Help me please (BC Calc Formulas)

d/dx(c)

0

d/dx(cx)

c

d/dx(tan x)

sec²x

d/dx(cot x)

-csc²x

d/dx(xⁿ)

nxⁿ⁻¹

d/dx(sec x)

sec x tan x

d/dx[f(x)g(x)]

f'g +g'f

d/dx(csc x)

-csc x cot x

d/dx(f/g)

(f'g-g'f)/g²

d/dx(sin⁻¹x)

1/√(1-x²)

d/dx(f(g(x))

f'(g(x))g'(x)

d/dx(cos⁻¹x)

-1/√(1-x²)

d/dx(sin x)

cos x

d/dx(cos x)

-sin x

d/dx(tan⁻¹x)

1/(1+x²)

d/dx(cot⁻¹x)

-1/(1+x²)

d/dx(csc⁻¹x)

1/|x|√(x²-1)

d/dx(sec⁻¹x)

-1/|x|√(x²-1)

d/dx(ln x)

1/x

d/dx(logₙx)

1/(ln n)x

d/dx(eˣ)

e^x

d/dx[aˣ]

(ln a)a^x

∫a dx

ax + C

∫xⁿ dx

xⁿ⁺¹/(n+1) +C

Integration by Parts

∫ u dv

uv- ∫v du

∫1/x dx

ln|x| +C

∫eˣ dx

eˣ+c

∫aˣ dx

a^x/(ln a) +c

∫sin x dx

-cos x +c

∫cos x dx

sin x +c

∫tan x dx

ln |sec x| +C

∫cot x dx

ln |sin x| +c

∫sec x dx

ln |sec x + tan x| +c

∫csc x dx

ln |csc x - cot x| + c

∫sec²x dx

tan x +c

∫csc²x dx

-cot x + c

∫sec x tan x dx

sec x + c

∫csc x cot x dx

-csc x + c

∫1/(x²+a²) dx

1/a(tan⁻¹(x/a)) +c

∫1/√(a²-x²) dx

sin⁻¹(x/a) + c

∫1/x√(x²-a²) dx

1/a(sec⁻¹|x/a|) + c

∫ln x dx

x ln x - x +c

√(a²-x²)

x=a sin θ

√(x²+a²)

x = a tan θ

√(x²-a²)

x = a sec θ

(use when power of cosine is odd) cos²x

1-sin²x, u=sin x

(use when power of sine is odd) sin²x

1-cos²x, u=cos x

(if both sin and cos have even powers) sin²x

1/2 -1/2(cos 2x)

(if both sin and cos have even powers) cos²x

1/2 +1/2(sin 2x)

trig identity sin x cos x

1/2 sin 2x

parametric dy/dx

dy/dt / dx/dt

parametric second derivative d²y/dx²

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

velocity vector v(t)

<x’(t), y’(t)>

acceleration vector a(t)

<x’’(t), y’’(t)>

parametric speed formula

√[(dy/dt)²+(dx/dt)²]

parametric arc length formula

∫ √[(dy/dt)²+(dx/dt)²] dt

Polar area

1/2 ∫r² dθ

polar derivative

dy = r' sin θ +r cos θ

dx = r' cos θ + r sin θ

polar x=

r cos θ

polar y=

r sin θ

r²

x²+y²

polar tan θ

y/x (sin x/cos x)

maclurin series e^x

1+x+ x²/2! + x³/3! + … + xⁿ/n!+…

maclurin series sin x

x- x³/3! +x⁵/5! -x⁷/7! +… + (-1)ⁿx²ⁿ⁺¹/(2n+1)!+…

maclurin series cos x

1- x²/2!+ x⁴/4! - x⁶/6! + … + (-1)ⁿx²ⁿ/(2n)!+…

maclurin series 1/(1-x)

1+x+x²+x³+…+ xⁿ+…

maclurin series ln(x+1)

x-x²/2+ x³/3- x⁴/4 +… + (-1)ⁿxⁿ/(2n)!+…

macurin series tan⁻¹ x

x- x³/3 +x⁵/5 -x⁷/7 +… + (-1)ⁿx²ⁿ⁺¹/(2n+1)+…

geometric series

a+ ar+ ar² +…+ arⁿ+…=∑arⁿ

diverge is |r| >= 1

converge is |r| <1

sum= a₁/(1-r)

harmonic series

∑1/n

always diverges

p-series

∑1/n^p

diverge if p<=1

converge if p>1

Nth Term Test (Divergence Test)

if lim(x→∞) aₙ ≠ 0, the series ∑aₙ diverges

limit= 0 does not mean converge, but keep testing

Telescoping series

Series such as (1- 1/2)+(1/2-1/3)+(1/3-1/4)+…

Collapses to one or the first few terms and this is the sum. Use a partial fraction to break down the telescoping series if necessary.

integral test

if ƒ is decreasing, continuous, and positive for x>=1 AND aₙ = f(x), then ∑aₙ and ∫(1→∞) f(x) dx either both converge or diverge.

