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40 Cards in this Set
 Front
 Back
slope

rise/run
change in y/change in x y2  y1/ x2 x1 

linear function

y=mx+b
m is the slope b is the vertical intercept 

exponential function

P=Poa^t
Po = initial quantity a = the factor by which P changes when t increases by 1 

continuous exponential function

P=Poe^kt


inverse function

a function has an inverse if its graph intersects any horizontal line at most once (horizontal line test)
graph is a reflection about the y=x line 

log10x=c

10^c=x


ln x = c

e^c = x


Properties of Natural Logs

ln(AB) = lnA + ln B
ln(A/B) = ln A  ln B ln(A^p) = p ln A ln e^x = x e^ln x = x ln 1 = o ln e = 1 

f(t) = A sin (Bt)

abs A = amplitude
2`/ abs B = period (in tangent period = `/abs B) 

inverse of a trig function

arc trig function


continuous function

no breaks, jumps or zeros
(don't pick up pencil) 

Intermediate Value Theorem

f is continous on closed interval A,B. If k is any number between f(a) and f(b), then there is at least one number c in A,B such that
f(c)=k 

average velocity

change in position/change in time
s(b)  s(a) / b  a 

instanteous velocity

1) at t=a
lim h>0 s(a+h)  s(a)/ h 2)the average velocity over an interval as the inverval shrinks around a 3) slope of the curve at a point(tangent line) 

Properties of Limits

lim k =k
lim x>c x = c 

limits with infinity

1)limit of 3x = infinity when x approaches infinity
2) limit of 1/3x = o when x approaches infinity 3) limit of 3x/4x = 3/4 when x approaches infinity 

average rate of change of f over the interval from a to a+h

f(a+h)  f(a)/h
(general formula while equation with s was specifically for height) 

derivative

instanteous rate of change
lim h>0 f(a+h)  f(a)/ h slope of the tangent line 

rules of derivatives

f'>0, f increasing
f'<0, f decreasing f(x) = k, f'(x) = 0 

power rule

f(x)=x^n, then f'(x) = nx^n1


interpretations of the derivative

dy/dx


second derivative

f">0, f' increasing, f concave up
f"<0, f' decreasing, f concave down 

d/dx(e^x)

e^x


d/dx(a^x)

(ln a)a^x


Product Rule

(fg)' = f'g + fg'


Quotient Rule

(f/g)' = f'g fg'/g^2


Chain Rule

d/dx(f(g(x)) = f'(g(x))*g'(x)


d/dx(sin x)

cos x


d/dx(cos x)

sin x


d/dx(tan x)

1/cos^2 x


d/dx(ln x)

1/x


d/dx(a^x)

(ln a)a^x


d/dx(arctan x)

1/1 + x^2


d/dx(arcsin x)

1/sqrt(1 x^2)


implicit functions

if there is a y use y'


tangent line approximation

f(x) = f(a) + f'(a)(xa)


Mean Value theorem

f'(c) = f(b)  f(a) / ba


sigma notation

definite integral
number at the top tells the number of intervals number at the bottom tells where to begin 

global max and min

overall maximum and minimum
highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest 

definite integral

F(b)  F(a) = integral from b to a F'(t) dt
