Middle term of a binomial

“The biggest coefficients cluster in the middle of the expansion.”

For (a+b)^n, there's one middle term if n is even, two if n is odd.

Memory trick: Count n+1 terms.

Sum of binomial coefficients

“Put x=1 and the whole expansion collapses to 2^n.”

The coefficients of (1+x)^n add up to 2^n.

ΣnCr = 2ⁿ

Memory trick: Set x=1 to sum them.

Section formula

“It's how mapmakers place a town a certain fraction along a road.”

A point dividing a segment in ratio m:n has coordinates ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

Memory trick: Weighted average of endpoints.

Distance of a point from a line

“The shortest hop from a point to a line is always perpendicular.”

d = |ax₀+by₀+c| / √(a²+b²).

d=|ax₀+by₀+c|/√(a²+b²)

Memory trick: Use the perpendicular distance.

Angle between two lines

“The formula quietly reveals parallel (θ=0) and perpendicular (denominator 0) cases.”

tanθ = |(m₁−m₂)/(1+m₁m₂)| from their slopes.

Memory trick: Slopes → angle.

Tangent to a circle

“A wheel touches the road along a tangent line.”

A tangent touches a circle at exactly one point and is perpendicular to the radius there.

Memory trick: Tangent ⊥ radius.

Ellipse equation

“Planets orbit the Sun in ellipses, not perfect circles.”

x²/a² + y²/b² = 1 describes an oval with two foci.

x²/a²+y²/b²=1

Memory trick: Sum of focal distances is constant.

Hyperbola equation

“A ship's LORAN navigation traced hyperbolas of constant time-difference.”

x²/a² − y²/b² = 1 is an open two-branch curve.

x²/a²−y²/b²=1

Memory trick: Difference of focal distances is constant.

Latus rectum

“It's a quick measure of how 'wide' a conic opens at the focus.”

The chord through a conic's focus, perpendicular to its axis.

Memory trick: Chord through focus ⊥ axis.

Derivative of sine and cosine

“Sine and cosine chase each other's derivatives in a four-step cycle.”

d/dx(sin x)=cos x and d/dx(cos x)=−sin x.

Memory trick: cos has the minus.

Derivative of e^x and ln x

“e^x is the only function that is its own derivative.”

d/dx(e^x)=e^x and d/dx(ln x)=1/x.

Memory trick: e^x is self-derivative.

Quotient rule

“The order in the numerator matters — top-derivative first.”

d/dx(u/v) = (u'v − uv')/v².

(u/v)'=(u'v−uv')/v²

Memory trick: (low·dHigh − high·dLow)/low².

Implicit differentiation

“It handles curves like circles that aren't neat y = f(x).”

Differentiate both sides treating y as a function of x, then solve for dy/dx.

Memory trick: Differentiate y, then isolate dy/dx.

Rolle's theorem

“If you return to the same height, somewhere your path was momentarily flat.”

If f is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then f'(c)=0 for some c.

Memory trick: Equal ends → a flat point.

Mean value theorem

“On any trip, at some instant your speed equalled your average speed.”

Somewhere on a smooth curve, the instantaneous slope equals the average slope.

Memory trick: Instant slope = average slope.

Integration by parts

“It trades a hard integral for an easier one, one factor at a time.”

∫u·dv = uv − ∫v·du.

∫u dv = uv − ∫v du

Memory trick: Choose u by ILATE.

Integration by substitution

“Spotting the right substitution is like finding the loose thread that unravels the knot.”

Replace a chunk with a new variable to simplify the integral.

Memory trick: Undo the chain rule.

Standard integral of 1/x

“The power rule breaks at n=−1 — and the logarithm rides to the rescue.”

∫(1/x)dx = ln|x| + C.

∫1/x dx=ln|x|+C

Memory trick: 1/x integrates to ln|x|.

Property of definite integrals

“This 'king' property often collapses a nasty integral in one line.”

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx.

Memory trick: Swap x → a+b−x.

Area between curves

“Integration measures the exact area of shapes no formula covers.”

The area between two curves is the integral of (top − bottom).

Memory trick: ∫(top − bottom).

Order and degree of a differential equation

“They classify a differential equation before you try to solve it.”

Order is the highest derivative present; degree is its power after clearing radicals.

Memory trick: Order = highest derivative.

Variable separable equations

“The simplest differential equations just ask you to sort x's and y's apart.”

If dy/dx = f(x)g(y), separate variables and integrate both sides.

Memory trick: Get x's and y's on opposite sides.

Vector addition (triangle law)

“Rowing across a river, your path is the resultant of boat and current.”

Two vectors add head-to-tail, the resultant closing the triangle.

Memory trick: Head-to-tail resultant.

Unit vector

“It strips away size and keeps only 'which way'.”

A vector of magnitude 1 giving pure direction, â = a/|a|.

â = a/|a|

Memory trick: Divide by its own length.