Calculus I - final review notecards
Terms
undefined, object
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- slope
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rise/run
change in y/change in x
y2 - y1/ x2 -x1 - linear function
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y=mx+b
m is the slope
b is the vertical intercept - exponential function
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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
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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 - log10x=c
- 10^c=x
- ln x = c
- e^c = x
- 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 - f(t) = A sin (Bt)
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abs A = amplitude
2`/ abs B = period
(in tangent period = `/abs B) - inverse of a trig function
- arc trig function
- continuous function
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no breaks, jumps or zeros
(don't pick up pencil) - 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 - average velocity
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change in position/change in time
s(b) - s(a) / b - a - 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) - Properties of Limits
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lim k =k
lim x->c x = c - 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 - 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) - derivative
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instanteous rate of change
lim h->0 f(a+h) - f(a)/ h
slope of the tangent line - rules of derivatives
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f'>0, f increasing
f'<0, f decreasing
f(x) = k, f'(x) = 0 - power rule
- f(x)=x^n, then f'(x) = nx^n-1
- interpretations of the derivative
- dy/dx
- second derivative
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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(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)(x-a)
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slope
rise/run
change in y/change in x -
y2 - y1/ x2 -x1