BC Calc Memorization Sheet Review

Flashcards for BC Calculus Memorization Sheet

Product Rule for Derivatives

uv' + vu'

Quotient Rule for Derivatives

(bt' - tb') / b^2

Integral of x^n

∫x^n dx = (x^(n+1))/(n+1) + C, n ≠ -1

∫(1/x) dx

ln|x| + C

ln|u| + C

∫(1/u) du

tan(x)

sin(x) / cos(x)

sin^2(x) + cos^2(x)

1

tan^2(x)

sec^2(x) - 1

Chain Rule

f'(u) * u'

d/dx (x^n)

n * x^(n-1)

d/dx (log_a(x))

1 / (x * ln(a))

d/dx (e^x)

e^x

d/dx (a^x)

a^x * ln(a)

∫e^x dx

e^x + C

∫a^x dx

(a^x / ln(a)) + C

∫cos(x) dx

sin(x) + C

∫sin(x) dx

-cos(x) + C

∫sec^2(x) dx

tan(x) + C

∫tan(x) dx

-ln|cos(x)| + C

∫cot(x) dx

ln|sin(x)| + C

∫sec(x)tan(x) dx

sec(x) + C

∫csc^2(x) dx

-cot(x) + C

∫csc(x)cot(x) dx

-csc(x) + C

∫(u'/(a^2 + u^2))du

(1/a) * arctan(u/a) + C

∫(u'/√(a^2 - u^2)) du

arcsin(u/a) + C

d/dx (arcsin(x))

1/√(1 - x^2)

d/dx (arctan(x))

1/(1 + x^2)

Second Fundamental Theorem of Calculus (d/dx ∫[u to v] f(t) dt)

f(v)v' - f(u)u'

Trapezoidal Rule

w/2 * (h0 + 2h1 + 2h2 + … + 2hn-1 + h_n)

Alternating Series Error Bound

error < |a_n+1|

d/dx (sin(x))

cos(x)

d/dx (cos(x))

-sin(x)

d/dx (tan(x))

sec^2(x)

d/dx (cot(x))

-csc^2(x)

d/dx (sec(x))

sec(x)tan(x)

d/dx (csc(x))

-csc(x)cot(x)

Integration by Parts

u*v - ∫vdu

∫lnx dx

xlnx - x + C

dP/dt in Logistic Growth

kP(M - P)

P(t) in Logistic Growth

M / (1 + Ce^(-kt))

Carrying capacity

M

e^x Series

1 + x + (x^2)/2! + (x^3)/3! + …

sin(x) Series

x - (x^3)/3! + (x^5)/5! - …

cos(x) Series

1 - (x^2)/2! + (x^4)/4! - …

Taylor Series

f(c) + f'(c)(x-c) + (f''(c)(x-c)^2)/2! + (f'''(c)(x-c)^3)/3! + …

Maclaurin Series

Taylor series with c = 0