative error: 3.797%.
2.Numerical methods
Answer: |E| = 0.06004
Numerical solution of y′ = f(t, y): Euler's method steps along the slope at the start of each step: yn+1 = yₙ + h·f(tₙ, yₙ). Heun's method (improved Euler) averages the slope at the start and at an Euler prediction: k₁ = f(tₙ, yₙ), k₂ = f(tₙ + h, yₙ + h·k₁), yn+1 = yₙ + (h/2)(k₁ + k₂). The classical Runge–Kutta method (RK4) combines four slopes: yn+1 = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄). The global error of Euler is proportional to h, Heun's to h², RK4's to h⁴.
Euler steps
y4 = −0.587328
Exact value
y(0.8) = −0.527292
Global error
|E| = |−0.527292 + 0.587328| = 0.06004
Graph
Answer
|E| = 0.06004
Halving h roughly halves the error of Euler's method.
3.Sequences & series
Answer: x = −1.718
Convergence: if the terms do not tend to 0 the series diverges (n-th term test). Σ1/kᵖ converges only for p > 1, and Σrᵏ only for |r| < 1. Compare a series with one of these when its terms behave like them for large k. Ratio test: L = lim |ak+1/ak|; root test: L = lim |ak|1/k. L < 1 converges, L > 1 diverges, L = 1 decides nothing. A power series Σck(x − x₀)ᵏ converges for |x − x₀| < R with R = lim |ck/ck+1|; an alternating series with terms decreasing to 0 converges.
Coefficients
ck/ck+1 = (1 + 1/k)ᵏ → e
Radius
R = 2.718
Interval
−1.718 < x < 3.718
x = −1.718
Answer
x = −1.718
The ratio test says nothing at the two end points; they have to be checked separately.
4.Sequences & series
Answer: |R₃| = 0.04167
Taylor polynomials: near x = a a smooth function is approximated by Pₙ(x) = f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + … + f⁽ⁿ⁾(a)(x − a)ⁿ/n!; for a = 0 this is the Maclaurin polynomial. Standard series: eˣ = Σxᵏ/k!, sin x = x − x³/3! + x⁵/5! − …, cos x = 1 − x²/2! + x⁴/4! − …, ln(1 + x) = x − x²/2 + x³/3 − …, 1/(1 − x) = Σxᵏ. Lagrange error bound: |f(x) − Pₙ(x)| ≤ M·|x − a|ⁿ⁺¹/(n + 1)!, where M is the largest value of |f⁽ⁿ⁺¹⁾| between a and x.
Bound on the derivative
M = 1
Lagrange bound
|R₃| ≤ 1·1⁴/4! = 0.04167
Answer
|R₃| = 0.04167
The true error is smaller: |f(1) − P₃(1)| = 0.0403.
5.Matrices & determinants
Answer: det A = 104
Matrices: the determinant of a 2×2 matrix is ad − bc. For 3×3, expand along a row: each entry times the determinant of the 2×2 matrix left after removing its row and column, with alternating signs + − +. Eigenvalues λ solve det(A − λI) = 0; for 2×2 that is λ² − (trace)·λ + det = 0.
2×2 minors of the first row
M₁₁ = −22, M₁₂ = −2, M₁₃ = −4
Cofactor expansion
det A = (−5)·(−22) − (−5)·(−2) + (−1)·(−4) = 104
Answer
det A = 104
6.Matrices & determinants
Answer: λ = 3
Matrices: the determinant of a 2×2 matrix is ad − bc. For 3×3, expand along a row: each entry times the determinant of the 2×2 matrix left after removing its row and column, with alternating signs + − +. Eigenvalues λ solve det(A − λI) = 0; for 2×2 that is λ² − (trace)·λ + det = 0.
Characteristic equation
λ² − (4)λ + (3) = 0
Solve
λ = (4 ± √(4))/2 → λ = 1, 3
λ = 3
Answer
λ = 3
7.Differential equations
Answer: T = 13.87 °C
First-order differential equations: dy/dt = k·y has the solution y = y₀·ekt (growth for k > 0, decay for k < 0); a half-life T gives k = ln 2/T. Newton's law of cooling, dT/dt = −k·(T − Tₐ), gives T(t) = Tₐ + (T₀ − Tₐ)·e−kt. A linear equation y′ + p(x)·y = q(x) is solved with the integrating factor μ = e∫p dx: (μy)′ = μq, so μy = ∫μq dx + C.
Solution of the cooling law
T(t) = 26 + (−20)·e−0.05t
Substitute t
T(10) = 26 + (−20)·0.6065 = 13.87 °C
Answer
T = 13.87 °C
8.Differential equations
Answer: y(0.25) = 2.54
Second-order linear equations: for y″ + b·y′ + c·y = 0 try y = ert: this gives the characteristic equation r² + b·r + c = 0. Two real roots r₁ ≠ r₂ give y = A·er₁t + B·er₂t (overdamped); a double root r gives y = (A + Bt)·ert (critically damped); complex roots −α ± jω give y = e−αt(A cos ωt + B sin ωt) (underdamped, oscillating). The initial values y(0) and y′(0) fix A and B. Written as y″ + 2ζωny′ + ωn²y = 0, ωn is the natural frequency and ζ the damping ratio: ζ < 1 oscillates. With a constant input, y″ + by′ + cy = c·K, the solution is the steady value K plus the free response.