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.
300 fundamentals
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
That's only true when f is continuous at a; limits can exist where the function is undefined.
“Your car's speedometer is a live derivative — the rate your distance is changing.”
A frequent error is confusing the derivative (a rate) with the function's value. In reality, the instantaneous rate of change of a function — the slope of the tangent at a point.
Memory trick: derivative = how fast the output changes as the input changes.
“Your car's speedometer is a live derivative — the rate your distance is changing.”
A frequent error is forgetting to multiply by the inner derivative. In reality, to differentiate a composite function, differentiate the outer function and multiply by the derivative of the inner.
Memory trick: for sin(2x), it's cos(2x)×2 — don't drop the ×2.
“Your car's speedometer is a live derivative — the rate your distance is changing.”
The derivative of a product is u'v+uv'; of a quotient is (u'v−uv')/v².
Memory trick: the derivative of a product is NOT the product of derivatives.
“Your car's speedometer is a live derivative — the rate your distance is changing.”
A frequent error is differentiating a product as just u'v'. In reality, the derivative of a product is u'v+uv'; of a quotient is (u'v−uv')/v².
Memory trick: the derivative of a product is NOT the product of derivatives.
“Your car's speedometer is a live derivative — the rate your distance is changing.”
The power rule. Use it when any real exponent n.
d/dx(xⁿ) = n·x^(n−1)
“Your car's speedometer is a live derivative — the rate your distance is changing.”
The chain rule for composite functions. Use it when y is a function of u, u of x.
dy/dx = dy/du · du/dx
“Your car's speedometer is a live derivative — the rate your distance is changing.”
The derivative dy/dx is a rate (slope); the differential dy = f'(x)dx is the small change it predicts for a small dx.
“Your car's speedometer is a live derivative — the rate your distance is changing.”
The product rule differentiates u·v as u'v+uv'; the quotient rule differentiates u/v as (u'v−uv')/v².
“Your car's speedometer is a live derivative — the rate your distance is changing.”
For composite functions you must multiply by the derivative of the inner function.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
Integration reverses differentiation; an indefinite integral always carries a constant of integration +C.
Memory trick: no +C means an incomplete indefinite integral.
“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”
A frequent error is forgetting to add +C to an indefinite integral. In reality, integration reverses differentiation; an indefinite integral always carries a constant of integration +C.
Memory trick: no +C means an incomplete indefinite integral.
“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.