Flashcards
C4: Definitions
Cards in this deck (26)
- 1Local MaximumLet f be a function
f has a local maximum at x = c, if there exists some neighbourhood (δ > 0), such that f (x) ≤ f(c) for all x ∈ (c − δ,c + δ)
(observe diagrams in notes) - 2Local MinimumLet f be a function
f has a local minimum at x = c, if there exists some neighbourhood (δ > 0), such that f (x) ≥ f(c)for all x ∈ (c − δ,c + δ) - 3A Local Extremum (conditions)Intuitively at a local extremum, the tangent line of a function must be horizontal or undefined.
- 4Stationary pointA point x = c in the domain of a function f is called a stationary point of f if f′(c) = 0
- 5Critical PointA point x = c in the domain of a function f is called a critical point of f if f′(c) = 0 or f′(c) does not exist.
- 6Inflexion Pointsnot every critical point is is a local extremum. Cases where f′(c) = 0 but which are not extremum are called inflexion points of f. (see later)
- 7Increasing/Decreasing FunctionsA function f is increasing (decreasing) on an interval I ⊆ R if f(b) > f(a) ( f(b) < f(a)) for all [a,b] ⊂ I. That is a,b ∈ I, b > a.
Also if f′(c) > 0 (or f′(c) < 0 ) then f is increasing (or decreasing) at x = c. - 8Indeterminate RatioAn indeterminate form of the type 0/0 , or ∞/∞
- 9Indeterminate ProductAn indeterminate form of the type 0 · ∞.
- 10Indeterminate DifferencesAn indeterminate form of the type ∞ − ∞ is a limit
- 11Indeterminate PowersAn indeterminate form of the type 0^0, 1^∞, and ∞^0 arises from the limit
- 12Approximation from Taylor/Maclaurin seriesIf f and g are both n-times differentiable functions and f(k)(c) =
g(k)(c) for k = 0,1,...,n, we say g approximates f near x = c to degree n. - 13Taylor PolynomialLet f be a function with an nth-order derivative at the point x0. We define the nth order Taylor polynomial for f about x = x0 to be the function
Tn(x) =
n
∑
k=0
(f^(k)(x0)) (x − x0)^k/k! - 14Maclaurin PolynomialWe see that the polynomial Tn(x) approximates the function f(x). If x0 = 0 we call (4.5) the Maclaurin polynomial for f with
Tn(x) =
n
∑
k=0
(f^(k)(0)) x^k/k! - 15The Binomial Coefficient for Binomial SeriesConsider the Maclaurin polynomial of f(x) = (1 + x)^m for integer m ∈ Z then
the binomial coefficient which for integer m > 0 can be
obtained using Pascal’s triangle. Thus for all x, here we have the equality (1 + x)m =
(see notes for this summation) - 16Taylor SeriesLet f be a function with derivatives of all orders at the point x0. We define the Taylor series for f about x = x0 to be the power series
∞
∑
k=0
(f^(k)(x0)) (x − x0)^k/k! - 17Maclaurin SeriesSame as the Taylor Series but with x0=0
- 18Geometric SeriesConsider now the Maclaurin series of
(1 −x)^−1
the Maclaurin series of (1 − x)−1 is the Geometric series, and we can write
1/ 1 − x
=
∞
∑x^k
k=0 |x| < 1 - 19The Remainder of a Functionfor some c ∈ (x,x0) the nth order remainder for f,
Rn(x) = f(x) − Tn(x) - 20Radius of ConvergenceIn general Rn(x) → 0 only for a range of values of x, and so we can write
f(x) =
∞
∑
k=0
(f^(k)(x0))(x − x0)^k/k!
for |x − x0| < ρ
where ρ is called the radius (interval) of convergence of the Taylor series. If ρ = ∞ the Taylor series converges for all x ∈ R, with f(x) being infinitely differentiable everywhere. - 21Concave upsuppose f and f′ are both differentiable on an interval I ⊂ R
We say f is concave up on I if f′ is increasing on I - 22Concave downSuppose f and f′ are both differentiable on an interval I ⊂ R
We say f is concave down on I if f′ is decreasing on I - 23Inflexion PointsWe define the inflexion points of a function f as the points in the domain of f
at which its concavity changes, that is the points where f′′ changes sign. - 24OptimisationMany real world problems involve optimization, the finding of global maximum and minimum values of a function on an interval.
- 25Global MaximumWe recall that x = c is the location of a global maximum of a function f on an interval I if f(c) ≥ f(x) for all x ∈ I. Sometimes local extrema are also global extrema and sometimes they are not.
- 26Newton's MethodThe method of approximating a root where f(x)=0. We use:
x = x0 −f(x0) /f′(x0)
the method works best when f′ is large.
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- Calculus (Chapter 4: Applications of Differentiation) Flashcards on C4: Definitions, created by Luke Danaher on 28/12/2016.
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- Luke Danaher
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