AP Calc AB Fluency quiz #!

Vertical asymptote (limit definition)

x = a is a vertical asymptote if lim x→a⁻ f(x) = ±∞ or lim x→a⁺ f(x) = ±∞ (at least one one-sided limit is infinite)

Horizontal asymptote (limit definition)

y = L is a horizontal asymptote if lim x→∞ f(x) = L or lim x→−∞ f(x) = L

Definition of derivative (h-form)

f′(x) = lim h→0 [f(x+h) − f(x)] / h

Definition of derivative at a point (x→a form)

f′(a) = lim x→a [f(x) − f(a)] / (x − a)

Definition of continuity at x = a

1) f(a) is defined; 2) lim x→a f(x) exists; 3) lim x→a f(x) = f(a)

Intermediate Value Theorem

If f is continuous on [a, b] and k is between f(a) and f(b), then there is at least one c in (a, b) with f(c) = k

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) near a and lim x→a g(x) = lim x→a h(x) = L, then lim x→a f(x) = L

Types of discontinuities

Removable (hole), Jump (left and right limits exist but differ), Infinite (vertical asymptote), Oscillating (e.g. sin(1/x) at 0)

Where a function is not differentiable

Discontinuity, Corner (e.g. |x| at 0), Cusp (e.g. x^(2/3) at 0), Vertical tangent (e.g. cube root of x at 0)

Average rate of change on [a, b]

[f(b) − f(a)] / (b − a), the slope of the secant line

Special trig limit: lim x→0 sin(x)/x

1

Special trig limit: lim x→0 (1 − cos x)/x

0

Special trig limit: lim x→0 sin(kx)/x

k

Constant rule

d/dx [c] = 0

Constant multiple rule

d/dx [c·f(x)] = c·f′(x)

Power rule

d/dx [xⁿ] = n·xⁿ⁻¹

Product rule

d/dx [f·g] = f′g + fg′

Quotient rule

d/dx [f/g] = (g·f′ − f·g′) / g² (low d-high minus high d-low, over low squared)

Chain rule

d/dx [f(g(x))] = f′(g(x))·g′(x)

d/dx [sin x]

cos x

d/dx [cos x]

−sin x

d/dx [tan x]

sec²x

d/dx [cot x]

−csc²x

d/dx [sec x]

sec x tan x

d/dx [csc x]

−csc x cot x

d/dx [eˣ]

eˣ

d/dx [aˣ]

aˣ · ln a

d/dx [eᵘ] (chain rule)

eᵘ · u′

d/dx [ln x]

1/x (also true for ln|x|)

d/dx [logₐ x]

1 / (x · ln a)

d/dx [ln u] (chain rule)

u′/u

d/dx [arcsin x]

1 / √(1 − x²)

d/dx [arccos x]

−1 / √(1 − x²)

d/dx [arctan x]

1 / (1 + x²)

d/dx [arccot x]

−1 / (1 + x²)

d/dx [arcsec x]

1 / (|x| · √(x² − 1))

d/dx [arccsc x]

−1 / (|x| · √(x² − 1))

Derivative of an inverse function

(f⁻¹)′(a) = 1 / f′(f⁻¹(a)). If f(b) = a, then (f⁻¹)′(a) = 1/f′(b)