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40 Cards in this Set
- Front
- Back
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slope
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rise/run
change in y/change in x y2 - y1/ x2 -x1 |
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linear function
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y=mx+b
m is the slope b is the vertical intercept |
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exponential function
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P=Poa^t
Po = initial quantity a = the factor by which P changes when t increases by 1 |
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continuous exponential function
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P=Poe^kt
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inverse function
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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 |
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log10x=c
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10^c=x
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ln x = c
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e^c = x
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Properties of Natural Logs
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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 |
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f(t) = A sin (Bt)
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abs A = amplitude
2`/ abs B = period (in tangent period = `/abs B) |
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inverse of a trig function
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arc trig function
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continuous function
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no breaks, jumps or zeros
(don't pick up pencil) |
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Intermediate Value Theorem
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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 |
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average velocity
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change in position/change in time
s(b) - s(a) / b - a |
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instanteous velocity
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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) |
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Properties of Limits
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lim k =k
lim x->c x = c |
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limits with infinity
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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 |
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average rate of change of f over the interval from a to a+h
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f(a+h) - f(a)/h
(general formula while equation with s was specifically for height) |
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derivative
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instanteous rate of change
lim h->0 f(a+h) - f(a)/ h slope of the tangent line |
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rules of derivatives
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f'>0, f increasing
f'<0, f decreasing f(x) = k, f'(x) = 0 |
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power rule
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f(x)=x^n, then f'(x) = nx^n-1
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interpretations of the derivative
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dy/dx
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second derivative
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f">0, f' increasing, f concave up
f"<0, f' decreasing, f concave down |
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d/dx(e^x)
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e^x
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d/dx(a^x)
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(ln a)a^x
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Product Rule
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(fg)' = f'g + fg'
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Quotient Rule
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(f/g)' = f'g -fg'/g^2
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Chain Rule
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d/dx(f(g(x)) = f'(g(x))*g'(x)
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d/dx(sin x)
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cos x
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d/dx(cos x)
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-sin x
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d/dx(tan x)
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1/cos^2 x
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d/dx(ln x)
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1/x
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d/dx(a^x)
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(ln a)a^x
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d/dx(arctan x)
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1/1 + x^2
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d/dx(arcsin x)
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1/sqrt(1- x^2)
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implicit functions
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if there is a y use y'
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tangent line approximation
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f(x) = f(a) + f'(a)(x-a)
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Mean Value theorem
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f'(c) = f(b) - f(a) / b-a
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sigma notation
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definite integral
number at the top tells the number of intervals number at the bottom tells where to begin |
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global max and min
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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 |
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definite integral
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F(b) - F(a) = integral from b to a F'(t) dt
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