Karteikarten
Calculus - Chapter Four
Karten in diesem Stapel (21)
- 1INTEGRATION What is the central idea of Riemann integration?It is that appropriately chosen improving underestimates and improving overestimates converge to the same value as the Riemann integral of the function.
- 2A function f : [a,b] tends to R is a) bounded above on [a,b] if ... b) bounded below on [a,b] if ...a) ... the set {f(x) : x element (a,b)} is bounded above, i.e. there exists K element R such that f(x)/k for all x element [a,b]
- 3c) bounded on [a,b] if ... If a set has a maximum element then ... If a set has a minimum element then ...c) ... the set {f(x) : x element [a,b]} is bounded, i.e there exists k,K element R such that k
- 4A bounded set will always have a ... but not necessarily a ...... least upper bound and a greatest lower bound ... maximum and/or minimum element
- 5A partition of a closed interval [a,b] is what? What two terms are used to mean the same thing?A finite set P of points x0,x1,x2,...,xn element [a,b] with a=x1
- 6Suppose that f is a bounded function on a closed interval [a,b] and that P={x0,x1,x2,...,xn} is a partition on [a,b]. For each i=1,2,...,n, let mi be the greatest lower bound of f on [xi-1,xi] and let Mi be the least upper bound of f on [xi-1,xi]. a) The lower sum L(P,f) = ? b) The upper sum U(P,f) = ?a) SIGMA(i=1,n)[mi(xi-xi-1)]
b) SIGMA(i=1,n)[Mi(xi-xi-1)] - 7Why do we simplify the notation to L(P) and U(P)?When the function referred to is unambiguous.
- 8Need to be able to recall the following three formulae: a) SIGMA(i=1,n0[i] = ? b) SIGMA(i=1,n)[i^2] = ? c) SIGMA(i=1,n)[i^3] = ?a) [n(n+1)]/2
b) [n(n+1)(2n+1)]/6
c) [n^2(n+1)^2]/4 - 9Suppose that f is a bounded function on [a,b] and P=(a=x0,x1,...,xn=b} a partition of [a,b]. Then m(b-a)Then m(b-a)
- 10Suppose that f is a bounded function a [a,b]. Considering all possible partitions P of [a,b] the lower sums L(P) are ... and the upper sums U(P) are ... PROOF ON SEPARATE SHEET!!... bounded above ... bounded below
- 11Suppose that P1 and P2 are partitions of [a,b]. We say that P2 is a refinement of P1 if ...... P1 is contained in P2, i.e. if every point of P1 appears in P2.
- 12Suppose that f is a bounded funtion on [a,b] and that Q is a refinement of P={x0,x1,x2,...,xn} obtained by adding a point y with xN-1/ ? b) U(Q)a) L(Q) >/ L(P)
b) U(Q) - 13Suppose that f is a bounded function on [a,b]. If P and Q are partitions of [a,b] then ... PROOF ON SEPARATE SHEET!!... L(P)
- 14RIEMANN INTEGRALS Suppose that f is a bounded function on [a,b]. The lower Riemann integral L i is defined to be ... The upper Riemann integral U is defined to be ...... the least upper bound of possible values L(R) with R a partition.
... the greatest lower bound of all possible values U(R) with R partition. - 15Suppose that f is a bounded function on [a,b]. Then LThen L
- 16Suppose that f is a bounded funcrion on [a,b]. We say that f is Riemann Integrable on [a,b] if ...... L=U. In this case we say that L=U=integral (from a to b)[f(x)] dx
- 17Suppose that f is a bounded function on [a,b] and that there are partitions Pn such that {L(Pn)} and {U(Pn)} are sequences converging to the same limit l, then f is ... PROOF ON SEPARATE SHEET!!... Riemann Integrable and the integral (from a to b)[f(x)] dx = l
- 18Suppose f is bounded on [a,b]. If there exists a sequence of partitions {Pn} such that ... then f is ... Moreover the integral (from a to b)[f(x)] dx = ?... U(Pn) - L(Pn) tends to 0 as n tends to infinity ... Riemann Integrable
L=U - 19Suppose f is bounded on [a,b] and that f is monotonic on [a,b]. Then f is ... PROOF ON SEPARATE SHEET!! What does it mean by monotonic?... Riemann Integrable.
Either increasing or decreasing - 20Increasing means ... Decreasing means ...... f(y)>/f(x) whenever a
- 21Suppose f is bounded on [a,b] and that a... f is Riemann Integrable on each of [a,c] and [c,b].
integral(from a to c) [f(x)] dx + integral(from c to b)[f(x)] dx
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- Karteikarten am Calculus - Chapter Four, erstellt von Cara Wales am 20/05/2016.
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