The Simple AP Calculus AB Cheat Sheet

The Simple AP Calculus AB Cheat Sheet Unit 1: Limits & Continuity (10–12%) Unit 2: Differentiation Basics (10–12%) Unit 3: Chain, Implicit, Inverse (9–13%) Unit 4: Contextual Applications (10–15%) Computing Limits Try direct substitution first. Plug in x = a. If 0/0 appears, factor or rationalize. L'Hôpital's rule: if you get 0/0 or ∞/∞, differen. top and bottom separately, try limit again. Type Strategy ∞/∞ rational Divide by highest power 0/0 rational Factor & cancel 0/0 with radicals Use conjugate to rationalize For piecewise functions: always check left limit and right limit separately. If different, limit does not exist. Continuity (3-Part Test) Function is continuous at x=a if all 3 are true: (1) limit exists, (2) f(a) is defined, (3) limit = f(a). Intermediate Value Theorem (IVT): if f continuous [a,b] and f(a) ≠ f(b), f hits all values between. Prove 'equation has solution.' Removable discontinuity: hole in graph, can redefine f(a) to fix. Jump discontinuity: left & right limits exist but differ. Infinite: vertical asymptote. Asymptotes Horizontal asymptote: as x → ∞, does f(x) approach a number? For rationals, compare degrees. Vertical asymptote: where denominator equals 0 but numerator doesn't. Oblique: use long division. ✓ Always justify limits and continuity claims for FRQ. Show which method you used. What Is a Derivative? Chain Rule: Most Important! Motion Problems Derivative f'(a) = lim(h→0) [f(a+h) - f(a)]/h. This measures the instantaneous rate of change (slope of tangent line). Think of it as: how fast is y changing with respect to x? Positive slope = increasing, negative = decreasing. Key Differentiation Rules For composite function y = f(g(x)): dy/dx = f'(g(x)) · g'(x). Or: outside' × inside'. Example: d/dx[sin(x²)] = cos(x²) · 2x. Derivative of sin is cos, times derivative of x². Stacked functions: if three layers, multiply three derivatives. d/dx[sin(e^(x²))] = cos(e^(x²))·e^(x²)·2x. Special exponentials & logs: d/dx[e^(g(x))] = e^(g(x))·g'(x). d/dx[ln(g(x))] = g'(x)/g(x). Implicit Differentiation Position x(t) tells location. Differentiate once → velocity v = dx/dt. Differentiate again → acceleration a = dv/dt = d²x/dt². Unit 5: Analytical Applications (15–18%) Unit 6: Integration & Accumulation (17– 20%) Finding Extrema (Max & Min) Critical points: solve f'(x) = 0 or find where f' is undef. These are candidates for local max/min. 1st Derivative Test: check sign of f' just left and right of critical point. + → - = local max. - → + = local min. 2nd Derivative Test: compute f''(c). If f'' < 0, it's a local max. If f'' > 0, it's a local min. Absolute max/min on [a,b]: eval f at critical pts + endpoints. Largest = abs max. Existence theorem: continuous function on closed [a,b] must have absolute max and min. Concavity & Inflection Points f'' > 0 means concave up (∪). f'' < 0 means concave down (∩). Inflection point: where f'' changes sign. To find inflections: set f''(x) = 0, check if sign changes on both sides. Curve Sketching Process: domain, intercepts, asymptotes, critical pts (use f'), inflections (use f''), sketch. Sketch workflow: plot f, f', f'' on number lines. Mark critical pts & inflections. Mean Value Theorem: if f continuous [a,b] & differentiable (a,b), then ∃c where f'(c) = [f(b)f(a)]/(b-a). ✓ Use sign charts (number lines) for f' and f'' to organize everything clearly. Rule Power Rule Product Rule Quotient Rule Formula d/dx[xⁿ] = nxⁿ⁻¹ (uv)' = u'v + uv' (u/v)' = (u'v - uv')/v² Tangent line equation at (a, f(a)): y - f(a) = f'(a) (x - a). Use this for linear approximations. Understanding f' and f'' If f'(x) > 0, f is increasing. If f'(x) < 0, f is decreasing. If f'(x) = 0, check if critical point (max/min). Second derivative f''(x) measures concavity. f'' > 0 = concave up (∪). f'' < 0 = concave down (∩). Differentiability: if diff at a point, function is continuous there. But continuous ≠ differentiable (e.g., |x| at 0). Not differentiable at: corners, cusps, vertical tangents, jumps, or any discontinuity. ⚠ Product rule: don't forget uv' term. Quotient: minus sign in numerator (u'v - uv'). Antiderivatives (Indefinite Integrals) Antiderivative F is a function where F'(x) = f(x). Symbol: ∫f(x) dx = F(x) + C (always add C!). Form xⁿ (n ≠ -1) 1/x eˣ sin(x) Antiderivative xⁿ⁺¹/(n+1) + C ln|x| + C eˣ + C -cos(x) + C Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Exception: ∫(1/x) dx = ln|x| + C (absolute