- College prep∫Calculus📐
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.
- High school∫Calculus👮
Squeeze theorem: sandwich to conclude
If v_n ≤ u_n ≤ w_n and v_n, w_n both tend to ℓ, so does u_n. A classic for tricky limits.
- High school∫Calculus📈
Product rule derivative: the « u'v + uv' » mnemonic
A little phrase to chant so you never mix it up again, and an example that clicks.
- High school∫Calculus📉
Increasing + bounded above ⇒ converges: the star theorem
Any increasing, bounded-above sequence converges (to a limit ≤ bound). The theorem that cracks the exam.
- High school∫Calculus🔓
The 5 logarithm properties to know 100%
$\ln$ turns products into sums, powers into products. Five formulas that unlock terminal-year calculus.
- College prep∫Calculus🌀
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.
- High school∫Calculus🚀
Limit at ±∞: keep only the dominant powers
At ±∞, only the highest-degree term matters. The rest gets crushed. A trick that simplifies 90% of polynomial and rational function limits.
- High school∫Calculus📐
Linear approximation: f(a+h) ≈ f(a) + h·f'(a)
Near a, the function behaves like its tangent. Engineers' mental-math trick.
- High school∫Calculus∫
Integration by parts: the formula and when to use it
∫ u'v = [uv] − ∫ uv'. The trick that cracks ∫ x·ln(x), ∫ x·eˣ, ∫ x·cos(x).
- High school∫Calculus🎯
Intermediate Value Theorem (IVT): the 3-step method
Continuous function + sign change = at least one solution. The classic exam problem.
- High school∫Calculus🪜
Mathematical induction: the 3-step method
Base case → Inductive step → Conclusion. The ritual that proves a property is true for every integer n.
- High school∫Calculus⚠️
Spot an indeterminate form: the 4 classics
$\infty - \infty$, $\dfrac{0}{0}$, $\dfrac{\infty}{\infty}$, $0 \times \infty$. The 4 traps to catch so you never say « the limit exists » too fast.