Source: SYMKC Series, Calculus Prep (Texas A&M University)
Tags: calculus, AP calculus, AB, BC, limits, continuity, derivatives, chain rule, product rule, quotient rule, integrals, fundamental theorem of calculus, mean value theorem, Rolle's theorem, extreme value theorem, intermediate value theorem, L'Hopital's rule, solids of revolution, disk method, washer method, arc length, integration by parts, Taylor series, Maclaurin series, Euler's method, polar curves, parametric equations, logistics
This is the Calculus AB/BC formula sheet you need to have memorised. It covers limits and continuity, the definition of the derivative, every major derivative rule (including trig and inverse trig), curve sketching, the Fundamental Theorem of Calculus, application problems (velocity, acceleration, displacement, distance), solids of revolution, and the BC-only material: integration by parts, Taylor/Maclaurin series, parametric derivatives, polar area, L'Hopital's rule, Euler's method, and logistic growth.
Limit
The value that f(x) approaches as x approaches a given value. Written lim(x→a) f(x).
Continuity
A function is continuous at x = a if three conditions hold: f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a).
Derivative
The instantaneous rate of change of f(x), defined as the limit of the difference quotient.
Critical point
A point where f'(x) = 0 or f'(x) does not exist.
Inflection point
A point where the concavity of f changes, i.e. where f''(x) changes sign.
Fundamental Theorem of Calculus (FTC)
Links differentiation and integration. Part 1: if F'(x) = f(x), then ∫[a to b] f(x)dx = F(b) – F(a). Part 2 (Corollary): d/dx ∫[a to g(x)] f(t)dt = f(g(x)) · g'(x).
Mean Value Theorem
If f is continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) where f'(c) = (f(b) – f(a)) / (b – a).
Notation:
Limit from the left of f(x) as x → a: lim(x→a⁻) f(x)
Limit from the right of f(x) as x → a: lim(x→a⁺) f(x)
Limit theorems (given lim(x→a) f(x) = F and lim(x→a) g(x) = G):
lim(x→a) [f(x) + g(x)] = F + G
lim(x→a) [f(x) – g(x)] = F – G
lim(x→a) [f(x) · g(x)] = F · G
lim(x→a) [f(x)]ⁿ = Fⁿ
lim(x→a) [f(x)/g(x)] = F/G, provided G ≠ 0
Special limits:
lim(x→0) sin(x)/x = 1
lim(x→0) (1 – cos x)/x = 0
A function is continuous at x = a if and only if:
f(a) is defined
lim(x→a) f(x) exists
lim(x→a) f(x) = f(a)
Critical points: where f'(x) = 0 or f'(x) is undefined
Global min: the smallest value of f on a closed interval; check critical points and endpoints
Global max: the largest value of f on a closed interval; check critical points and endpoints
Point of inflection: where f''(x) changes sign (concavity changes)
If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on [a, b].
Definition of derivative:
d/dx f(x) = lim(h→0) [f(x + h) – f(x)] / h
Alternate form at a point x = a:
f'(a) = lim(x→a) [f(x) – f(a)] / (x – a)
Chain rule:
d/dx [f(u)] = f'(u) · u', where u is a function of x
Product rule:
d/dx (uv) = u'v + uv'
Quotient rule:
d/dx (u/v) = (u'v – uv') / v²
where u and v are functions of x.
Power rule and basic:
d/dx (xⁿ) = nxⁿ⁻¹
Trig derivatives:
d/dx (sin u) = cos u · u'
d/dx (cos u) = –sin u · u'
d/dx (tan u) = sec²u · u'
d/dx (cot u) = –csc²u · u'
d/dx (sec u) = sec u tan u · u'
d/dx (csc u) = –csc u cot u · u'
Exponential and logarithmic:
d/dx (ln u) = u'/u
d/dx (eᵘ) = eᵘ · u'
d/dx (aᵘ) = aᵘ · ln a · u'
d/dx (log_a u) = u' / (u · ln a)
Inverse trig:
d/dx (sin⁻¹u) = u' / √(1 – u²)
d/dx (cos⁻¹u) = –u' / √(1 – u²)
d/dx (tan⁻¹u) = u' / (1 + u²)
d/dx (cot⁻¹u) = –u' / (1 + u²)
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that:
f'(c) = [f(b) – f(a)] / (b – a)
In plain terms, there is at least one point where the instantaneous rate of change equals the average rate of change over the interval.
A special case of the MVT: if f(a) = f(b), then there exists some c in (a, b) where f'(c) = 0.
If f is continuous on [a, b] and N is any value between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.
If F is an antiderivative of f on [a, b]:
∫[a to b] f(x) dx = F(b) – F(a)
d/dx ∫[a to g(x)] f(t) dt = f(g(x)) · g'(x)
The trapezoidal rule approximates the integral using trapezoids:
∫[a to b] f(x) dx ≈ (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
If f is continuous on [a, b]:
Average value = (1/(b – a)) ∫[a to b] f(x) dx
Given position function s(t), or parametric position ⟨x(t), y(t)⟩:
Velocity: v(t) = s'(t), or the velocity vector ⟨x'(t), y'(t)⟩
Acceleration: a(t) = v'(t) = s''(t), or ⟨x''(t), y''(t)⟩
Speed (rectangular and parametric): |v(t)| = √((x'(t))² + (y'(t))²)
Displacement: ∫[a to b] v(t) dt
Total distance (rectangular and parametric): ∫[a to b] |v(t)| dt = ∫[a to b] √((x')² + (y')²) dt
Average velocity: (1/(b – a)) ∫[a to b] v(t) dt, or equivalently [s(b) – s(a)] / (b – a)
Disk method (rotation about an axis, no hole):
V = π ∫[a to b] [R(x)]² dx
Washer method (rotation about an axis, with a hole):
V = π ∫[a to b] ([R(x)]² – [r(x)]²) dx
where R is the outer radius and r is the inner radius.
