BC Calculus — Reference Card

Everything on one sheet. Print landscape. Companion to the full guide — if a formula here looks unfamiliar, that's the section to reread.

Limits — the procedure

Substitute. Number? Done.
c/0 → infinite. Not indeterminate.
0/0 → hidden factor. Factor · rationalize · combine fractions · spot a difference quotient.
x→±∞ → divide by highest denominator power.
Other forms → rewrite, then L'Hôpital.

Special limits

sin x / x	→ 1
(1−cos x)/x	→ 0
(1−cos x)/x²	→ 1/2
tan x / x	→ 1
(eˣ−1)/x	→ 1
(1+x/n)ⁿ	→ eˣ

All at x→0 except the last. Rebuild sin x/x from the sector squeeze: sin x ≤ x ≤ tan x.

Growth hierarchy

ln x ≪ xᵖ ≪ eˣ ≪ x! ≪ xˣ

Keep only the dominant term top and bottom, then compare. Also governs series convergence and improper integrals.

Degrees of a rational function

Top > bottom → ±∞ (slant asymptote if by exactly 1)
Top < bottom → 0
Equal → ratio of leading coefficients
√(x²) = |x| = −x when x<0

Derivative — definition

f′(a) = lim_h→0[f(a+h)−f(a)]/h

= the multiplier in the best linear approximation. Differentiable ⟹ continuous, not conversely. Fails at corners, cusps, vertical tangents, discontinuities.

Rules

(cf)′	= cf′
(f±g)′	= f′±g′
(xⁿ)′	= nxⁿ⁻¹
(fg)′	= f′g + fg′
(f/g)′	= (f′g−fg′)/g²
(f∘g)′	= f′(g)·g′
(f⁻¹)′(b)	= 1/f′(f⁻¹(b))

Forgot the quotient rule? Set Q = f/g, so f = Qg, product rule, solve for Q′.

Derivative table

sin	cos
cos	−sin
tan	sec²
cot	−csc²
sec	sec·tan
csc	−csc·cot
eˣ	eˣ
aˣ	aˣ ln a
ln x	1/x
arcsin	1/√(1−x²)
arctan	1/(1+x²)

Every "co-" function carries a minus. Chain forms: (ln u)′ = u′/u, (√u)′ = u′/2√u.

Trig you must have

sin²+cos² = 1
÷cos²: tan²+1 = sec²
÷sin²: 1+cot² = csc²
sin(A+B)=sinAcosB+cosAsinB
cos(A+B)=cosAcosB−sinAsinB
sin2θ = 2sinθcosθ
cos2θ = cos²θ−sin²θ
sin²θ = (1−cos2θ)/2
cos²θ = (1+cos2θ)/2

Derive all but the first two lines from the first. Last two are required for ∫sin², ∫cos², and polar area.

Techniques

Implicit: every y gives a y′; collect, factor, solve. Horizontal tangent: numerator = 0. Vertical: denominator = 0. For y″, sub y′ back in.
Log diff: variable in base and exponent, or a big product. ln both sides, differentiate, ×y.
Related rates: sketch → relation → eliminate unknown-rate variables → differentiate in t → then substitute.

MVT and friends

EVT: continuous on closed [a,b] ⟹ max and min exist
IVT: continuous, N between f(a),f(b) ⟹ f(c)=N
MVT: f′(c) = [f(b)−f(a)]/(b−a)
MVT ⟹ f′>0 means increasing; f′=0 means constant; same derivative ⟹ differ by a constant (the +C)

Shape

Critical point: f′=0 or undefined, in the domain
1st test: + to − = max, − to + = min
2nd test: f″<0 = max, f″>0 = min, f″=0 = inconclusive
f″>0 ⟺ concave up ⟺ tangents below curve
Inflection needs a sign change in f″
Closed interval: critical points plus both endpoints

Motion

s → v = s′ → a = v′; speed = |v|
Speeding up ⟺ v, a same sign
Direction change needs v to change sign
displacement = ∫v · distance = ∫|v|
position = s(a) + ∫_a^bv

Approximation

L(x) = f(a) + f′(a)(x−a)
Concave up ⟹ underestimate; down ⟹ over
Newton: x_n+1 = x_n − f(x_n)/f′(x_n)
Euler: y_n+1 = y_n + f(x_n,y_n)h

All three are the tangent line, reused.

L'Hôpital

Only 0/0 or ∞/∞. Recheck the form each time.
Differentiate top and bottom separately
0·∞ → make a fraction
∞−∞ → common denominator
1^∞, 0⁰, ∞⁰ → ln, limit, then exponentiate
Fails by: wrong form · cycling · f′/g′ having no limit

Integral — definition

∫_a^bf dx = lim Σf(x_i*)Δx

Accumulates a product with one varying factor. Units of integrand × units of dx. Below the axis counts negative.

