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 ∫02 eˣ dx with the trapezoid rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |T − I|.
f(x) = eˣ[a, b] = [0, 2]n = 2
|E| =
2.Numerical methodsHard
Solve y′ = 1.5y, y(0) = −1 with Euler's method and step h = 0.1 up to t = 0.5. The exact solution is y = y₀e1.5t. Find the global error |E| = |y(0.5) − y5|.
y′ = 1.5yy(0) = −1h = 0.1
|E| =
3.Sequences & seriesMedium
Exactly one of the four series converges. Click it.
A: Σ 1/√kB: Σ k!/2ᵏC: Σ 1/(k² + 9)D: Σ (7/4)ᵏ
1A: Σ 1/√k
2B: Σ k!/2ᵏ
3C: Σ 1/(k² + 9)
4D: Σ (7/4)ᵏ
4.Sequences & seriesHard
Use the Maclaurin polynomial of degree 3 of f(x) = sin(−x) to approximate f(−1.5). Give P₃(−1.5).
f(x) = sin(−x)n = 3x = −1.5
P₃(−1.5) =
5.Matrices & determinantsMedium
Find the determinant det A of the 3×3 matrix A shown.
A = [−3 1 5; 3 −5 2; 2 −4 4]
det A =
6.Matrices & determinantsHard
Find the larger eigenvalue λ of the matrix A shown.
A = [4 −1; 4 −1]
λ =
7.Differential equationsMedium
A can of soda at 3 °C is taken out of the fridge into a room at 29 °C. After 9 min its temperature is 12 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.
T0 = 3 °CTa = 29 °CT(9) = 12 °C
k = min⁻¹
8.Differential equationsHard
Solve the initial value problem y′ + 2y/x = 4x for x > 0, with y(1) = −2. Find y(2).
y′ + 2y/x = 4xy(1) = −2
y(2) =
3Solutions
1.Numerical methods
Answer: |E| = 0.5238
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 = 6.389056
Trapezoid rule
T = (1/2)·(1 + 2·2.718282 + 7.389056) = 6.91281
Error
|E| = |6.91281 − 6.389056| = 0.5238
Answer
|E| = 0.5238
Relative error: 8.198%.
2.Numerical methods
Answer: |E| = 0.1056
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
y5 = −2.011357
Exact value
y(0.5) = −2.117
Global error
|E| = |−2.117 + 2.011357| = 0.1056
Graph
Answer
|E| = 0.1056
Halving h roughly halves the error of Euler's method.
3.Sequences & series
Answer: C: Σ 1/(k² + 9)
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 1/√k → p-series with p = 1/2 ≤ 1: diverges.
B: Σ k!/2ᵏ → Ratio test: L = lim (k + 1)/2 = ∞, diverges.
C: Σ 1/(k² + 9) → Smaller than 1/k², a convergent p-series: converges by comparison.
D: Σ (7/4)ᵏ → Geometric with r = 7/4 > 1: the terms grow, so it diverges.
The answer
C: Σ 1/(k² + 9)
Answer
C: Σ 1/(k² + 9)
4.Sequences & series
Answer: P₃(−1.5) = 0.9375
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.
Maclaurin polynomial
P₃(x) = −x + 0.1667x³
Substitute
P₃(−1.5) = 1.5 − 0.5625 = 0.9375
Answer
P₃(−1.5) = 0.9375
The exact value is f(−1.5) = 0.997495; the error 0.059995 shrinks when you take more terms or a point closer to 0.
5.Matrices & determinants
Answer: det A = 18
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 = −12, M₁₂ = 8, M₁₃ = −2
Cofactor expansion
det A = (−3)·(−12) − 1·8 + 5·(−2) = 18
Answer
det A = 18
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 − (3)λ + (0) = 0
Solve
λ = (3 ± √(9))/2 → λ = 0, 3
λ = 3
Answer
λ = 3
7.Differential equations
Answer: k = 0.04721 min⁻¹
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.
Ratio of temperature differences
e−9k = -17/-26
Solve for k
k = ln(-26/-17)/9 = 0.04721 min⁻¹
Answer
k = 0.04721 min⁻¹
8.Differential equations
Answer: y(2) = 3.25
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.