Foundation

Algebra & Pre-Calculus

Pre-calculus builds the toolkit for everything that follows. Mastery here pays dividends through Calc III and beyond. Students who struggle in calculus are often missing a pre-calc concept, not a calculus one.

Topics Covered

Functions: domain, range, composition, inverse
Polynomials: factoring, synthetic division, remainder theorem
Rational functions: asymptotes, holes, end behavior
Exponentials and logarithms: change of base, natural log
Trigonometry: unit circle, identities, inverse trig
Complex numbers: rectangular and polar form, Euler's formula

Key Relationships to Memorize

ln(e^x) = x and e^(ln x) = x
sin²(θ) + cos²(θ) = 1
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) − sin²(θ)
e^(iπ) + 1 = 0 (Euler's identity)
log_b(xy) = log_b(x) + log_b(y)

Common Errors

Log rules: log(x + y) ≠ log(x) + log(y). The product rule is log(xy) = log x + log y
Trig signs by quadrant: memorize ASTC (All, Sin, Tan, Cos) for signs in quadrants I–IV
Function composition direction: f(g(x)) means apply g first, then f — not the other way
Inverse functions: f²(x) means [f(x)]², but f−¹(x) means the inverse function, not 1/f(x)

Worked Example 1

Solve: log₂(x + 3) + log₂(x − 1) = 3

Show Full Solution

Apply the product rule: log₂[(x+3)(x−1)] = 3

Convert: (x+3)(x−1) = 2³ = 8

Expand: x² + 2x − 3 = 8 → x² + 2x − 11 = 0

Quadratic formula: x = [−2 ± √(4 + 44)] / 2 = [−2 ± 4√3] / 2 = −1 ± 2√3

Check domain: x − 1 > 0 requires x > 1, so x = −1 + 2√3 ≈ 2.46 is the only valid solution.

Worked Example 2

Find all θ in [0, 2π) satisfying: 2sin²(θ) − sin(θ) − 1 = 0

Show Full Solution

Treat as quadratic in u = sin(θ): 2u² − u − 1 = 0

Factor: (2u + 1)(u − 1) = 0 → u = −1/2 or u = 1

sin(θ) = 1: θ = π/2

sin(θ) = −1/2: θ = 7π/6 and θ = 11π/6 (Quadrants III and IV)

Solutions: θ = π/2, 7π/6, 11π/6

Worked Example 3

If f(x) = x² + 1 and g(x) = √(x − 1), find the domain of f(g(x)).

Show Full Solution

f(g(x)) = f(√(x−1)) = (√(x−1))² + 1 = (x − 1) + 1 = x

But the domain must respect the domain of g(x): x − 1 ≥ 0 → x ≥ 1

Domain of f(g(x)): [1, ∞) — even though the simplified form x has no restriction, the composition inherits g's constraint.

Calculus I

Limits & Derivatives

Calculus I introduces the two central ideas of calculus: the limit (what a function approaches) and the derivative (rate of change). These concepts underpin all of physics, engineering, and modern data science.

Essential Derivative Rules d/dx[x²] = nx^(n−1)     (Power Rule)
d/dx[f·g] = f'g + fg'     (Product Rule)
d/dx[f/g] = (f'g − fg') / g²     (Quotient Rule)
d/dx[f(g(x))] = f'(g(x)) · g'(x)     (Chain Rule)
d/dx[e^x] = e^x      d/dx[ln x] = 1/x      d/dx[sin x] = cos x

Key Concepts

Formal limit definition: lim[x→a] f(x) = L
L'Hôpital's rule for 0/0 and ∞/∞ indeterminate forms
Continuity: limit exists, f(a) is defined, and they are equal
Implicit differentiation for equations not solved for y
Related rates: differentiate both sides with respect to time
Critical points, concavity, and the First and Second Derivative Tests

Applied Uses Dr. Preston Uses Daily

Criticality calculations: neutron flux gradients in reactor cores
ML optimization: gradient descent uses the derivative of a loss function
Signal processing: rate of change of detected radiation counts
Related rates: neutron population growth as a function of time

Common Errors

Chain rule: d/dx[sin(x²)] = cos(x²) · 2x — forgetting the "inner derivative" 2x is the most common Calc I mistake
Product rule: d/dx[fg] ≠ f'g'. Both terms are required: f'g + fg'
Implicit differentiation: when differentiating y², the result is 2y · dy/dx, not just 2y
L'Hôpital's rule applies only to 0/0 or ∞/∞ — verify the form before applying

Worked Example 1

Differentiate: f(x) = x³ sin(x)

Show Full Solution

Use the Product Rule: f'(x) = (x³)' sin(x) + x³ (sin x)'

= 3x² sin(x) + x³ cos(x)

f'(x) = 3x² sin x + x³ cos x

Worked Example 2

Find dy/dx if x² + y² = 25 (implicit differentiation).

