Flashcards
C5: Definitions
Cards in this deck (25)
- 1PartitionSuppose we split up the interval [a,b] into n subintervals [xi−1,xi] for i = 1,2,...,n, each of width δxi = xi − xi−1. The set of points a = x0 < x1 < ··· < xn = b is called a partition Pn of the interval [a,b].
- 2NormWe define the norm of the partition by
|Pn| = max
1≤i≤n
δxi = max
1≤i≤n
(xi − xi−1)
as the length of the largest interval. - 3Riemann Left SumSuppose f is a function defined on [a,b] and that Pn is a partition of [a,b] into n subintervals [xi−1,xi], with δxi = xi − xi−1 then
(a) The Riemann left sum for f on [a,b] is
n
∑f(xi−1)δxi
i=1 - 4Riemann Right SumSuppose f is a function defined on [a,b] and that Pn is a partition of [a,b] into n subintervals [xi−1,xi], with δxi = xi − xi−1 then
The Riemann right sum for f on [a,b] is
n
∑f(xi)δxi
i=1 - 5Riemann Upper SumSuppose f is a function defined on [a,b] and that Pn is a partition of [a,b] into n subintervals [xi−1,xi], with δxi = xi − xi−1 with Mi and mi the maximum and minimum value of f on [xi−1,xi], then
The Riemann upper sum for f on [a,b] is given by
Uf (Pn) =
n
∑Mk(xk − xk−1)
k=1 - 6Riemann Lower SumSuppose f is a function defined on [a,b] and that Pn is a partition of [a,b] into n subintervals [xi−1,xi], with δxi = xi − xi−1 with Mi and mi the maximum and minimum value of f on [xi−1,xi], then
The Riemann lower sum for f on [a,b] is given by
Lf (Pn) =
n
∑mk(xk − xk−1)
k=1 - 7Riemann Integrablefunction f is said to be Riemann integrable (or simply inte-grable) on the interval [a,b] if there is a unique number I that satisfies the inequality
Lf (P) ≤ I ≤ Uf (P) for all partitions P of [a,b]
where Uf and Lf are the upper and lower Riemann sums of f on [a,b]. - 8Value of Integral (I)The number
I is called the definite (Riemann) integral of f from a to b - 9f(x) being integratedThe function f(x) is called the integrand
- 10a, b in the integral boundsa and b are
called the lower and upper limits of integration. - 11Geometric AreaFor any integrable function f on [a,b], the geometric area be-tween the graph of f, the x-axis, and the line x = a and x = b is given by
b
∫ |f(x)| dx
a - 12Signed AreaIf a function f takes both positive and negative values, then the definite integral corresponds to the signed area between the graph of f and the interval [a,b] along the x-axis, the area above the x-axis minus the area below the x-axis.
- 13Absolute Area Between graphsIf f and g are integrable functions, then the (absolute) area be-tween the graphs of f and g from x = a to x = b is the definite integral
b
∫ |f(x) − g(x)| dx
a - 14AntiderivativeA function F is called an antiderivative of a function f on a interval I ⊂ R if F′(x) = f(x) for all x ∈ I.
- 15Indefinite IntegralThe indefinite integral of a continuous function f is defined to be the family of antiderivatives, denoted by
∫f(x)dx = F(x) + C where F′(x) = f(x) - 16Area Accumulation FunctionLet f be continuous on [a,b]. Then, for x ∈ [a,b], the area accu-
mulation function for f on [a,b] is
A(x) =
x
∫ f(t)dt
a - 17The Natural Logarithm FunctionThe natural logarithm function is the function defined by
lnx :=
x
∫ 1/t dt for x ∈ (0,∞)
1 - 18The Exponential FunctionThe inverse of the natural logarithm exp : R → (0,∞) is called the exponential function.Thus
y = lnx ⇐⇒ x = exp(y) - 19Logarith base change formulaIf b > 0 and r is a real number then we define
b^r = e^(r ln b),
logb x =lnx/ lnb - 20Pointwise ConvergenceA sequence of real-valued functions {fn(x)} is said to converge pointwise to a function f(x) on a set I = [a,b] if and only if for all x ∈ I and for all ε > 0 there exists a positive integer N(ε,x) such that
|fn(x) − f(x)| < ε for all n > N(ε,x) - 21Uniform ConvergenceA sequence of real-valued functions {fn(x)} is said to converge uniformly to a function f(x) on a set S if and only if for all ε > 0 there exists a positive integer N(ε), independent of x, such that
|fn(x) − f(x)| < ε for all n > N(ε) - 22Upper BoundIf f is a bounded real-valued function on an interval [a,b] then there exists a real number M, called an upper bound of f, such that |f(x)| ≤ M
for all x ∈ [a,b]. - 23Uniform normFurther the uniform norm of f on the interval is defined to be
∥f∥∞ = sup{|f(x)| : a ≤ x ≤ b} - 24Supremumthe supremum is the least upper bound.
- 25Uniform PartitionA uniform partition consists of subintervals of equal width with |Pn| = δx = (b − a)/n with xi = a + iδx. Since f is continuous we may assume f(x) ≈ f(xi) for x ∈ [xi−1,xi] for a sufficiently small δxi and approximate the region R by rectangles.
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- Calculus (Chapter 5: Integration) Flashcards on C5: Definitions, created by Luke Danaher on 29/12/2016.
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