AP Calculus AB/BC — Premium Practice Exam

📏 1. Limits & Continuity

A limit \(\lim_{x\to c}f(x)=L\) exists iff left-hand and right-hand limits are equal and equal \(L\).
A function is continuous at \(c\) if: defined, limit exists, and \(f(c)=\lim_{x\to c}f(x)\).
Squeeze Theorem: if \(g(x)\le f(x)\le h(x)\) and \(\lim g=\lim h=L\), then \(\lim f=L\).

\(\lim_{x\to 0}\dfrac{\sin x}{x}=1\) | \(\lim_{x\to\infty}\left(1+\tfrac{1}{x}\right)^x=e\)

Example: \(\lim_{x\to 0}\dfrac{\sin 3x}{x}=3\cdot\lim_{x\to0}\dfrac{\sin 3x}{3x}=3\cdot1=3\) ✓

📐 2. Definition & Rules of Differentiation

Definition: \(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\)
Power Rule: \(\dfrac{d}{dx}x^n=nx^{n-1}\)
Product Rule: \((uv)'=u'v+uv'\)
Quotient Rule: \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\)
Chain Rule: \(\dfrac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)\)

\(\tfrac{d}{dx}\sin x=\cos x\) | \(\tfrac{d}{dx}\cos x=-\sin x\) | \(\tfrac{d}{dx}\tan x=\sec^2 x\)
\(\tfrac{d}{dx}e^x=e^x\) | \(\tfrac{d}{dx}\ln x=\tfrac{1}{x}\) | \(\tfrac{d}{dx}a^x=a^x\ln a\)

Example: \(f(x)=x^3\sin x\Rightarrow f'(x)=3x^2\sin x+x^3\cos x\) (Product Rule)

📈 3. Applications of Derivatives

Critical points: \(f'(c)=0\) or \(f'(c)\) undefined.
First Derivative Test: sign change of \(f'\) determines local min/max.
Second Derivative Test: \(f''(c)>0\Rightarrow\) local min; \(f''(c)<0\Rightarrow\) local max.
Inflection point: \(f''\) changes sign at \(c\).
Mean Value Theorem (MVT): \(\exists\, c\in(a,b)\) s.t. \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\)

L'Hôpital's Rule: if \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), then \(\lim\tfrac{f}{g}=\lim\tfrac{f'}{g'}\)

Example: \(f(x)=x^3-3x\): \(f'(x)=3x^2-3=0\Rightarrow x=\pm1\). \(f''(1)=6>0\Rightarrow\) local min at \(x=1\).

∫ 4. Integration — Antiderivatives & Definite Integrals

Fundamental Theorem Part 1: \(\dfrac{d}{dx}\int_a^x f(t)\,dt=f(x)\)
Fundamental Theorem Part 2: \(\int_a^b f(x)\,dx=F(b)-F(a)\)
u-substitution: change variable to simplify integrand.
Integration by Parts (BC): \(\int u\,dv=uv-\int v\,du\)

\(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C\,(n\neq-1)\) | \(\int e^x\,dx=e^x+C\)
\(\int\tfrac{1}{x}\,dx=\ln|x|+C\) | \(\int\sin x\,dx=-\cos x+C\)

Example: \(\int_0^1 2x\,dx=\big[x^2\big]_0^1=1-0=1\) ✓

📊 5. Applications of Integration

Area between curves: \(\int_a^b[f(x)-g(x)]\,dx\) where \(f(x)\ge g(x)\).
Disk/Washer Method (rotation about x-axis): \(\pi\int_a^b[f(x)]^2\,dx\)
Shell Method: \(2\pi\int_a^b x\cdot f(x)\,dx\)
Net displacement: \(\int_a^b v(t)\,dt\); total distance: \(\int_a^b|v(t)|\,dt\)

Average value: \(\bar{f}=\dfrac{1}{b-a}\int_a^b f(x)\,dx\)

Example: Area between \(y=x^2\) and \(y=x\): \(\int_0^1(x-x^2)\,dx=\tfrac{1}{2}-\tfrac{1}{3}=\tfrac{1}{6}\)

🔄 6. Differential Equations

Separable DE: separate variables, integrate both sides.
Slope fields: diagram where each point \((x,y)\) shows \(\frac{dy}{dx}\).
Exponential growth/decay: \(\dfrac{dy}{dt}=ky\Rightarrow y=Ce^{kt}\)
Euler's Method (BC): \(y_{n+1}=y_n+h\cdot f'(x_n,y_n)\)

Logistic (BC): \(\dfrac{dP}{dt}=kP\!\left(1-\dfrac{P}{M}\right)\Rightarrow\) S-curve with carrying capacity \(M\)

Example: \(\tfrac{dy}{dx}=2xy\Rightarrow\int\tfrac{dy}{y}=\int 2x\,dx\Rightarrow \ln|y|=x^2+C\Rightarrow y=Ae^{x^2}\)

∞ 7. Infinite Series (BC Only)

Geometric series: \(\sum_{n=0}^{\infty}ar^n=\dfrac{a}{1-r}\) for \(|r|<1\).
p-series: \(\sum\dfrac{1}{n^p}\) converges iff \(p>1\).
Ratio Test: converges if \(\lim\left|\dfrac{a_{n+1}}{a_n}\right|<1\).
Alternating Series Test, Integral Test, Comparison Test.

Taylor series: \(f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n\)
\(e^x=\sum_{n=0}^{\infty}\dfrac{x^n}{n!}\) | \(\sin x=\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x=\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n}}{(2n)!}\) | \(\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^n,\,|x|<1\)

Example: \(\sum_{n=1}^{\infty}\dfrac{1}{n^2}\) converges (p-series, \(p=2>1\)).

🌀 8. Parametric, Polar & Vector (BC)

Parametric slope: \(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\)
Parametric arc length: \(L=\int_a^b\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2}\,dt\)
Polar area: \(A=\tfrac{1}{2}\int_\alpha^\beta[r(\theta)]^2\,d\theta\)
Speed of particle: \(|\mathbf{v}(t)|=\sqrt{[x'(t)]^2+[y'(t)]^2}\)

Polar to Rect: \(x=r\cos\theta,\;y=r\sin\theta,\;r^2=x^2+y^2\)

Example: Area inside \(r=2\cos\theta\): \(\tfrac{1}{2}\int_0^{\pi}(2\cos\theta)^2\,d\theta=\pi\)