Show Full Solution

Differentiate both sides with respect to x:

2x + 2y(dy/dx) = 0

Solve: 2y(dy/dx) = −2x

dy/dx = −x/y (the derivative of a circle, giving the slope of the tangent at any point (x,y))

Worked Example 3

Evaluate: lim[x→0] (sin x) / x

Show Full Solution

Direct substitution gives 0/0 — apply L'Hôpital's rule: differentiate numerator and denominator separately.

d/dx[sin x] = cos x d/dx[x] = 1

lim[x→0] cos(x)/1 = cos(0) = 1

Calculus II

Integration & Series

Calculus II covers the integral (accumulation of change) and infinite series. This is where many STEM students hit their first significant wall — a systematic approach to technique selection makes the difference.

Core Integration Techniques ∫ u dv = uv − ∫ v du     (Integration by Parts: LIATE for u)
∫ f(g(x)) g'(x) dx → let u = g(x)     (u-Substitution)
∫ sin²(x) dx = x/2 − sin(2x)/4 + C     (Power Reduction)
Taylor: f(x) = ∑ f^(n)(a)/n! (x−a)^n     (Taylor Series at x=a)

Key Techniques

Antiderivatives and the Fundamental Theorem of Calculus
u-substitution: reverse chain rule
Integration by parts: ∫ u dv = uv − ∫ v du
Trig substitution: x = a sinθ, a tanθ, or a secθ
Partial fractions for rational functions
Improper integrals: limits at infinity or discontinuities
Taylor and Maclaurin series with convergence tests

Convergence Tests

Divergence test: if lim a_n ≠ 0, the series diverges
Integral test: compare ∑ a_n to ∫ f(x) dx
Ratio test: L = lim |a_{n+1}/a_n|; converges if L < 1
Alternating Series Test: |a_{n+1}| ≤ |a_n| and lim = 0
Comparison and limit comparison tests

Common Errors

Forgetting +C for indefinite integrals — worth partial credit on most exams and essential for ODEs
Wrong u-sub choice: u should simplify the integral, not complicate it
Alternating series sign errors: ensure the series alternates sign and terms decrease to zero
Applying the Ratio Test to alternating series without taking absolute value

Worked Example 1

Evaluate: ∫ x e^x dx

Show Full Solution

Integration by Parts. LIATE: choose u = x (algebraic), dv = e^x dx

Then du = dx and v = e^x

∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C

= e^x(x − 1) + C

Worked Example 2

Evaluate: ∫ dx / (x² − 4)

Show Full Solution

Factor: x² − 4 = (x−2)(x+2). Use partial fractions:

1/[(x−2)(x+2)] = A/(x−2) + B/(x+2)

Solve: A = 1/4, B = −1/4

∫ [1/4 · 1/(x−2) − 1/4 · 1/(x+2)] dx

= (1/4) ln|x−2| − (1/4) ln|x+2| + C = (1/4) ln|(x−2)/(x+2)| + C

Worked Example 3

Find the Taylor series for f(x) = e^x centered at x = 0.

Show Full Solution

The nth derivative of e^x is e^x for all n. At x = 0: f^(n)(0) = 1 for all n.

Taylor formula: e^x = ∑[n=0 to ∞] x^n / n!

= 1 + x + x²/2! + x³/3! + x&sup4;/4! + …

Converges for all real x (radius of convergence = ∞)

Calculus III

Multivariable Calculus

Multivariable calculus extends single-variable ideas into multiple dimensions. This is the mathematical language of physics, fluid dynamics, electromagnetism, and reactor theory.