value!). Linearity: ∫[af + bg] dx = a∫f dx + b∫g dx. Split by addition/subtraction. Definite Integrals & FTC Theorem Part 1: ∫ᵃᵇ f(x) dx = F(b) - F(a). Find antiderivative, evaluate at bounds. Theorem Part 2: d/dx[∫ᵃˣ f(t) dt] = f(x). The derivative of accumulation = original function. U-substitution: let u = g(x), du = g'(x) dx. Rewrite integral in terms of u, integrate, substitute back. Area under curve: ∫ᵃᵇ f dx if f ≥ 0. If f<0, area = -∫ᵃᵇ f dx. Take absolute value.Between curves: ∫ᵃᵇ |f-g| dx. If cross, split interval at intersection pts. ✓ Always check: differentiate your antiderivative. You should get back the original function. For equations like x² + y² = 25 (circle): differentiate both sides with respect to x. Treat y as a function of x: d/dx[y²] = 2y · dy/dx. Then solve for dy/dx algebraically. Related rates problems: use implicit differentiation, but differentiate with respect to time t instead of x. Inverse Functions Key derivatives: d/dx[arcsin x] = 1/√(1-x²), d/dx[arctan x] = 1/(1+x²), d/dx[ln x] = 1/x. If y = f⁻¹(x), then f(y) = x. Diff: 1 = f'(y) · dy/dx → dy/dx = 1/f'(y). Evaluate at a point: To find (f −1 )′ (a), first solve f (x) = a for x. Then calculate 1 . f ′ (x) ✓ Related rates: draw a diagram, label all variables, write constraint, then differentiate. Condition v > 0, a > 0 v > 0, a < 0 v < 0, a < 0 v=0 Interpretation Speeding up, moving right Slowing down, moving right Speeding up, moving left Momentary stop (might reverse) Total distance ≠ displacement. Find when v = 0, split interval, sum |Δx| of each piece. Speed = |v|, always ≥ 0. Displacement = x(b) x(a), can be negative. Related Rates & Optimization Related rates setup: identify variables, write constraint equation (like V = πr²h), differentiate w.r.t. time t, solve. Maximize/minimize: set dP/dx = 0 to find critical points, use 2nd derivative test or sign analysis. Marginal cost/revenue: dC/dx and dR/dx. Profit P = R - C. Maximize where dP/dx = 0. Applied context: 'at minimum' means dP/dt=0 & d²P/dt²>0. Always justify. ⚠ Always include units in your final answer. ✓ Translate words to math. 'Rate increasing' = second derivative > 0. Unit 7: Differential Equations (6–12%) Unit 8: Applications & FRQ Tips (10–15%) Slope Fields & Visualization Area Between Curves Slope field: at each point (x,y), draw a short segment with slope = dy/dx. Solution curves follow these segments. Reading slope fields: horizontal segments mean dy/dx = 0 (equilibrium solutions). Steeper = larger |dy/dx|. Euler's method: yₙ₊₁ = yₙ + f(xₙ, yₙ)·Δx. Approximate solution by small steps. Solving: Separation of Variables Area = ∫ᵃᵇ |f(x) - g(x)| dx. First identify which function is on top. If curves cross, split at intersection. If dy/dx = f(x) · g(y): separate variables. Move g(y) to left: (1/g(y)) dy = f(x) dx. Then integrate both sides: ∫(1/g(y)) dy = ∫f(x) dx. Solve for y if possible. Use initial condition: plug in y(x₀) = y₀ to find C. Euler's method: yₙ₊₁ = yₙ + f(xₙ,yₙ)·Δx. Approx solution by stepping. Standard Models Method Disk/Washer Shell Average Value Setup V = π∫[R(x)]² dx or π∫[R²-r²] dx V = 2π∫ (radius)(height) dx f_avg = (1/(b-a))∫ᵃᵇ f dx Choose disk/washer if rotating about x- or yaxis. Choose shell if easier. Work integral: W = ∫ F · ds. Spring force: W = ∫₀ˣ kx dx. FRQ Strategy (High-Value Tips) Part A (calculator allowed, 30 min): usually 1 differential equation + 1 area/volume. Show setup clearly. Part B (no calc, 60 min): 4 problems. Justify each claim: 'f'=0 at x=3, f''(3)<0 → local max.' Exponential growth/decay: dy/dt = ky → solution Rubric: setup (1-2 pts), execution (1-2 pts), y = Ae^(kt). If k > 0 grow, if k < 0 decay. answer (1 pt). Total 6 pts/problem. Logistic equation: dy/dt = ky(L-y). Has carrying Always: define variables, label units, link to realcapacity L. S-shaped solution curve. world context, draw diagrams. Newton's cooling: dT/dt = k(T - Tₐ). T Partial credit: setup alone = some points. Perfect approaches Tₐ exponentially. process wrong answer = fewer points. Particular vs general: general has arbitrary C; ⚠ Traps: wrong bounds, forget ±√, forget +C, particular uses IC to find C. ⚠ Initial condition determines C. Forgetting loses points on FRQ. Part A: 30 no-calc MC + 15 calc MC. Part B: 2 calc FRQ + 4 no-calc FRQ. Formula sheet provided. Graphing calc on calc sections. | www.albert.io don't evaluate limits, wrong sign of 2nd deriv. ✓ Link to context. Explain every step for credit.