MATH 15200 - Midterm 2

new things, no repeats i think

Exponential Growth/Decay Formula

P(t) = P₀ e^kt

P(t) = Population

P₀ = Initial Population

k = rate of growth (negative for decay)

t = time

d/dx sin^-1(x)

1 / sqrt(1-x²)

d/dx cos^-1(x)

- 1 / sqrt(1-x²)

int( 1 / sqrt(a² - x²) )dx

sin^-1(x/a) + c

d/dx tan^-1(x)

1 / 1 + x²

int ( 1 / a² + x²) dx

1/a tan^-1(x/a) + c

d/dx cot^-1(x) 

-1 / 1+ x²

d/dx sec^-1(x)

1 / |x|sqrt(x² - 1)

d/dx csc^-1(x)

-1/ |x|sqrt(x² - 1)

Integration by parts

int u dv = dv - int(v)du OR
int(u(x)*v’(x)dx) = u(x)v(x) - int(u’(x)v(x) dx)

Integral with a² + b²

x = a tan(u)

Integral with sqrt(a² - x²)

x = a sin(u)

sin²x + cos²x

= 1

1 + tan²x

= sec²x

1 + cot²x

= csc²x

sin(a + b)

sin(a)cos(b) + cos(a)sin(b)

sin(a - b)

sin(a)cos(b) - cos(a)sin(b)

cos(a + b)

cos(a)cos(b) - sin(a)sin(b)

cos(a - b)

cos(a)cos(b) + sin(a)sin(b)

sin2x

2sinxcosx

cos2x

cos²x - sin²x

sin²x

½ - ½ cos2x

cos²x

½ + ½ cos2x

sin(a)sin(b)

(cos(a-b) - cos(a+b)) / 2

cos(a)cos(b)

(cos(a-b) + cos(a+b)) / 2

sin(a)cos(b)

(sin(a+b) + sin(a+b)) / 2

Integral with sqrt(x²-a²)

x = a sec(u)

integral of secx

ln|secx + tanx|

d/dx (cosx)

-sin x

d/dx(sin x)

cos x

d/dx tan x

sec²x

d/dx secx

secx + tanx

d/dx cscx

-cscxcotx

d/dx cot

-csc²x

d/dx (p^x)

= p^x * ln(p) * x’

integral p^x dx

= 1 / ln(p) * p^x + c

log_px

ln(x)/ ln(p)

log_pp^t

= t

d/dx (log_px)

1 / x*ln(p)

half-life

kT = -ln2

integral of 1/ sqrt(a² + x²)

= ln | x + sqrt(a² + x²) | + c