Karteikarten
C3: Definitions
Karten in diesem Stapel (14)
- 1The Tangent LineThe tangent line to the curve y = f(x) at the point A: (c,f(c))
is the straight line y = mx + c with slope
m = lim
h→0
[f(c + h) − f(c)]/h
provided this limit exists. - 2The Derivative of a function at a pointLet a function f be defined in a neighbourhood Nδ (c) : = (c − δ,c + δ), for some δ > 0, then the derivative of f at the point x = c is the number
f′(c) ≡df/ dx(c): = lim
h→0
f(c + h) − f(c)/h
provided the limit, which is an indeterminate form 0/0 , exists - 3Left DerivativeLet a function f be defined on (c − δ,c], for some δ > 0, then the left
derivative f′− is
f′−(c): = lim
h→0−
[f(c + h) − f(c)]/h - 4Right DerivativeLet a function f be defined on [c ,c+ δ), for some δ > 0, then the right
derivative f′+ is
f′+(c): = lim
h→0+
[f(c + h) − f(c)]/h - 5The Derivative of a FunctionThe derivative of a function f is the function f′(x), defined by
x → f′(x) ≡
df/dx(x) := lim
h→0
[f(x + h) − f(x)]/h
where the domain of f′ is the set of all x for which the limit exists. - 6DifferentiableA function f is said to be differentiable at x = c if the limit (on previous definition)
for f′(x) exists at x = c. - 7Left/right differentiablefunction f is said left-differentiable or right-differentiable
at x = c, if its left derivative f′- (c) or right derivative f′+ (c) exists.
Note we must distinguish between f′
± (c) and f′ (c±) = limx→c± f′(x) - 8Equation of a tangent line at a specific point cL(x) ≡ f(c) + f′(c)(x − c)
- 9Linear Approximation via tangent lineFor values near x = c the function f(x) can be approximated by this tangent line Indeed we define the linear approximation to f at x = c, by
f(x) ≈ f(c) + f′(c)(x − c) - 10Differentialswhen the linear approximation is valid, that is δx is sufficiently small that δy ≈ f′(x)δx , then
dy = f′(x)dx - 11Differentiably Continuous FunctionsLet Cn (I) denote the set of functions whose nth derivative is con-tinuous on the interval I ⊂ R, so
f ∈ Cn [I] ⇐⇒ f(n) ∈ C [I].
Thus the set C1 is known as the set of differentiably continuous functions. - 12Parametric EquationsParametric equations in R2 are the pair of functions x = x(t), y = y(t), which can regarded as a mapping x : R → R2 given by t → x = (x(t),y(t)). A parameterised curve is the set of points in the plane
S = {x(t),y(t)|t ∈ I} ⊂ R2 - 13The Exponential Functionhe exponential function expx is the sum of the infinite series
expx =
∞
∑
k=1
x^k/k!
= lim
n→∞
n
∑
k=1
xk/k!= 1 + x +
x^2/2!
+ ··· +
x^n/n!
+ ··· - 14Unique Continuous FunctionThe exponential function expx is the unique continuous function f(x) satisfying f′(x) = f(x) for all x ∈ R and f(0) = 1.
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- Calculus (Chapter 3: Differentiation) Karteikarten am C3: Definitions, erstellt von Luke Danaher am 27/12/2016.
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- Luke Danaher
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