BC Calculus Yellow Board

AB and BC Topics included

d/dx arcsin =

1/√1-u² * u’

d/dx arccos =

-1/√1-u² * u’

d/dx arctan =

1/1+u² * u’

d/dx arccot =

-1/√1+u² * u’

d/dx arcsec =

1/u√u²-1 * u’

d/dx arccsc =

-1/u√u²-1 * u’

d/dx log_au =

1/(u ln a) * u’

d/dx a^u=

a^ulna*u’

_a∫^bk^p =

∑ᵇ⁻ᵃ⁄ₙ(a+⁽ᵇ⁻ᵃ⁾ᵏ⁄ₙ)^p

sin²x+cos²x=

1

tan²x+1=

sec²x

cot²x+1=

csc²x

Disk formula

π∫(r²)

Washer Formula

π∫(big r)²-(small r)²

Shells Formula

2π∫r*h

Shells with gap

2π∫r(big height-small height)

Semicircle cross sections

∫½πr²; ᶠ⁽ˣ⁾⁻ᵍ⁽ˣ⁾⁄₂

equilateral triangle cross sections

∫(√3)/4 s²; s=f(x)-g(x)

Isosceles Right Triangle cross sections

∫½s²; s=f(x)-g(x)

Logistic Growth Formula

ky(1-y/L) or ky/L (L/y)

Euler’s Method

y_new=y_old-y’∆x

Integration by Parts

∫udv=uv-∫vdu

Arc Length

∫√1+(f’(x))²

Parametric slope

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

parametric 2nd derivative

(d/dx dy/dx)/dx/dt

parametric arc length

∫√(dy/dt)²+(dx/dt)²

parametric speed

√(dy/dt)²+(dx/dt)²

parametric position

x(t)+∫x’(t) , y(t)+∫y’(t)

polar x coordinate conversion

x=rcosθ

polar y coordinate conversion

y=rsinθ

rose curve formula

r=asin(nθ) or r=acos(nθ)

#of petals

if n is odd petals=n; if n is even petals=2n

circle formulas

r=2asinθ or r=2acosθ

limacon formula

r=a±bsinθ or r=a±bcosθ

a/b<1

inner loop

a/b=1

cardioid

1<a/b<2

dimpled

a/b>2

convex

lemniscate formula

r²=a²sin(2θ) or r²=a²cos(2θ)

archimedian spiral formula

r=aθ

polar area

½∫r² dθ

polar area b/w curves

½∫(r₁²-r₂²) dθ

polar arc length

∫√r²+(dr/dθ)²

nth term test

take limit as k→∞, if lim=0, continue testing; if lim≠0, diverge

geometric series

∑ar^k if r<1 converge, if r≥1 diverge; a/1-r gives convergence value

p series

∑1/k^p ; p>1 converge p≤1 diverge

Integral Test

must be continuous, decreasing, and positive. ∑u=∫f(x)

ratio test

∑u_k; take lim k→∞ u_k+1/u_k = ρ; ρ<1 converge, p>1 diverge, ρ=1 indeterminate

Root test

same as ratio, take 1/k power

Comparison test

compare to smaller series that diverges or larger series that converges

limit comparison test

a_k=series you want to prove, b_k=known divergent/convergent, a_k/b_k=ρ, ρ>0 means valid comparison

Alternating Series

identify: (-1)^k ; a_k+1<a_k, lim k→∞ a_k=0 means converge

Interval of Convergence

1. ratio test

2. set ρ<1

3. check endpoints

Maclaurin series

f(0)+f’(0)x/1!+f”(0)x²/2!+f”’(0)x³/3!…

Taylor series

f©+f’©(x-c)/1!+f”©(x-c)²/2!…

La Grange Error Bound

|R_n|≤max|x-a|ⁿ⁺¹/(n+1)! ; max value between a and x

Alternating Series Error Bound

|S-S_n|=|R_n|≤|a_n+1|

Power series 1/1-x

∑x^k ; 1+x²+x³…

power series 1/1+x²

∑(-1)^k*x^2k ; 1-x²+x⁴-x⁶…

power series e^x

∑x^k/k! ; 1+x+x²/2!+x³/3!…

power series sinx

∑(-1)^kx^2k+1/(2k+1)! ; x-x³/3!+x⁵/5!-x⁷/7!…

power series cosx

∑(-1)^kx^2k/(2k)! ; 1-x²/2!+x⁴/4!-x⁶/6!