Differentiation Flashcards
When is a function differentiable on an open interval?
if it is differentiable at every point on (a,b)
if a function is differentiable at x = a it is continuous at x = a
if a function is differentiable at x = a it is continuous at x = a
What is an equivalent form of differentiation by first principles
(f(x)-f(a))/x-a
as x -> a
what’s the derivative of x^a
a^x * ln(a)
what is leibnez’s rule for repeated products?
y’ = f’g + g’f
y’’ =f’‘g + 2f’g’ + fg’’
if y = f(x) what is the equation for the tangent line at the curve given at any point across the curve
(y - f(a))/x-a = f’(a)
lim f (a + h) − L(a + h) = 0.
h→0
lim f (a + h) − L(a + h) = 0.
h→0 h
Definition 4.27 A function f has a local maximum at c if f (c) ≥ f (x) for all x in some interval containing c. Similarly, f has a local minimum at c if f (c) ≤ f (x) for all x in some open interval containing c.
Definition 4.27 A function f has a local maximum at c if f (c) ≥ f (x) for all x in some interval containing c. Similarly, f has a local minimum at c if f (c) ≤ f (x) for all x in some open interval containing c.
what’s the extreme value theorem
Theorem 4.29 — Extreme Value Theorem. If f (x) is continuous at every point on [a,b] then f takes both its maximum and minimum values on this interval. In other words, there are points xmax ∈ [a, b] and xmin ∈ [a, b] such that
f(xmin) ≤ f(x) ≤ f(xmax) for all x ∈ [a, b].
when is f convex or concave
We say f is convex if any chord joining two points lies above the graph. Similarly it is concave if any chord lies below the graph.w
What is l’hopitals rule
Theorem 4.39 — L’Hôpital’s Rule. In general, sup- pose that f (a) = g(a) = 0 or lim f (x) = lim g(x) = ∞.
Then
x→a x→a lim f(x) = lim f′(x).
x→a g(x) x→a g′(x) assuming the derivatives exist and g′(a) ̸= 0.