General volume equation (cross-sections of known shape):
V = ∫[a to b] A(x) dx, where A(x) is the cross-sectional area
L = ∫[a to b] √(1 + (dy/dx)²) dx
If lim(x→a) f(x)/g(x) gives the indeterminate form 0/0 or ∞/∞, then:
lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x)
(provided the right-hand limit exists)
A numerical method for approximating solutions to differential equations. Starting from a known point (x₀, y₀) with step size Δx:
yₙ₊₁ = yₙ + f(xₙ, yₙ) · Δx
∫ u dv = uv – ∫ v du
Choose u and dv using the LIATE priority: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
dP/dt = kP(1 – P/L)
where L is the carrying capacity and k is the growth constant. The population grows fastest at P = L/2.
dy/dx = (dy/dt) / (dx/dt)
d²y/dx² = [d/dt(dy/dx)] / (dx/dt)
Area: A = (1/2) ∫[α to β] r² dθ
Slope: dy/dx = (dy/dθ) / (dx/dθ), where x = r cos θ and y = r sin θ
The Taylor series for f(x) centred at x = a:
f(x) = Σ (n=0 to ∞) [f⁽ⁿ⁾(a) / n!] · (x – a)ⁿ
Know these by heart:
eˣ = 1 + x + x²/2! + x³/3! + ... = Σ xⁿ/n!
cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + ... = Σ (–1)ⁿ x²ⁿ/(2n)!
sin x = x – x³/3! + x⁵/5! – x⁷/7! + ... = Σ (–1)ⁿ x²ⁿ⁺¹/(2n+1)!
1/(1 – x) = 1 + x + x² + x³ + ... = Σ xⁿ, for |x| < 1
ln(x + 1) = x – x²/2 + x³/3 – x⁴/4 + ... = Σ (–1)ⁿ⁺¹ xⁿ/n, for |x| ≤ 1 (x ≠ –1)
Use convergence tests (ratio test, root test, comparison test, integral test, alternating series test, p-series test) to determine whether a series converges. The Lagrange error bound for Taylor polynomials:
|Rₙ(x)| ≤ M · |x – a|ⁿ⁺¹ / (n + 1)!
where M is the maximum value of |f⁽ⁿ⁺¹⁾(c)| on the interval between a and x.
Definition of derivative: f'(x) = lim(h→0) [f(x + h) – f(x)] / h
FTC: ∫[a to b] f(x)dx = F(b) – F(a)
Corollary to FTC: d/dx ∫[a to g(x)] f(t)dt = f(g(x)) · g'(x)
MVT: f'(c) = [f(b) – f(a)] / (b – a)
Disk: V = π ∫ R² dx
Washer: V = π ∫ (R² – r²) dx
Arc length: L = ∫ √(1 + (y')²) dx
Integration by parts: ∫ u dv = uv – ∫ v du
Taylor: f(x) = Σ [f⁽ⁿ⁾(a)/n!](x – a)ⁿ
⚠️ The two special limits (sin x / x → 1 and (1 – cos x)/x → 0 as x → 0) are used constantly in derivative proofs and limit evaluation.
⚠️ For the chain rule, the most common error is forgetting to multiply by the derivative of the inner function (u').
⚠️ d/dx (cos u) = –sin u · u'. The negative sign is easy to drop.
⚠️ d/dx (ln u) = u'/u, not 1/u. The chain rule applies; do not forget u'.
⚠️ The FTC Corollary requires multiplying by g'(x). If the upper limit is something other than plain x, you must account for it.
⚠️ Displacement (net change) and total distance are different. Displacement is ∫ v dt; total distance is ∫ |v| dt.
⚠️ L'Hopital's Rule only applies to 0/0 or ∞/∞. You cannot use it on, say, 1/0.
⚠️ For Maclaurin series: sin x uses odd powers, cos x uses even powers. Mixing them up is a classic error.
⚠️ The alternating series error bound says the error is less than the absolute value of the first omitted term. This is BC-specific and comes up frequently.
Q: State the derivative of ln(3x²).
A: d/dx [ln(3x²)] = (6x)/(3x²) = 2/x.
Q: Evaluate lim(x→0) sin(5x)/x.
A: Rewrite as 5 · [sin(5x)/(5x)]. As x → 0, sin(5x)/(5x) → 1. So the limit is 5.
Q: Find the average value of f(x) = x² on [0, 3].
A: (1/3) ∫[0 to 3] x² dx = (1/3)(9) = 3.
Q: What is the Maclaurin series for eˣ, up to the x³ term?
A: 1 + x + x²/2 + x³/6.
Q: Using the disk method, set up the integral for the volume when y = √x is rotated about the x-axis from x = 0 to x = 4.
A: V = π ∫[0 to 4] (√x)² dx = π ∫[0 to 4] x dx.
Q: State the three conditions for continuity at a point x = a.
A: f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a).
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