AP Calculus Study Guide
1. Limits & Continuity
Limit: value a function approaches as x → a.
Notation: limx→af(x)=L\lim_{x \to a} f(x) = Llimx→af(x)=L.
One-sided limits: approach from left (−) or right (+).
Continuity: function is continuous at x=a if:
1. f(a) is defined.
2. limx→af(x)\lim_{x \to a} f(x)limx→af(x) exists.
3. limx→af(x)=f(a)\lim_{x \to a} f(x) = f(a)limx→af(x)=f(a).
Indeterminate forms: 0/0, ∞/∞ → use L’Hôpital’s Rule.
2. Derivatives
Derivative = slope of tangent line, instantaneous rate of change.
Definition: limh→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h) - f(x)}
{h}limh→0hf(x+h)−f(x).
Rules:
o Power Rule: ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-
1}dxd[xn]=nxn−1.
o Product Rule: (fg)’ = f’g + fg’.
o Quotient Rule: fg’=f’g−fg’g2\frac{f}{g}’ = \frac{f’g - fg’}
{g^2}gf’=g2f’g−fg’.
o Chain Rule: (f(g(x)))’ = f’(g(x))·g’(x).
Applications: slopes, velocity, acceleration, optimization, related
rates.
3. Applications of Derivatives
Critical points: f’(x)=0 or undefined.
First Derivative Test: determines local max/min.
Second Derivative Test: concavity (f’’ > 0 concave up, f’’ < 0
concave down).
Inflection Points: where concavity changes.
Optimization: maximize/minimize real-world quantities.
Motion:
o Velocity = derivative of position.
o Acceleration = derivative of velocity.
4. Integrals
Indefinite Integral: antiderivative + C.
Definite Integral: area under curve between a and b.
o ∫abf(x)dx\int_a^b f(x) dx∫abf(x)dx.
Fundamental Theorem of Calculus:
o Part 1: Derivative of ∫ f(x) = f(x).
o Part 2: ∫abf(x)dx=F(b)−F(a)\int_a^b f(x) dx = F(b) - F(a)∫ab
f(x)dx=F(b)−F(a).
Basic Rules:
o ∫ k dx = kx + C.
o ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1.
o ∫ eˣ dx = eˣ + C.
o ∫ 1/x dx = ln|x| + C.
5. Applications of Integrals
Area under a curve: ∫ f(x) dx.
Area between curves: ∫ [top − bottom] dx.
Volumes (Solids of Revolution):
o Disk Method: π∫R(x)2dxπ ∫ R(x)² dxπ∫R(x)2dx.
o Washer Method: π∫[R2−r2]dxπ ∫ [R² − r²] dxπ∫[R2−r2]dx.
Average Value of a Function:
1b−a∫abf(x)dx\frac{1}{b-a} ∫_a^b f(x) dxb−a1∫abf(x)dx.
Work & Accumulation Problems: total work = ∫ F(x) dx.
6. Series & Sequences (AP Calc BC only)
Convergence Tests: nth-term test, ratio test, comparison test.
Taylor/Maclaurin Series: polynomial approximation of functions.
Geometric Series: converges if |r| < 1.
7. Key Graphs & Concepts
Derivative graphs (f vs f’ vs f’’).
Position-velocity-acceleration connections.
Areas between curves.
Slope fields & differential equations (Euler’s method).
Logistic growth & separable DEs (BC).
