PART I Concept Review & Memorization
1 Limits & Continuity
The foundation of calculus — understanding behavior as inputs approach a value.
★ Memorize
\(\lim_{x \to a} f(x) = L\) iff \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\)
L'Hôpital's Rule: If \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\):
\(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\)
L'Hôpital's Rule: If \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\):
\(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\)
Example
Evaluate \(\displaystyle\lim_{x \to 0} \frac{\sin x}{x}\)
Answer: 1 (fundamental trig limit)
2 Differentiation Rules
Core rules for computing derivatives analytically.
★ Memorize
Power: \(\frac{d}{dx}[x^n] = nx^{n-1}\)
Product: \((fg)' = f'g + fg'\)
Quotient: \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\)
Chain: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Product: \((fg)' = f'g + fg'\)
Quotient: \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\)
Chain: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Example
Find \(\frac{d}{dx}[\sin(x^2)]\)
Answer: \(2x\cos(x^2)\) — Chain Rule
3 Trig & Inverse Trig Derivatives
Essential derivatives — every single one appears on the exam.
★ Memorize All
\(\frac{d}{dx}[\sin x] = \cos x\) | \(\frac{d}{dx}[\cos x] = -\sin x\)
\(\frac{d}{dx}[\tan x] = \sec^2 x\) | \(\frac{d}{dx}[\sec x] = \sec x \tan x\)
\(\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}\)
\(\frac{d}{dx}[\tan x] = \sec^2 x\) | \(\frac{d}{dx}[\sec x] = \sec x \tan x\)
\(\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}\)
Example
Find \(\frac{d}{dx}[\arctan(2x)]\)
Answer: \(\dfrac{2}{1+4x^2}\)
4 MVT, IVT, & EVT
Foundational theorems that guarantee existence of values.
★ Memorize
MVT: If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\),
\(\exists\, c \in (a,b)\) : \(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)
IVT: If \(f\) is continuous on \([a,b]\), it takes every value between \(f(a)\) and \(f(b)\).
\(\exists\, c \in (a,b)\) : \(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)
IVT: If \(f\) is continuous on \([a,b]\), it takes every value between \(f(a)\) and \(f(b)\).
Example
\(f(x)=x^3\) on \([0,2]\). Find \(c\) guaranteed by MVT.
Answer: \(c = \frac{2}{\sqrt{3}}\)
5 Related Rates & Optimization
Applying derivatives to real-world changing quantities.
★ Strategy
Differentiate both sides with respect to time \(t\).
Use given rates; substitute known values after differentiating.
Critical Points: \(f'(c)=0\) or DNE
Absolute extrema occur at critical points or endpoints.
Use given rates; substitute known values after differentiating.
Critical Points: \(f'(c)=0\) or DNE
Absolute extrema occur at critical points or endpoints.
Example
Area of a circle: \(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\)
If \(\frac{dr}{dt}=3\) and \(r=5\): \(\frac{dA}{dt} = 30\pi\)
6 Definite Integrals & FTC
The bridge between accumulation and differentiation.
★ Memorize
FTC Part 1: \(\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\)
FTC Part 2: \(\int_a^b f(x)\,dx = F(b) - F(a)\)
Chain Rule version: \(\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)\)
FTC Part 2: \(\int_a^b f(x)\,dx = F(b) - F(a)\)
Chain Rule version: \(\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)\)
Example
\(\frac{d}{dx}\int_0^{x^2} \cos(t)\,dt\)
Answer: \(2x\cos(x^2)\)
7 Integration Techniques
Methods to evaluate antiderivatives.
★ Memorize
u-substitution: Let \(u=g(x)\), \(du=g'(x)\,dx\)
Integration by Parts (BC):
\(\int u\,dv = uv - \int v\,du\)
Partial Fractions (BC): decompose rational functions
Integration by Parts (BC):
\(\int u\,dv = uv - \int v\,du\)
Partial Fractions (BC): decompose rational functions
Example
\(\int x e^{x^2}\,dx\)
Let \(u=x^2\): Answer: \(\frac{1}{2}e^{x^2} + C\)
8 Differential Equations
Equations relating a function to its derivatives.
★ Memorize
Separable: \(\frac{dy}{dx} = f(x)g(y)\)
Separate: \(\frac{dy}{g(y)} = f(x)\,dx\), then integrate.
Exponential Model: \(\frac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}\)
Euler's Method (BC): \(y_{n+1} = y_n + h\cdot f'(x_n, y_n)\)
Separate: \(\frac{dy}{g(y)} = f(x)\,dx\), then integrate.
Exponential Model: \(\frac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}\)
Euler's Method (BC): \(y_{n+1} = y_n + h\cdot f'(x_n, y_n)\)
Example
Solve \(\frac{dy}{dx} = 2xy\), \(y(0)=3\)
Answer: \(y = 3e^{x^2}\)
9 Area & Volume of Revolution
Geometric applications of integration.
★ Memorize
Area between curves: \(\int_a^b [f(x)-g(x)]\,dx\)
Disk Method: \(V = \pi\int_a^b [f(x)]^2\,dx\)
Washer Method: \(V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx\)
Shell Method (BC): \(V = 2\pi\int_a^b x f(x)\,dx\)
Disk Method: \(V = \pi\int_a^b [f(x)]^2\,dx\)
Washer Method: \(V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx\)
Shell Method (BC): \(V = 2\pi\int_a^b x f(x)\,dx\)
Example
Volume of \(y=\sqrt{x}\) on \([0,4]\) rotated about \(x\)-axis
Answer: \(\pi\int_0^4 x\,dx = 8\pi\)
10 Infinite Series (BC Only)
BC ONLY
Convergence tests and power series representations.
★ Memorize All Tests
Geometric: \(\sum_{n=0}^\infty ar^n = \frac{a}{1-r}\) \((|r|<1)\)
Ratio Test: \(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| < 1\) → converges
Taylor Series: \(f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n\)
Maclaurin for \(e^x\): \(\sum_{n=0}^\infty \frac{x^n}{n!}\)
Ratio Test: \(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| < 1\) → converges
Taylor Series: \(f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n\)
Maclaurin for \(e^x\): \(\sum_{n=0}^\infty \frac{x^n}{n!}\)
Example
Sum of \(\sum_{n=0}^\infty \left(\frac{2}{3}\right)^n\)
Answer: \(\frac{1}{1-\frac{2}{3}} = 3\)
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PART II Practice Exam — 20 Questions
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PART III Complete Solutions & Explanations
Each solution is written with full College Board–level mathematical justification. Study these carefully.