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34 Cards in this Set
- Front
- Back
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Antiderivative
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function F is antiderivative of f on an interval I if F'(x)=f(x) for all x in I
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Antiderivative
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F'(x) = f(x) for all x in I
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Indefinite integral
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All antiderivatives of f w/ respect to x
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Indefinite integral
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|f(x) or |F'(x)
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First "n" squares
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(n(n+1)(2n+1))/6
...... (Think Summation k^2) |
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First "n" cubes
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((n(n+1))/2)^2
(Think Summation k^3) |
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Definite Integral
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Given e>0 there exists a # d>0 such that for every partition
P= {x0, x1,... , xn} of [a,b] w/ ||P||<d and any choice ck in [x(k-1), xk] we have |Summation f(ck) /\xk -J|<e |
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Definite Integral (summed up)
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J as lim-->infinity, summation:
f(ck)((b-a)/n) |
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/\xk
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(b-a)/n
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Interchangable continuous function
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if a function is continuous over [a,b] then an definite integral exists and is integrable over [a,b]
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Average Value
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av(f)= 1/(b-a)* a|b(f(x)dx)
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Mean Value Theorem
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If f is continuous on [a,b], then at some point c in [a,b]:
f(c) = (1/b-a)* a|b(f(x)dx) |
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Fundamental Theorem of Calculus Part 1 (FTC PT1)
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If f is continuous on [a,b], then F(x)=(a|x)(f(t)dt) is cont on [a,b] and differentiable on (a,b) and its derivative f(x) is
F'(x)=(d/dx)(a|x)(f(t)dt)= f(x) |
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FTC Part 1
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F'(x)= (d/dx)(a|x)(f(t)dt)=f(x)
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Fundamental Theorem of Calculus Part 2 (FTC PT2)
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If f is cont on [a,b] and F is the antiderivative of f on [a,b] then
(a|b)f(x)dx = F(b)-F(a) |
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FTC Part2
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(a|b)f(x)dx = F(b)-F(a)
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Net Change
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Net of a function F(x) over [a,b] is integral of its Rate of Change
F(b)-F(a) =(a|b)F'(x)dx |
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Rate of Change
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F(b)-F(a) =(a|b)F'(x)dx
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Substitution Method
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If u=g(x) is a differentiable function whose range is interval I and f is cont on I then:
|(f(g(x))g'(x)dx = |f(u)du |
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Substitution of Definite Integral
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Remember to Plug a and b values into u for new values corresponding to u!!
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In the context of Riemann sums, a partition (P) of [a,b] is a set:
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P = {x0, x1, x2,......,x(n-1), x(n)}
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the k-th subinterval is denoted as:
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[x(k-1), x(k)]
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Its norm ||P|| is defined as
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the largest of all subinterval widths
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MVT Eqn
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f(c)=(1/b-a)(a|b)(f(x)dx)
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Closed Formula
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summation (f(a+((b-a)/n)(i-1))/\x)
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lnx =
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(1|x)(1/t)dt x>0
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Exponential growth and decay
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y =y(o)e^(kt)
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Integration by parts
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|(f(x)g'(x)dx) = f(x)g(x) - |f'(x)g(x)dx
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Integration by parts (short)
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|u dv = uv- |vdu
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sin^2(x) =
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(1-cos(2x))/2
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cos^2(x)=
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(1+cos2x)/2
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Partial Fractions method 1
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factor the denominator and :
(A/(denom factor))+(B/(denom factor) |
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Partial Fractions method 2
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If denominator is to the power of:
(A1/(x-r))+ (A2/(x-r)^2)+ .... (Am/(x-r)^m) |
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Partial Fractions method 3
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((B1x+C)/(x^2+px+q))+....((Bnx + Cn)/(x^2+px+q)^n)
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