Δx = (b−a)/n
Σi = n(n+1)/2 · Σi² = n(n+1)(2n+1)/6

Over / under

Increasing f: left under, right over
Concave up: trapezoid over, midpoint under
Midpoint ≈ twice as accurate as trapezoid

FTC

∫_a^bf = F(b) − F(a)
d/dx ∫_a^xf(t)dt = f(x)
d/dx ∫_a^g(x)f = f(g)·g′
Both limits vary: f(up)·up′ − f(low)·low′
f_avg = [1/(b−a)]∫_a^bf

Antiderivatives

xⁿ	xⁿ⁺¹/(n+1), n≠−1
1/x	ln\|x\| ← bars
eˣ	eˣ
aˣ	aˣ/ln a
sin	−cos
cos	sin
sec²	tan
sec·tan	sec
1/(1+x²)	arctan x
1/√(1−x²)	arcsin x
tan x	ln\|sec x\|

Integration techniques

u-sub — a function and its derivative present. Definite: change the limits.
By parts ∫u dv = uv − ∫v du. LIATE for u. Single ln or arctan → dv = dx. eˣsin x → twice, then solve for the original.
Partial fractions — factor, decompose, substitute the roots.
Rewrite first — identities, long division, splitting fractions.

Missing constant is fixable. Missing variable is fatal.

Improper integrals

Write the limit explicitly — scored step
∫₁^∞dx/x^p converges ⟺ p > 1
Check for asymptotes inside the interval first

Applications of the integral

Area: ∫(top−bottom)dx or ∫(right−left)dy
Cross-sections: ∫A(x)dx
Disk: π∫R² · Washer: π∫(R²−r²)
Never (R−r)². Radius = distance from the axis.
Arc length: ∫√(1+(y′)²)dx
Square s² · semicircle (π/8)s² · equilateral (√3/4)s²
Amount = initial + ∫rate. State units.

Differential equations

Separate → integrate → one +C → apply the initial condition immediately
y′ = ky ⟹ y = y₀e^kt
doubling T: k = ln2/T · half-life: k = −ln2/T
Cooling: T = T_a + Ce^kt
Logistic: P′ = kP(1−P/M)
Logistic: limit M · equilibria 0, M · fastest at M/2
Slope field: equilibria are rows of flat ticks

Parametric & vector

dy/dx = (dy/dt)/(dx/dt)
d²y/dx² = [d/dt(dy/dx)]/(dx/dt)
speed = √(x′²+y′²)
arc length = ∫speed dt
Horizontal tangent: y′(t)=0. Vertical: x′(t)=0.
Speed is a scalar; velocity is a vector.

Polar

x = r cosθ · y = r sinθ · r² = x²+y²
Area = ½∫_α^β r²dθ (sectors, not rectangles)
Limits: find r = 0; watch retracing; use symmetry
Slope: convert to x(θ), y(θ). dr/dθ is not a slope.
Rose r = a cos nθ: n petals if n odd, 2n if even

Series — the standards

Geometric Σarⁿ = a/(1−r), |r|<1
p-series Σ1/n^p converges ⟺ p>1
Harmonic Σ1/n diverges (like ln n)
Alternating harmonic → ln 2
Σ1/n² = π²/6

Series — test order

aₙ → 0? No → diverges, stop.
Geometric or p-series? → done.
Factorial or nth power? → ratio (or root)
Alternating? → check |aₙ| first, then AST
Rational-looking? → limit comparison, dominant powers
Integrable? → integral test
Else → direct comparison

Ratio inconclusive at L = 1. Integral test needs positive, continuous, decreasing — and its value is not the sum.

Taylor

f(x) = Σ f⁽ⁿ⁾(a)(x−a)ⁿ/n!
eˣ = 1 + x + x²/2! + x³/3! + ⋯
sin x = x − x³/3! + x⁵/5! − ⋯
cos x = 1 − x²/2! + x⁴/4! − ⋯
1/(1−x) = 1 + x + x² + ⋯ , |x|<1
Build others by substituting, differentiating, integrating, multiplying — never by taking derivatives
Coefficient of xⁿ × n! = f⁽ⁿ⁾(0)
Radius by ratio test; test both endpoints by hand
Error ≤ max|f⁽ⁿ⁺¹⁾|·|x−a|ⁿ⁺¹/(n+1)!
Alternating: error < first omitted term

Calculator — the only four things

Graph in a window you choose.
Solve f(x) = 0 numerically.
Derivative at a point, numerically.
Definite integral, numerically.

Those four are the whole permitted list. Write the setup anyway — the equation being solved, or the derivative or integral being evaluated. A bare calculator answer earns the answer point and nothing else.

Use any other feature and you must show the full mathematical steps; the calculator result alone scores zero.

Store values, don't retype them. Round once, at the end, to three decimals. Check you're in radians.

Phrases that score

"f′ changes from positive to negative at c, therefore f has a local maximum."
"f is continuous on [a,b] and differentiable on (a,b), so by the Mean Value Theorem…"
"f is continuous, so by the Intermediate Value Theorem…"
Always state units on accumulation answers.
Write the limit for any improper integral.
Answer the question actually asked.

Top ten point-losers

Endpoints omitted on a closed interval
Substituting numbers before differentiating in related rates
Forgetting to exponentiate after a log limit
π∫(R−r)² instead of π∫(R²−r²)
Not changing limits in a definite u-sub
Distance vs. displacement (missing the |v|)
Initial condition left out of an accumulation
Missing ln|x| absolute value
Interval-of-convergence endpoints untested
Justification sentence never written