The value of the integral is not the sum of the series

Alternating series test

if aₙ > 0, then the alternating series ∑(-1)ⁿaₙ converges if both of the both following conditions are satisfied

a) lim (n→∞) aₙ = 0 (Nth term test)

b) {aₙ} is decreasing

common alternators: (-1)ⁿ, (-1)ⁿ⁺¹, cos (πn)

direct comparison test

if aₙ>= 0 and bₙ>=0

a) if ∑bₙ converges and 0<= aₙ <= bₙ, then ∑aₙ converges

b) if ∑aₙ diverges and 0<=aₙ,<=bₙ, then ∑bₙ diverges

limit comparison test

if aₙ>=0 and bₙ>= 0, and lim(n→∞) aₙ/bₙ = L or lim(n→∞) bₙ/aₙ =L, where L is both positive and finite, then the two series ∑aₙ and ∑bₙ, either both converge or diverge

absolute convergence

If the series ∑|aₙ| converges then ∑aₙ also converges. Such a series is called absolutely convergent.

If the series ∑|aₙ| diverges and ∑aₙ converges, the series is conditionally convergent

ratio test

∑aₙ converges if lim(n→∞) |aₙ₊₁/aₙ|

root test

the series ∑aₙ converges if lim(n→∞) ⁿ√|aₙ|

Taylor polynomial Tₙ(x)

f(c) + f'(c)(x-c) + f''(c)/2! (x-c)² + … + fⁿ(c)/n! (x-c)ⁿ

Maclurin Polynomial Mₙ(x)

(Taylor polynomial where c=0)

f(0) + f'(0)x + f''(0)x²/2! + … + fⁿ(0)xⁿ/n!

Lagrange error bound

If a function ƒ is differentiable through order n+ in an interval containing the center x=c, then for each x=a in the interval, there exists a number x=z between and c such that Tₙ(x)= f(c ) +f'(c)(a-c) + f''(c)/2! (a-c)² +…+ fⁿ(c)/n! (a-c)ⁿ + Rₙ(a)

Rₙ(a)<=

<=|maxƒⁿ⁺¹(z)/(n+1)! (a-c)ⁿ⁺¹|

Alternating series error bound

Suppose an alternating series converges and that the series has a sum S, then |Rₙ|=

|S-Sₙ| <= |aₙ₊₁|

Power Series

∑aₙ(x-c)ⁿ is a power series centered at x=c, where c is a constant.

a) The series converges at x=c (ALL POWER SERIES CONVERGE AT THEIR CENTER)

the radius is 0

Power Series

∑aₙ(x-c)ⁿ is a power series centered at x=c, where c is a constant.

b) The series converges for all x

the radius is ∞

Power Series

∑aₙ(x-c)ⁿ is a power series centered at x=c, where c is a constant.

c) there exists an R>0 such that the series converges for |x-c| < R and diverges for |x-c| > R

the corresponding domain [(c-R, c+R)], is called the interval of convergence

Exponential growth

Dy/dx=

Y=

dy/dx= ky

y= Ce^kt

Logistic Models

Dy/dx=

Y=

dy/dx = k/L y(L-y) = ky(1-y/L)

y=L/1+Ce^-kt

Euler's method

x | y | dy/dx|(x,y) | ∆y= m∆x | y new = y+∆y

∆x= (b-a)/n

Separation of variables

put y with dy and x with dx then integrate

area under the curve

∫a→b f(x) dx or ∫a→b f(y) dy

Disk method

π∫a→b R² dx or dy

Washer method

π∫a→b R²-r² dx or dy

arc length

∫a→b √(1+(dy/dx)²) dx

cross section

∫a→b A(x) dx

cross section square

s²

cross section semicircle

1/2 πr²

cross section equilateral triangle

√3/4 s²

cross section isosceles right triangle

1/2 x²