Ap Calc Quater One Flash Cards

Derivative of e^x

e^x

derivative of lnX

1/x

Derivative of a^x

a^x ln(a)

derivative of LOGa X

1/xlna

IVT (Intermediate Value Theorem)

if f(x) is continuous on [a,b], there exists a point "c" between a and b such that f(a) ≤ c ≤ f(b)

average rate of change

f(b)-f(a)/b-a

F(x) B continuous at x=c if

1- F(c) exists

2- lim F(x)as x->c exists

3- lim f(x) as x-> c = F(c)

differentiability implies

continuity

Definition of the derivative F'(x)=

Lim as X->0 f(x+h) -f(x) / h

Alternate definition of derivative f'(c)

lim x-> c f(x+h)-f(x) / h

A function is not differentiable at x=c if

1- there is a cusp of sharp point at x=c

2- there is a vertical tangant at x=c

3- there is a discontinuity at x=c

Lim f(x) as x->0

find horizontal asymptote

Average rate of change vs Instantaneous rate of change

AROC= f(b)-f(a)/b-a

IROC= f'(c)

squeeze theorem

If f(x) ≤ g(x) ≤ h(x) for all x ̸= a and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L

sin^2 x + cos^2 X

= 1

Sec^2 x -1

= tan^2 x

cscθ

1/sin θ

cotθ

cosθ/sinθ

Derivative of sinx

cosx

Derivative of cosx

sinx

Derivative of tanx

sec^2x

Derivative of secx

secx tanx

Derivative of cscx

-cscx cotx

Derivative of cotx

-csc^2x