Topics

Partial derivatives: ∂f/∂x holds all other variables constant
Gradient: ∇f points in the direction of steepest ascent
Divergence: ∇·F measures outward flux per unit volume
Curl: ∇×F measures rotational tendency of a field
Double and triple integrals with change of coordinates
Stokes’ theorem: &iint; (∇×F)·dS = ∮ F·dr

Real Applications

Reactor neutron diffusion: the Laplacian ∇²φ appears in the diffusion equation
Gradient descent in ML: ∇L guides parameter updates
Maxwell’s equations: ∇×E and ∇·B govern electromagnetic fields
Heat equation: ∂T/∂t = α∇²T describes thermal diffusion

Common Errors

Mixing up the gradient (∇f, a vector) and the Laplacian (∇²f, a scalar) — the Laplacian is the divergence of the gradient
Integration order: when switching order of integration, the limits must be re-derived from the region, not just swapped
Forgetting the Jacobian when changing coordinates: dA = r dr dθ in polar, not just dr dθ

Worked Example 1

Find ∇f if f(x, y, z) = x²yz + z³

Show Full Solution

∂f/∂x = 2xyz ∂f/∂y = x²z ∂f/∂z = x²y + 3z²

∇f = ⟨2xyz, x²z, x²y + 3z²⟩

Worked Example 2

Evaluate &iint;_R x dA where R is the region 0 ≤ x ≤ 2, 0 ≤ y ≤ x.

Show Full Solution

&iint; x dA = ∫[0 to 2] ∫[0 to x] x dy dx

Inner integral: ∫[0 to x] x dy = xy|[0 to x] = x²

Outer integral: ∫[0 to 2] x² dx = x³/3|[0 to 2] = 8/3

Answer: 8/3

Linear Algebra

Linear algebra is the language of data science, quantum mechanics, and structural engineering. The concepts here — eigenvalues, SVD, transformations — are the mathematical backbone of modern ML.

Topics

Matrix operations: addition, multiplication, transpose, inverse
Determinants: Laplace expansion, properties
Systems of equations: row reduction and RREF
Eigenvalues: det(A − λI) = 0
Eigenvectors: (A − λI)v = 0
Singular Value Decomposition (SVD): A = UΣV¹
Vector spaces: span, basis, dimension, rank
Linear transformations: kernel and image

Why It Matters

SVD underlies PCA (dimensionality reduction) in ML
Eigenvalues of the neutron transport matrix determine reactor criticality
Neural network weight matrices are linear transformations
Least squares regression: solve (A¹A)x = A¹b
Quantum states are vectors in Hilbert space

Common Errors

Matrix multiplication order: AB ≠ BA in general. Multiplication is not commutative
Eigenvalue vs. eigenvector: the eigenvalue λ is a scalar; the eigenvector v is the nonzero vector satisfying Av = λv
Singular vs. invertible: a matrix is singular (non-invertible) if and only if its determinant is zero
Rank-nullity theorem: rank(A) + nullity(A) = number of columns — often forgotten on exams

Worked Example 1

Find the eigenvalues of A = [[3, 1], [0, 2]].

Show Full Solution

Solve det(A − λI) = 0:

det([[3−λ, 1], [0, 2−λ]]) = (3−λ)(2−λ) − 0 = 0

(3−λ)(2−λ) = 0 → λ = 3 or λ = 2

Note: for upper triangular matrices, eigenvalues are always the diagonal entries.

Worked Example 2

Is the set {[1, 2, 0], [0, 1, 3], [1, 3, 3]} linearly independent?

Show Full Solution

Compute the determinant of the matrix formed by the vectors as rows.

det = 1(1·3 − 3·3) − 2(0·3 − 3·1) + 0(0·3 − 1·1)

= 1(3−9) − 2(0−3) + 0 = −6 + 6 = 0

det = 0 → linearly dependent. Note that [1,3,3] = [1,2,0] + [0,1,3].

Worked Example 3

Find the inverse of A = [[2, 1], [5, 3]].

Show Full Solution

det(A) = 2·3 − 1·5 = 1

A²¹ = (1/det) · [[d, −b], [−c, a]] = [[3, −1], [−5, 2]]

Verify: A · A²¹ = [[2·3+1·(−5), 2·(−1)+1·2], [5·3+3·(−5), 5·(−1)+3·2]] = [[1,0],[0,1]] ✓

A²¹ = [[3, −1], [−5, 2]]

Differential Equations

Differential equations are the mathematical description of how things change. Every physics law of motion, every reactor kinetics model, every population growth equation is a differential equation.

