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
The trapezoid rule is used for ∫02 eˣ 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) = eˣ[a, b] = [0, 2]ε = 0.0005
n =
2.Numerical methodsHard
Use Heun's method (improved Euler) with h = 0.5 for y′ = 1 + sin(ty), y(0) = 1 to estimate y2 ≈ y(1).
y′ = 1 + sin(ty)y(0) = 1h = 0.5
y2 =
3.Sequences & seriesMedium
Exactly one of the four series diverges. Click it.
A: Σ 5/k⁴B: Σ 1/(k(k + 2))C: Σ (3/2)ᵏD: Σ k²/3ᵏ
1A: Σ 5/k⁴
2B: Σ 1/(k(k + 2))
3C: Σ (3/2)ᵏ
4D: Σ k²/3ᵏ
4.Sequences & seriesHard
Find the coefficient c₂ of (x − 8)² in the Taylor series of f(x) = ∛x about a = 8.
f(x) = ∛xa = 8k = 2
c₂ =
5.Matrices & determinantsMedium
Let C = AB for the matrices shown. Find the entry c12 (row 1, column 2).
A = [−5 4; 1 −3]B = [−2 0; −6 2]
c12 =
6.Matrices & determinantsHard
Solve the system −3x + 3y = 2 and −4x − 6y = −12. Find x.
−3x + 3y = 2−4x − 6y = −12
x =
7.Differential equationsMedium
Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 0. Find y(1.5).
y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 0
y(1.5) =
8.Differential equationsHard
A tank holds 100 L of water in which 10 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 25 min?
V = 100 Lm0 = 10 kgc = 0.2 kg/Lr = 5 L/min
m = kg
3Solutions
1.Numerical methods
Answer: n = 100
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.
Second derivative bound
M = e2 = 7.389
Solve for n
n² > 8·7.389/(12·0.0005) = 9852
n > 99.26 → n = 100
Answer
n = 100
2.Numerical methods
Answer: y2 = 2.497
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.
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.
Test each series
A: Σ 5/k⁴ → p-series with p = 4 > 1: converges.
B: Σ 1/(k(k + 2)) → Smaller than 1/k²: converges by comparison.
C: Σ (3/2)ᵏ → Geometric with r = 3/2 > 1: the terms grow, so it diverges.
D: Σ k²/3ᵏ → Ratio test: L = 1/3 < 1, converges.
The answer
C: Σ (3/2)ᵏ
Answer
C: Σ (3/2)ᵏ
4.Sequences & series
Answer: c₂ = −0.003472
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.
Derivative f⁽²⁾
f⁽²⁾(x) = −0.2222·x1/3 − 2
Coefficient
c₂ = −0.2222·81/3 − 2/2! = −0.003472
c₂ = −0.003472
Answer
c₂ = −0.003472
5.Matrices & determinants
Answer: c12 = 8
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.
Row times column
c12 = (−5)·0 + 4·2 = 8
Answer
c12 = 8
6.Matrices & determinants
Answer: x = 0.8
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.
Determinant of the coefficient matrix
det A = (−3)·(−6) − 3·(−4) = 30
Replace the x column
det Aₓ = 2·(−6) − 3·(−12) = 24
x = 24/30 = 0.8
Answer
x = 0.8
y = 1.467
7.Differential equations
Answer: y(1.5) = 0.1222
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.
Double root
(r + 3)² = 0 → r = −3
Constants
A = y(0) = 2
y′(0) = B − 3A → B = 0 + 6 = 6
Evaluate
y(1.5) = (2 + 6·1.5)·e−4.5 = 0.1222
Answer
y(1.5) = 0.1222
8.Differential equations
Answer: m = 17.13 kg
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.
Set up the equation
dm/dt = 5·0.2 − (5/100)·m = 1 − 0.05·m
Solve
The integrating factor is ert/V; the solution approaches the equilibrium c·V.