Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
The definite integral gives the (signed) area under a curve between two limits.
Memory trick: split the interval at the roots if you need total (unsigned) area.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
A frequent error is ignoring that area below the x-axis counts as negative. In reality, the definite integral gives the (signed) area under a curve between two limits.
Memory trick: split the interval at the roots if you need total (unsigned) area.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
Substitution, integration by parts, and partial fractions handle most integrals.
Memory trick: try substitution first; use by-parts for products like x·e^x.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
A frequent error is reaching for by-parts when a simple substitution works. In reality, substitution, integration by parts, and partial fractions handle most integrals.
Memory trick: try substitution first; use by-parts for products like x·e^x.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
The power rule for integration. Use it when n ≠ −1.
∫xⁿ dx = x^(n+1)/(n+1) + C
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
The integral of 1/x. Use it when x ≠ 0.
∫(1/x) dx = ln|x| + C
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
A definite integral has limits and gives a number (area); an indefinite integral has no limits and gives a family of functions (+C).
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
Differentiation finds a rate/slope from a function; integration reverses it to recover the function or find accumulated area.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
Every indefinite integral needs the arbitrary constant of integration.
“Weather forecasts, insurance and cricket predictions all run on probability.”
Probability = favourable outcomes ÷ total outcomes, always between 0 and 1.
Memory trick: if you get P > 1, recount the sample space.
“Weather forecasts, insurance and cricket predictions all run on probability.”
A frequent error is giving a probability greater than 1. In reality, probability = favourable outcomes ÷ total outcomes, always between 0 and 1.
Memory trick: if you get P > 1, recount the sample space.
“Weather forecasts, insurance and cricket predictions all run on probability.”
A frequent error is treating mutually exclusive events as independent. In reality, independent events don't affect each other; mutually exclusive events cannot both happen.
Memory trick: mutually exclusive events ARE dependent — one happening rules the other out.
“Weather forecasts, insurance and cricket predictions all run on probability.”
A frequent error is assuming P(A|B) equals P(B|A). In reality, P(A|B) is the probability of A given that B has occurred, equal to P(A∩B)/P(B).
Memory trick: conditioning direction matters — Bayes' theorem flips it correctly.
“Weather forecasts, insurance and cricket predictions all run on probability.”
The addition rule. Use it when any two events.
P(A∪B) = P(A)+P(B)−P(A∩B)
“Weather forecasts, insurance and cricket predictions all run on probability.”
Conditional probability. Use it when P(B) > 0.
P(A|B) = P(A∩B)/P(B)
“Weather forecasts, insurance and cricket predictions all run on probability.”
Independent events don't influence each other's probability; mutually exclusive events can't co-occur (so they're dependent).
“Weather forecasts, insurance and cricket predictions all run on probability.”
These conditional probabilities ask different questions and are generally unequal — Bayes' theorem relates them.
“Weather forecasts, insurance and cricket predictions all run on probability.”
Mutually exclusive events cannot both occur, so they are dependent, not independent.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
A frequent error is expecting the dot product to be a vector. In reality, a·b = |a||b|cosθ gives a scalar and is zero when the vectors are perpendicular.
Memory trick: dot product → scalar; it measures how much two vectors align.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
A frequent error is expecting the cross product to be a scalar. In reality, a×b = |a||b|sinθ·n̂ gives a vector perpendicular to both a and b.
Memory trick: cross product → vector; its length is the parallelogram's area.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
The components that fix a line's or vector's direction in 3D space.
Memory trick: direction cosines are direction ratios divided by the magnitude.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
A frequent error is confusing direction ratios with direction cosines (which are normalised). In reality, the components that fix a line's or vector's direction in 3D space.
Memory trick: direction cosines are direction ratios divided by the magnitude.
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
Dot product (scalar). Use it when θ is the angle between the vectors.
a·b = |a||b|cosθ
“A pilot adds wind and engine thrust as vectors to know where the plane actually goes.”
Magnitude of the cross product (vector). Use it when θ is the angle between the vectors.
|a×b| = |a||b|sinθ