AP BC Calculus Flash Cards

The domain of y=lnx is the set of…

all positive numbers x>0

The range of y=lnx is the set of…

all real numbers -∞<y<∞

y=lnx characteristic on domain

continuous and increasing everywhere on its domain

ln(ab)=

lna + lnb

ln(a/b)=

lna - lnb

lna^r =

rlna

y= lnx < 0 if…

0< x < 1

lim x→+∞ lnx =

+∞

lim →0+ lnx =

-∞

log_ax =

lnx/lna

What is the unit circle?

lim x→0 sinx/x =

1

Definition of Derivative

m= f’(x) =

lim h→0 f(x+h) - f(x) / h

lim h→0 e^h-1 / h =

1

Alternative definition of derivative at a point

f’(a) =

lim x→a f(x) - f(a) / x - a

lim h→0 cos h - 1 / h =

0

d/dx e^x =

e^x

Power rule

d/dx xⁿ =

nx^n-1

d/dx a^x =

a^xlna

d/dx lnx=

1/x

Product Rule

d/dx (uv) =

u’ v + u v’

Quotient Rule

d/dx (u/v) =

u’ v - u v’ / v²

df/dx = d/dx (c) =

0

Constant Multiple Rule

d/dx (cu) =

c (du / dx)

Sum and Difference Rule

d/dx (u+- v) =

du/dx +- dv/dx

d/dx log_b x =

1/xlnb

Chain rule

d/dx f(g(x))=

f’ (g(x)) g’(x)

d/dx (sin x) =

cosx

d/dx (cosx) =

-sinx

d/dx (tanx)=

sec²x

d/dx (cscx) =

-cscxcotx

d/dx (secx)=

secxtanx

d/dx (cotx)=

-csc²x

d/dx sin^-1x=

1/√1-x²

d/dx cos^-1x=

-1/√1-x²

d/dx tan^-1x=

1/x²+1

d/dx cot^-1x=

-1/x²+1

d/dx sec^-1x=

1/|x|√x²-1

d/dx csc^-1x=

-1/|x|√x²-1

Derivative of Inverses

If g(x) is the inverse of f(x) and f(a) = b then

g’(b) = 1/f’(a)

Trapezoidal Approximations

∫ f(x)dx=

h/2 (y₀+2y₁+2y₂+…2y_n-1+y_n)