#studynotes — Public Fediverse posts

Alright, future engineers!
**Trapezoidal Rule:** Approximates definite integrals by summing areas of trapezoids under the curve.
Ex: For `y=f(x)`, area over `[a,b]` is `~ (b-a)/2 * (f(a)+f(b))`.
Pro-Tip: Use more trapezoids (smaller `h`) for better accuracy!
#NumericalIntegration #CalcHacks #STEM #StudyNotes

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**Interleaving:** Mixing different problem types or topics in one study session.
Ex: Study Ch 1 Calc, then Ch 3 Physics, then Ch 2 Diff Eq.
Pro-Tip: Boosts problem-solving flexibility by forcing your brain to discriminate strategies!
#StudySmart #BrainHacks #STEM #StudyNotes

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**Spaced Repetition:** Reviewing material at increasing intervals to move info to long-term memory.
Ex: Study a topic today, then in 3 days, 7 days, 14 days.
Pro-Tip: Use flashcard apps (Anki!) to automate your personalized review schedule!
#StudySmart #MemoryHacks #STEM #StudyNotes

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**Work (W):** Energy transferred when a force causes displacement.
Ex: `W = F * d * cos(theta)`. Pushing a box.
Pro-Tip: Only the force component *parallel* to the displacement does work! Perpendicular forces do none.
#Mechanics #EnergyTransfer #STEM #StudyNotes

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**Determinant:** A single scalar value derived from a square matrix.
Ex: For `[[a,b],[c,d]]`, `det = ad - bc`.
Pro-Tip: If `det = 0`, the matrix is NOT invertible (no unique solution for systems!).
#LinearAlgebra #MatrixTips #STEM #StudyNotes

Alright, future engineers!
**Determinant:** A single scalar value derived from a square matrix.
Ex: For `[[a,b],[c,d]]`, `det = ad - bc`.
Pro-Tip: If `det = 0`, the matrix is NOT invertible (no unique solution for systems!).
#LinearAlgebra #MatrixTips #STEM #StudyNotes

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**Newton's Third Law:** For every action, there is an equal & opposite reaction force.
Ex: You push a wall, the wall pushes back on you.
Pro-Tip: These forces act on *different* objects, never on the same one! #Physics #N3L #STEM #StudyNotes

Alright, future engineers!
**Newton's Third Law:** For every action, there is an equal & opposite reaction force.
Ex: You push a wall, the wall pushes back on you.
Pro-Tip: These forces act on *different* objects, never on the same one! #Physics #N3L #STEM #StudyNotes

Alright, future engineers!
**Newton-Raphson:** An iterative method to efficiently find the roots (where f(x)=0) of a function.
Ex: `x_n+1 = x_n - f(x_n)/f'(x_n)`
Pro-Tip: Your initial guess is CRUCIAL! A poor one can lead to divergence or the wrong root.
#NumericalMethods #RootFinding #STEM #StudyNotes

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**Bayes' Theorem:** Updates an event's probability (hypothesis) based on new evidence.
Ex: P(H|E) = [P(E|H) * P(H)] / P(E).
Pro-Tip: Essential for diagnostic tests & updating beliefs with new data! Use it to refine predictions.
#ProbStats #Bayesian #STEM #StudyNotes

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**Torque (τ):** A twisting force that causes rotation.
Ex: `τ = r * F * sin(theta)`. Use a long wrench (large r) for easier turning!
Pro-Tip: Align the force perpendicular to the lever arm (theta=90) for max effect!
#Physics #Rotational #STEM #StudyNotes

Alright, future engineers!
**Torque (τ):** A twisting force that causes rotation.
Ex: `τ = r * F * sin(theta)`. Use a long wrench (large r) for easier turning!
Pro-Tip: Align the force perpendicular to the lever arm (theta=90) for max effect!
#Physics #Rotational #STEM #StudyNotes

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**Feynman Technique:** Explaining a tough concept simply, as if to a child.
Ex: Learn integration -> explain it to your little sibling or a rubber duck.
Pro-Tip: Gaps in your explanation? That's where to focus your next study session!
#StudyHacks #FeynmanTech #STEM #StudyNotes

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**Matrix Addition:** Combine matrices by adding their corresponding elements.
Ex: If A=[1,2;3,4] & B=[5,6;7,8], then A+B = [6,8;10,12].
Pro-Tip: They MUST have the same dimensions to add them!
#LinearAlgebra #MatrixMath #STEM #StudyNotes

Alright, future engineers!
**Matrix Addition:** Combine matrices by adding their corresponding elements.
Ex: If A=[1,2;3,4] & B=[5,6;7,8], then A+B = [6,8;10,12].
Pro-Tip: They MUST have the same dimensions to add them!
#LinearAlgebra #MatrixMath #STEM #StudyNotes