Key Solution Forms Separable: ∫ dy/g(y) = ∫ f(x) dx
First-order linear: multiply by integrating factor μ = e^(∫ P dx)
2nd-order homogeneous (const coeff): try y = e^(rx) → characteristic equation
Undetermined coefficients: guess y_p based on form of g(x)
Laplace: L{y'} = sY − y(0) L{y''} = s²Y − sy(0) − y'(0)

Topics

First-order: separable, linear, exact equations
Second-order homogeneous with constant coefficients
Method of undetermined coefficients
Variation of parameters
Laplace transforms and inverse transforms
Systems of first-order ODEs
Phase plane analysis and stability

Real-World Applications

Reactor point kinetics: dN/dt = (ρ − β)/Λ N + ∑ λ_i C_i
Newton’s law of cooling: dT/dt = −k(T − T_∞)
RLC circuits: L d²q/dt² + R dq/dt + q/C = V(t)
Population dynamics: logistic equation dP/dt = rP(1 − P/K)

Common Errors

Particular vs. homogeneous solution: the general solution is y = y_h + y_p. Students often forget to add both parts
Sign errors in Laplace transforms: L{e^(at)} = 1/(s−a), not 1/(s+a)
Characteristic equation: for y'' + py' + qy = 0, the equation is r² + pr + q = 0. Sign errors here propagate through the entire problem
Initial conditions are applied to the general solution, not to y_h or y_p separately

Worked Example 1

Solve the separable ODE: dy/dx = xy, with y(0) = 2.

Show Full Solution

Separate: dy/y = x dx

Integrate both sides: ln|y| = x²/2 + C

Exponentiate: y = Ae^(x²/2) where A = e^C

Apply IC: 2 = Ae^0 = A → A = 2

y = 2e^(x²/2)

Worked Example 2

Solve: y'' − 5y' + 6y = 0

Show Full Solution

Characteristic equation: r² − 5r + 6 = 0 → (r−2)(r−3) = 0

Roots: r = 2, r = 3 (real, distinct)

y = C&sub1;e^(2x) + C&sub2;e^(3x)

Worked Example 3

Use Laplace transforms to solve: y' + 3y = e^(−2t), y(0) = 0.

Show Full Solution

Take the Laplace transform: sY − y(0) + 3Y = 1/(s+2)

With y(0) = 0: (s+3)Y = 1/(s+2)

Y = 1/[(s+2)(s+3)]

Partial fractions: Y = 1/(s+2) − 1/(s+3)

Inverse Laplace: y(t) = e^(−2t) − e^(−3t)

Recommended Books

Mathematics Textbooks

Dr. Preston's field-tested recommendations. Amazon links use the fissionlab-20 affiliate tag — your purchase supports free content at no additional cost.

Calculus

James Stewart

The standard university calculus text. Clear explanations, excellent problem sets from routine to challenging. Used in Calc I through III at most universities.

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Linear Algebra Done Right

Sheldon Axler

The best theoretical linear algebra book available. Builds intuition for vector spaces and linear maps before matrices. Highly recommended for CS and math majors.

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Introduction to Probability

Blitzstein & Hwang

Exceptional treatment of probability with real intuition, story proofs, and connections to statistics and ML. The accompanying Harvard Stat 110 lectures are free on YouTube.

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Discrete Mathematics

Kenneth Rosen

The standard reference for discrete math: logic, proofs, graph theory, combinatorics, and number theory. Essential for CS majors and anyone studying algorithms.

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Mathematics —From Fundamentalsto Advanced Topics

Algebra & Pre-Calculus

Topics Covered

Key Relationships to Memorize

Common Errors

Limits & Derivatives

Key Concepts

Applied Uses Dr. Preston Uses Daily

Common Errors

Integration & Series

Key Techniques

Convergence Tests

Common Errors

Multivariable Calculus

Topics

Real Applications

Common Errors

Linear Algebra

Topics

Why It Matters

Common Errors

Differential Equations

Topics

Real-World Applications

Common Errors

Mathematics Textbooks

Study With Other STEM Students

Ready for 1:1 Math Tutoring?

Nuclear Engineer Explains

Mathematics —
From Fundamentals
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