AP Calculus AB/BC | Premium Practice Exam

📚 Concept Review & Key Formulas to Memorize

★ Must Memorize

A function \(f\) is continuous at \(x=c\) iff all three hold:
① \(f(c)\) exists ② \(\lim_{x\to c}f(x)\) exists ③ they are equal.

\(\lim_{x\to 0}\frac{\sin x}{x}=1\qquad \lim_{x\to 0}\frac{1-\cos x}{x}=0\)

★ Must Memorize

The derivative as a limit of a difference quotient. Know both forms.

\(f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\)

★ Must Memorize

Power, Product, Quotient, Chain rules. Know derivatives of all trig, exp, log functions.

\(\frac{d}{dx}[\ln x]=\frac{1}{x},\quad \frac{d}{dx}[e^x]=e^x\)

★ Must Memorize

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c\) with \(f'(c)=\frac{f(b)-f(a)}{b-a}\).

★ Must Memorize

Part 1: \(\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\)
Part 2: \(\int_a^b f(x)\,dx = F(b)-F(a)\)

★ Must Memorize

u-substitution, integration by parts (\(\int u\,dv = uv-\int v\,du\)), and partial fractions.

\(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad (n\neq -1)\)

BC Topic

Replace infinite or discontinuous bounds with a limit. Converges if the limit is finite.

\(\int_1^{\infty}\frac{1}{x^p}dx\) converges iff \(p>1\)

BC Topic

Geometric series \(\sum ar^{n-1}=\frac{a}{1-r}\) if \(|r|<1\). Use Ratio Test, Integral Test, Comparison, AST.

BC Topic

Key series: \(e^x=\sum\frac{x^n}{n!}\), \(\sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\), \(\frac{1}{1-x}=\sum x^n\) for \(|x|<1\).

BC Topic

Slope: \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\). Arc length, area in polar \(A=\frac{1}{2}\int r^2\,d\theta\).

📝 Worked Example

Find \(\dfrac{d}{dx}\displaystyle\int_0^{x^2}\cos(t^3)\,dt\).

By FTC Part 1 + Chain Rule:
\(\cos\!\left((x^2)^3\right)\cdot 2x = 2x\cos(x^6)\)

Key: substitute the upper limit into the integrand, then multiply by the derivative of the upper limit.