BC memorization

lim x→a f(x)

lim x→a- f(x) = lim x→a+ f(x)

f(x) is continuous at x = a

f(a) = lim x→a f(x)

MVT: f(x) is cont on [a,b] and diff on (a,b), there is at least one value c such that

f’(c) = f(b)-f(a)/b-a

def of derivative: lim x→a f(x)-f(a)/x-a

lim h→0 f(x+h)-f(x)/h

ARC

f(b)-f(a)/x-a

Avg Val

1/(b-a) ∫ f(x)

If (a,b) is on the graph of f(x) and g(x) = f^-1(x) then g’(b) = _______

1/f’(a)

f(x) has POI

f’’(x) changes sign

area of trapezoid

w((h1+h2)/2)

FTC: d/dx ∫ f(t)dt

f(x)

FTC extension: d/dx ∫ f(t)dt

f(g(x)) × g’(x)

Logistic growth: carrying capacity = M, y(t) grows fastest when

y(t) = M/2

Arc length (rect)

∫ √1+[f’(x)]²dx

Nth term test diverges

lim n→∞ aₙ ≠ 0

Nth term test is inconclusive

lim n→∞ aₙ = 0

P series converges

p > 1

P series diverges

p ≤ 1

Geo test converges

|r| <1

Geo test diverges

|r| ≥ 1

Ratio test converges

L < 1

Ratio test diverges

L > 1

Slope of tangent line (parametric)

dy/dx = (dy/dt)/dx/dt

area between polar curve and origin

½ ∫ r²dθ

parametric 2nd deriv: d2y/dt2

(deriv dy/dx)/dx/dt

Total distance = arc length (parametric) for <x(t), y(t)>

∫ √[x’(t)]²+[y’(t)]²dt

Logistic Differential

dy/dx = ky(1-y/M)

Slope of tangent line (polar)

dy/dx = (dy/dθ)/dx/dθ

y (polar)

rsinθ

x (polar)

rcosθ⁹⁶