**Conditional Probability:** The likelihood of an event occurring *given* another event has already occurred.
Ex: P(A|B) = P(A and B) / P(B).
Pro-Tip: The given event B *defines* your new, reduced sample space. Focus only on outcomes where B happened!
#ProbStats #DecisionMaking #STEM #StudyNotes

Alright, future engineers!
**Quadratic Formula:** Solves `ax^2+bx+c=0` for x.
Ex: `x = [-b +/- sqrt(b^2-4ac)] / 2a`
Pro-Tip: The discriminant `(b^2-4ac)` tells you if roots are real or complex & how many there are!
#Algebra #MathTips #STEM #StudyNotes

Alright, future engineers!
**Quadratic Formula:** Solves `ax^2+bx+c=0` for x.
Ex: `x = [-b +/- sqrt(b^2-4ac)] / 2a`
Pro-Tip: The discriminant `(b^2-4ac)` tells you if roots are real or complex & how many there are!
#Algebra #MathTips #STEM #StudyNotes

Alright, future engineers!
**Dot Product:** A scalar indicating how much two vectors point in the same direction.
Ex: For `v=[1,2]` & `u=[3,4]`, `v.u = (1*3) + (2*4) = 11`.
Pro-Tip: If `v.u = 0`, the vectors are orthogonal (perpendicular)!
#LinearAlgebra #VectorMath #STEM #StudyNotes

Alright, future engineers!
**Dot Product:** A scalar indicating how much two vectors point in the same direction.
Ex: For `v=[1,2]` & `u=[3,4]`, `v.u = (1*3) + (2*4) = 11`.
Pro-Tip: If `v.u = 0`, the vectors are orthogonal (perpendicular)!
#LinearAlgebra #VectorMath #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term is a function of its preceding terms.
Ex: Fibonacci: `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for analyzing algorithms & modeling dynamic systems!
#DiscreteMath #Algorithms #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term is a function of its preceding terms.
Ex: Fibonacci: `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for analyzing algorithms & modeling dynamic systems!
#DiscreteMath #Algorithms #STEM #StudyNotes

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**Active Recall:** Retrieving info from memory *without* notes.
Ex: After studying, close your book & list all key concepts.
Pro-Tip: It's harder than rereading, but *way* more effective for long-term retention!
#StudyHacks #RetrievalPractice #STEM #StudyNotes

Alright, future engineers!
**Impulse:** The change in an object's momentum, or force applied over time.
Ex: `Impulse = F_avg * dt`. Catching a ball softly increases `dt`, reducing `F_avg`.
Pro-Tip: Great for collision analysis—it links force, time, & momentum change!
#Mechanics #Collisions #STEM #StudyNotes

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**Mean Value Theorem:** For continuous, differentiable `f` on `[a,b]`, there's a `c` where `f'(c)` equals the avg rate of change.
Ex: `f'(c) = (f(b)-f(a))/(b-a)`
Pro-Tip: Guarantees a tangent line parallel to the secant line!
#Calculus #MVT #STEM #StudyNotes

Alright, future engineers!
**Derivative:** The instantaneous rate of change of a function, or the slope of the tangent line at any point.
Ex: If `f(x)=x^2`, then `f'(x)=2x`.
Pro-Tip: Always think slope! It reveals how fast something is changing *right now*.
#Calculus #Derivatives #STEM #StudyNotes

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**Limit:** The value a function *approaches* as its input gets infinitely close to a point.
Ex: `lim (x->2) (x+1) = 3`.
Pro-Tip: It's about the value *near* the point, not necessarily *at* it! Crucial for derivatives.
#Calculus #Limits #STEM #StudyNotes

**Work (W):** Energy transferred when a force causes displacement.
Ex: `W = F * d` (when force & displacement are parallel).
Pro-Tip: Only the force component parallel to displacement does work! #Physics #EnergyTransfer #STEM #StudyNotes

**Work (W):** Energy transferred when a force causes displacement.
Ex: `W = F * d` (when force & displacement are parallel).
Pro-Tip: Only the force component parallel to displacement does work! #Physics #EnergyTransfer #STEM #StudyNotes

Alright, future engineers!
**Pigeonhole Principle:** If you have more items than categories, at least one category must contain more than one item.
Ex: Place 5 socks into 4 drawers. At least one drawer has 2+ socks.
Pro-Tip: Identify your items & categories to apply it effectively!
#DiscreteMath #LogicPuzzles #STEM #StudyNotes

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**Expected Value (E[X]):** The long-run average outcome of a random variable.
Ex: For discrete X, `E[X] = Sum(x * P(x))`. Avg winnings in a game!
Pro-Tip: Key for decision-making in uncertainty & risk assessment.
#ProbStats #RiskAnalysis #STEM #StudyNotes

