midterm to know apcalc

def of continuity

1) f(a) exists
2) lim as x approaches a of f(x) exists
3)lim as x approaches a of f(x) \= f(a)

if f(x) grows faster than g(x) in f(x)/g(x)

lim as x approaches infinity \= infinity

if f(x) grows slower than g(x) in f(x)/g(x)

lim as x approaches infinity\=0

if f(x) grows at same rate as g(x) in f(x)/g(x)

lim as x approaches inf is a finite number m, g(x) cannot equal 0

Squeeze Theorem

If f(x)

r\>0 and rational: lim x-\>inf 1/x^r

\=0

r\>0, rational, x^r defined for all x, lim x-\>-inf 1/x^r

\=0

continuity rules: if f and g are continuous at a and c is a constant then following functions are also continuous at a

f+g, f-g, cf, fg, f/g if g(a) doesnt equal 0

functions continuous at every number in domain

polynomials, trig func, exponential func, rational func, inverse trig func, log func, root func

Intermediate Value Theorem

1) f is continuous on closed interval [a,b]
2) N is any number between f(a) and f(b)
3) then number c in (a,b) such that f(c)\=N

lim x-\>0 (sin(x))/x

\=1

lim x-\>0 (cos(x)-1)/x

\=0

lim x-\>inf 1/x

\=0

lim x-\>0- 1/x

\=-inf

lim x-\>0+ 1/x

\=inf

lim x-\>-inf 1/x

\=0

lim x-\>0 1/x^2

\=inf

lim x-\>0 (abs:x)/x

\=DNE

lim x-\>0 sin(1/x)

\=DNE

lim x-\>0 1/x

DNE

instantaneous rate of change at a point is

lim h-\>0 f(a+h)-f(a)/h (derivative at that point)

average rate of change over an interval [a,b]

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

deriv of inverse function

d/dx (f^-1(x))\= 1/f'(f^-1(x))

normal line

line perpendicular to tangent line

y\=loga(b)

a^y\=b

change of base

loga(x)\=lnx/lna

deriv of sin(x)

cos x

deriv of cos x

-sin x

deriv of tan x

sec^2 x

deriv of csc x

-csc x cot x

deriv of secant

sec x tanx

deriv of cot x

-csc^2 x

deriv of e^x

e^x

deriv of ln x

1/x

deriv of a^x

ln(a) (a^x)

deriv of absval (x)

(absvalx)/x if x is not 0

deriv of arcsin(x)

1/sqrt(1-x^2)

deriv of arccos(x)

-1/sqrt (1-x^2)

deriv of arctan (x)

1/(1+x^2)

deriv of arccsc (x)

-1/(abs:x(sqrt (x^2-1)))

deriv of arcsec (x)

1/(abs: x(sqrt(x^2-1))

deriv of arccot x

-1/1+x^2

logarithmic differentiation

take natural log of both sides, derive implicitly, solve for dy/dx, remember your natural log

linear approximation formula

f(b)\=f(a)+f'(a)(x-a)

velocity function

p'(t)

acceleration function

v'(t)\=p''(t)

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum f(c) and an absolute minimum f(d) at some numbers c and d on [a,b].

Fermat's Theorem

If f has a local maximum or minimum at c, and if f'(c) exists, then f'(c)\=0

Rolle's Theorem

1) f is cont on closed interval [a,b], 2)f is differentiable on open interval (a,b), 3) f(a)\=f(b), then there is at least one number c in (a,b) such that f'(c)\=0

mean value theorem

1) f is cont on [a,b], 2) f is differentiable on (a,b), Then there exists a number such that f'(c)\=f(b)-f(a).b-a

L'Hopital's Rule

if limit of f(x)/g(x) as x-\>a produces the forms 0/0 or inf/inf then lim x-\>a f(x)/g(x)\= limx-\>a f'(x)/g'(x)