- College prepβ«Calculusπ
Taylor expansions at 0: the 6 essentials for prep classes
$e^x$, $\ln(1+x)$, $\sin$, $\cos$, $(1+x)^\alpha$, $\dfrac{1}{1-x}$ to order 3. The table that unlocks all 0-limits.
- College prepπ€Algebraπ’
Matrices: addition, multiplication, transpose
The 3 basic matrix operations. Dimension rules, non-commutativity, transpose.
- College prepβ«Calculusπ
Linear ODE y' = ay + b: the one-formula solution
Any first-order linear ODE with constant coefficients solves in 30 seconds with this formula. Bonus: initial condition.
- College prepβ«CalculusβΎοΈ
Why the harmonic series diverges (but $\sum 1/n^2$ doesn't)
$\sum 1/n$ diverges, while $\sum 1/n^2$ converges. Elegant proof by grouping, plus the Riemann criterion.
- College prepπ€Algebraβ‘
Diagonalize a 2Γ2 matrix in 60 seconds
Trace, determinant, discriminant. Three calculations and boom, your eigenvalues. Then eigenvectors in three lines each.
- College prepπ€AlgebraπΊ
$(a+b)^n$: binomial theorem and Pascal's triangle
Expand $(a+b)^n$ without multiplying $n$ times: coefficients come from Pascal's triangle, powers follow a simple rule.