Alright, future engineers!
**Trapezoidal Rule:** Approximates definite integrals (area under a curve) by summing trapezoid areas.
Ex: For 1 segment, `Area ~ (f(a)+f(b))*h/2`
Pro-Tip: Smaller step size (h) yields greater accuracy, but at a computational cost!
#NumericalIntegration #MathTips #STEM #StudyNotes

Alright, future engineers!
**Graph:** A set of vertices (nodes) connected by edges.
Ex: Social networks: people are nodes, friendships are edges.
Pro-Tip: Vertices can exist without edges (isolated nodes)!
#GraphTheory #DiscreteMath #STEM #StudyNotes

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**Combinations:** Choosing items where order *doesn't* matter.
Ex: Selecting 3 friends from 5 for a team. C(5,3) = 10 ways.
Pro-Tip: Think groups or sets. Changing the order doesn't create a new combination!
#Combinatorics #ProbStats #STEM #StudyNotes

Alright, future engineers!
**Feynman Technique:** Explain complex ideas in simple terms, as if to a child.
Ex: Teach a difficult theorem out loud without notes. If you stumble, review & simplify.
Pro-Tip: It instantly exposes your knowledge gaps, forcing true understanding, not just memorization!
#StudyHacks #DeepLearning #STEM #StudyNotes

Alright, future engineers!
**Kinetic Energy (KE):** Energy an object possesses due to its motion.
Ex: `KE = 0.5 * m * v^2`. A 1kg mass moving at 2m/s has 2J KE.
Pro-Tip: Speed (v) has a squared effect on KE—a small v change means huge KE change!
#Physics #Energy #STEM #StudyNotes

Alright, future engineers!
**Kinetic Energy (KE):** Energy an object possesses due to its motion.
Ex: `KE = 0.5 * m * v^2`. A 1kg mass moving at 2m/s has 2J KE.
Pro-Tip: Speed (v) has a squared effect on KE—a small v change means huge KE change!
#Physics #Energy #STEM #StudyNotes

Alright, future engineers!
**Power Set (P(S)):** The set of all possible subsets of a given set S, including the empty set & S itself.
Ex: If S={1,2}, P(S) = {{}, {1}, {2}, {1,2}}. Size is 2^n.
Pro-Tip: Every element in P(S) is itself a set!
#SetTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Indefinite Integral:** The reverse of differentiation; finding the function whose derivative is given.
Ex: `int(2x dx) = x^2 + C`
Pro-Tip: Don't forget the `+C`! It represents any constant that vanished during differentiation.
#Calculus #Integration #STEM #StudyNotes

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**Normal Distribution:** A common bell-shaped probability distribution where most data clusters around the mean.
Ex: Heights or test scores often follow this curve. `X ~ N(mu, sigma^2)`
Pro-Tip: The 68-95-99.7 Rule is your shortcut for understanding data spread!

#ProbStats #DataScience #STEM #StudyNotes

Alright, future engineers!
**Inverse Functions:** A function that undoes another function, effectively reversing its operation.
Ex: If `f(x)=2x+1`, its inverse `f^-1(x)=(x-1)/2`.
Pro-Tip: To find it, swap `x` and `y`, then solve for the new `y`!
#Algebra #Functions #STEM #StudyNotes

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**Power Rule for Exponents:** To raise a power to another power, you multiply the exponents.
Ex: `(x^a)^b = x^(a*b)`. So, `(x^2)^3 = x^6`.
Pro-Tip: Don't confuse it with `x^a * x^b = x^(a+b)`! Totally different rules!
#Algebra #ExponentRules #STEM #StudyNotes

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**Rational Expressions:** Fractions with polynomials in the numerator and/or denominator.
Ex: `(x^2 - 4) / (x + 2)`
Pro-Tip: Always factor first to simplify & identify values that make the denominator zero!
#Algebra #Simplify #STEM #StudyNotes

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**Factoring Quadratics:** Expressing `ax^2 + bx + c` as a product of linear terms, often `(x-r1)(x-r2)`.
Ex: `x^2 + 7x + 10 = (x+2)(x+5)`
Pro-Tip: Look for two numbers that multiply to 'c' and add to 'b'. Simplifies finding roots!
#Algebra #MathSkills #STEM #StudyNotes

Alright, future engineers!
**Linear Combination:** Sum of vectors multiplied by scalars.
Ex: `c1*v1 + c2*v2`
Pro-Tip: Fundamental for understanding span & basis for vector spaces!
#LinearAlgebra #VectorMath #STEM #StudyNotes

