AP Calc AB - ultimate review

equations, definitions, and theorems

L’Hospital’s Rules

if lim f/g = 0/0 or ∞/∞, then lim f/g = lim f'/g' (differentiate top & bottom separately).

Piecewise/graphs

lim(x→a) f(x) may ≠ f(a). Left limit ≠ right limit → limit DNE.

Derivative

Continuous at x = a if

(1) lim(x→a) f(x) exists, (2) f(a) defined, (3) lim = f(a). All 3 needed.

Removable discontinuity

hole (can redefine to fix). Jump: both sides exist but ≠. Infinite: vert asymptote.

Intermediate value theorem (IVT)

f continuous [a,b] & f(a) ≠ f(b) → f hits every value between.

✓ IVT for proving solutions exist. FRQ staple: "show that equation has solution on interval."

Horizontal Asymptote

lim(x→±∞) f(x) = L → y = L. Rationals: compare degrees. Top degree higher → none.

Vertical Asymptote

where denom = 0 & numer ≠ 0. From graph: f→ ±∞ at x = a.

oblique/slat asymptote

use polynomial long division if numer degree = denom degree + 1

Never say 'limit = ∞'. S a y ' d i v e r g e s t o ∞' (∞ is not a number)

Piecewise limits

eval left & right separately. Example: f(x)={x² if x<2; 5 if x≥2}. lim(x→2⁻)=4, lim(x→2⁺)=5 → DNE. x

If Limit is ∞/∞ rational

Divide by highest power g'(x)/g(x) Formula Momentary stop (check direction change)

If limit is 0/0 rational

Factor, cancel, re-evaluate

If limit is 0/0 radicals

Rationalize (conjugate) multiply the numerator or denominator by its conjugate

Derivative definition

f'(a) = lim(h→0) [f(a+h) - f(a)]/h.

Instantaneous rate of change; slope of tangent line.

Alternative derivative

f'(a) = lim(x→a) [f(x) - f(a)]/(x - a). Use when x-form given in problem.

Right deriv, Left deriv

f'₊(a) = lim(h→0⁺) [...], f'₋(a) = lim(h→0⁻) [...] , Both must exist & equal,

Power rule

d/dx[xⁿ] = nxⁿ⁻¹

Product Rule

(uv)' = u'v + uv'

Squeeze Theorem

f ≤ g ≤ h & lim f = lim h

Quotient Rule

(u/v)' = (u'v - uv')/v²

d/dx of e^x

e^x

d/dx of ln x

1/x

d/dx of cos

-sin

d/dx of sin

cos

d/dx of a consant mult: [cf’]

cf’

d/dx of sum [f + g]’

f’ + g’

Ta n g e n t l i n e a t ( a , f ( a ) ) :

y - f(a) = f'(a)(x - a)

Linear approx.

f(x) ≈ f(a) + f'(a)(x - a) for x near a.

Differentiability

if diff at a, then continuous at a. Converse FALSE. Not diff at corners, cusps, vert tangents.

⚠ Check diff at boundaries & piecewise points. Continuous ≯ differentiable.

Second derivative:

f''(x) = d/dx[f'(x)]. Use for concavity & acceleration. May be messy from quotient/product.

From graph to deriv:

f' tells slope at each x. If f increasing at x=2, then f'(2)>0. Steeper = larger |f'|.