Unit 1 — Limits & Continuity
Memory Point
DIVIDE top & bottom by highest power of x → leading coefficients give the limit
Quick Example
\(\displaystyle\lim_{x\to\infty}\frac{3x^2+1}{x^2-5} = \frac{3}{1} = 3\) (divide by \(x^2\))
Evaluate: \(\displaystyle\lim_{x \to \infty} \frac{5x^3 - 2x + 7}{2x^3 + 4x^2 - 1}\)
Memory Point
0/0 or ∞/∞? → L'Hôpital: diff top AND bottom SEPARATELY — not the quotient rule!
Quick Example
\(\displaystyle\lim_{x\to 0}\frac{\sin x}{x} \overset{L'H}{=} \frac{\cos x}{1}\bigg|_{x=0} = 1\)
Evaluate: \(\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\)
Unit 2 — Differentiation
Memory Point
OUTSIDE × INSIDE' — diff outer, keep inner unchanged, multiply by inner's derivative
Let \(f(x) = \sin(3x^2 + 1)\). Find \(f'(x)\).
Memory Point
Every time you diff a y-term, attach dy/dx → collect all dy/dx on one side and factor
Given \(x^2 + y^2 = 25\), find \(\dfrac{dy}{dx}\) at the point \((3,\,4)\).
Memory Point
ln rule: \(\tfrac{d}{dx}[\ln u] = \tfrac{u'}{u}\) — always divide by the inside function
Find \(\dfrac{d}{dx}\bigl[\ln(\cos x)\bigr]\).
Unit 3 — Integration Techniques
Memory Point
SPOT inner + its derivative nearby → set u = inside, swap du, integrate, sub back
Quick Example
\(\int 2x\cos(x^2)\,dx\): let \(u=x^2,\; du=2x\,dx\) → \(\int\cos u\,du = \sin(x^2)+C\)
Evaluate: \(\displaystyle\int 3x^2 e^{x^3}\,dx\)
Memory Point
LIATE — pick u as: Log › Inverse trig › Algebraic › Trig › Exponential
Evaluate: \(\displaystyle\int x e^x\,dx\)
Memory Point
FTC Part 2: \(\int_a^b f(x)\,dx = F(b)-F(a)\) — TOP minus BOTTOM, no +C needed
Evaluate: \(\displaystyle\int_0^{\pi/2} \cos x\,dx\)
Unit 4 — Applications of Integration
Memory Point
TOP minus BOTTOM → \(\int_a^b[\text{top}-\text{bottom}]\,dx\). Find intersections for limits first!
Find the area of the region enclosed by \(y = x^2\) and \(y = x\).
Memory Point
FTC 1 + Chain Rule: \(\dfrac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)\) — plug in, then multiply by derivative of upper limit
Find \(\dfrac{d}{dx}\displaystyle\int_1^{x^2} \sin(t)\,dt\).
Unit 5 — Differential Equations
Memory Point
SEPARATE → INTEGRATE → SOLVE for y — all y's left side, all x's right side, then exponentiate
Solve: \(\dfrac{dy}{dx} = 2xy\), with initial condition \(y(0) = 3\).
Memory Point
NEW = OLD + step × slope: \(y_{n+1} = y_n + h\cdot f(x_n,y_n)\) — repeat for each step
Using Euler's Method with step size \(h=0.1\), starting at \((0,1)\) with \(\dfrac{dy}{dx} = y\), approximate \(y(0.2)\).
Unit 6 — Sequences & Series
Memory Point
Sum \(= \dfrac{a}{1-r}\) only when \(|r|<1\). Check |r| first — if |r| ≥ 1, it DIVERGES!
Find the sum: \(\displaystyle\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n\)
Memory Point
L < 1 → converge | L > 1 → diverge | L = 1 → inconclusive (try another test!)
For \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^n}\), what is \(L = \displaystyle\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|\)?
Memory Point
Big 3: \(e^x=\sum\tfrac{x^n}{n!}\) · \(\sin x=\sum\tfrac{(-1)^n x^{2n+1}}{(2n+1)!}\) · \(\cos x=\sum\tfrac{(-1)^n x^{2n}}{(2n)!}\)
The Maclaurin series for \(e^{-x^2}\) begins with which terms?
Unit 7 — Parametric & Polar Curves
Memory Point
dy/dx = (dy/dt) ÷ (dx/dt) — diff each w.r.t. t, then divide. Never diff parametric w.r.t. x directly!
A curve is defined by \(x = t^2\) and \(y = t^3 - 3t\). Find \(\dfrac{dy}{dx}\) at \(t = 2\).
Memory Point
Polar Area = ½∫r² dθ — the ½ is ALWAYS there. Find bounds where the loop lives first.
Find the area enclosed by the polar curve \(r = 2\cos\theta\) for one complete loop.
Unit 8 — BC Special Topics
Memory Point
Replace ∞ with limit b → ∞ — write \(\lim_{b\to\infty}\int_1^b\), evaluate, then take the limit
Determine whether \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\) converges. If so, find its value.
Memory Point
Arc Length: \(\int_a^b\sqrt{1+[f'(x)]^2}\,dx\) — square the derivative, add 1, then root
Set up (do not evaluate) the arc length of \(f(x) = x^{3/2}\) from \(x=0\) to \(x=4\).
Memory Point
Radius R = 1/L from Ratio Test → converges for \(|x-\text{center}| < R\). Always check endpoints separately!
Find the radius of convergence of \(\displaystyle\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}\).
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