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**Logarithms:** The exponent a base needs to become a number.
Ex: `log_2(8)=3` means `2^3=8`.
Pro-Tip: `log` IS the exponent! Essential for solving `x` in `b^x=N`.
#Algebra #MathTips #STEM #StudyNotes

Alright, future engineers!
**Impulse (J):** Change in an object's momentum.
Ex: `J = F_avg * delta_t`. A car crash reduces force with longer impact time (airbag!).
Pro-Tip: Same momentum change, but longer time means less average force! Think safety.
#Physics #Mechanics #STEM #StudyNotes

Alright, future engineers!
**Dot Product:** A scalar result measuring how much two vectors point in the same direction.
Ex: For `v=[a,b], w=[c,d]`, `v.w = ac+bd`.
Pro-Tip: If `v.w = 0`, vectors are orthogonal (perpendicular)!
#LinearAlgebra #VectorMath #STEM #StudyNotes

Alright, future engineers!
**Quadratic Formula:** Solves for `x` in `ax^2 + bx + c = 0`.
Ex: `x = [-b +- sqrt(b^2 - 4ac)] / 2a`
Pro-Tip: The `b^2 - 4ac` part (discriminant) tells you if you have 0, 1, or 2 real solutions!
#Algebra #MathSkills #STEM #StudyNotes

Alright, future engineers!
**Newton's 2nd Law:** Net Force (F) equals Mass (m) x Acceleration (a). (F=ma)
Ex: 10N on 2kg mass -> 5 m/s^2 accel.
Pro-Tip: F is the *vector sum* of ALL forces! Direction matters for `a`.
#Physics #Dynamics #STEM #StudyNotes

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**Newton's 1st Law (Inertia):** An object's velocity won't change unless a net force acts on it.
Ex: A space probe coasts at constant velocity because no friction or air drag.
Pro-Tip: Mass is the measure of inertia! More mass = harder to change its motion.

#Physics #Mechanics #STEM #StudyNotes

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**Logarithm:** Finds the exponent you need to raise a base to get a number.
Ex: `log_b(x) = y` means `b^y = x`.
Pro-Tip: Use change of base: `log_b(x) = ln(x)/ln(b)` for calculations!
#Algebra #Logarithms #STEM #StudyNotes

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**Linear Independence:** Vectors are LI if no vector can be expressed as a linear combo of the others.
Ex: For `c1*v1 + c2*v2 = 0`, only `c1=c2=0` works.
Pro-Tip: LI vectors are unique & don't introduce redundant info. Key for basis & unique solutions!
#LinearAlgebra #VectorSpaces #STEM #StudyNotes

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**Standard Deviation (SD):** Quantifies how spread out data points are from the mean.
Ex: Low SD means data is tight; high SD means wide range.
Pro-Tip: Essential for gauging precision & consistency in measurements & models!
#Stats #DataScience #STEM #StudyNotes

**Work:** Energy transferred by a force causing displacement.
Ex: Lifting a `10N` weight `2m` up means `W = 10N * 2m = 20J`.
Pro-Tip: Only the force component *parallel* to motion does work! Perpendicular force does *zero* work.
#Physics #Mechanics #STEM #StudyNotes

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**Eigenvector:** A non-zero vector whose direction doesn't change when transformed by a matrix. It only scales!
Ex: `Av = λv`, where `v` is the eigenvector & `λ` its eigenvalue.
Pro-Tip: They uncover the natural directions of a matrix's action. Essential for system stability!
#LinearAlgebra #EigenStuff #STEM #StudyNotes

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**Power Set:** The set of ALL possible subsets of a given set.
Ex: For `A = {1, 2}`, `P(A) = { {}, {1}, {2}, {1, 2} }`.
Pro-Tip: If a set has `n` elements, its power set has `2^n` elements!
#SetTheory #DiscreteMath #STEM #StudyNotes

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**Factoring:** Breaking a polynomial into simpler expressions (factors) that multiply back to the original.
Ex: `x^2 - 4 = (x-2)(x+2)`.
Pro-Tip: Always look for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

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**Matrix Multiplication:** Produces a new matrix by combining rows of the first with columns of the second.
Ex: For `C = AB`, `C_ij = sum(A_ik * B_kj)`.
Pro-Tip: Inner dimensions MUST match! `(m x n) * (n x p)` gives `(m x p)`.
#MatrixMath #LinearAlgebra #STEM #StudyNotes

Alright, future engineers!
**Dot Product:** A scalar value indicating how much two vectors point in the same direction.
Ex: For `u=[1,2]`, `v=[3,4]`, `u.v = (1*3) + (2*4) = 11`.
Pro-Tip: If `u.v = 0`, the vectors are orthogonal (perpendicular)! Key for geometric analysis!
#VectorMath #LinearAlgebra #STEM #StudyNotes