Math (Winter Exam 2024) Flashcards
What is the formal definition of continuity?
lim f(x) as x —> c = f(c)
When is a function continuous?
F(x) is continuous at x=c if and ONLY if f(c) is defined AND lim f(x) as x —> c = f(c).
Can a function be continuous in one direction?
A function can be continuous at lim f(x) as x—> c+ OR c-.
Describe the Intermediate Value Theorem.
F(x) is continuous on the closed interval [a, b] for any d ∈ ℝ if either
i) f(a) ≤ d ≤ f(b)
ii) f(a) ≥ d ≥ f(b)
this means that some c ∈[a, b] exists such that f(c) = d.
Formal definition of a derivative (text)
For a function f(x) and a point x=c where f(x) is defined, the DERIVATIVE of f(x) at x=c is the limit (if it exists).
Formal definition of a derivative (equation)
lim as h —> 0 = [f(c+h) - f(c)]/ [(c+h) - c]
What are the two versions of derivative notation?
f ‘ (c) = Newton
df/dx (c) = Leibniz
Define sum rule (Leibniz notation and Newton)
d/dx (fx + gx) = df/dx + dg/dx
(Fx + gx) ‘ = fx’ + gx’
Define product rule (Leibniz and Newton)
d/dx [(ax)(bx)] = da/dx (b) + db/dx (a)
[(ax)(bx)] = (ax)’(bx) + (ax)(bx)’
Define chain rule in Leibniz and Newton notation
d/dx f(g(x)) = [df/dx (g(x))][dg/dx]
f(g(x))’ = [f ‘ (g(x))][g ‘ (x)]
Define quotient rule (Leibniz and Newton)
d/dx (a/b) = [b(da/dx) - a (db/dx)]/b^2
(a/b) ‘ = [ba’ - ab’]/ b^2
describe how to do log/ln differentiation
f ‘(x)= f(x) * d/dx (ln(f(x)))
given the function c = a + b, how would you set up the equation for implicit differentiation?
d/dx (c) = d/dx (a) + d/dx (b)
describe implicit differentiation
Implicit differentiation is a way to find the derivative in equations involving x and y without solving for y. Treat y as a function of x and use the chain rule.
what is the equation for linear approximation of f(x) and point a?
L(x) = f(a) + f ‘ (a)(x-a)
what is linear approximation?
Linear approximation estimates the value of a function near a point using the tangent line, assuming the function behaves like a straight line close to that point.
What is the formal definition of an ABSOLUTE MAX (let c∈D)
f has an absolute max at x=c iff ∀x∈D, f(x) ≤ f(c)
what is the formal definition of an ABSOLUTE MIN (let c∈D)
f has an absolute min at x=c iff ∀x∈D, f(x) ≥ f(c)
what is the formal definition of a RELATIVE MAX (let c∈D, f: D -> ℝ)
f has a local max at x=c iff ∃ε>0 such that ∀x∈(c-ε, c+ε) f(x) ≤ f(c).
what is the formal definition of a RELATIVE MIN (let c∈D, f: D -> ℝ)
f has a local max at x=c iff ∃ε>0 such that ∀x∈(c-ε, c+ε) f(x) ≥ f(c).
we can say that c∈D is a critical point of f, iff any of the following are true:
1) f ‘(c) = 0
2) f ‘(c) DNE
3) c is on the boundary of D (on the edge of the domain)
describe the intermediate value theorem
if f: D -> ℝ AND d is closed/bounded, then f has BOTH an absolute max and min on D. this must occur at a critical value.
what is the mean value theorem?
suppose f: [a, b] -> ℝ which is continuous on [a, b] and differentiable on (a, b). Then ∃c∈(a, b) such that f ‘(c) = (f(b) - f(a))/b - a.
what is an inflection point? how is it found?
point where concavity switches. second derivative test.
what is L’Hôpital’s Rule?
If lim x→ c of [f(x)/g(x)] = 0/0 or ∞/∞, then lim x–>c [f(x)/g(x)] = lim x→ c [f’(x)/g’(x)]
What is the epsilon-delta definition of continuity?
∀ ε > 0, ∃ δ>0 such that ∀x (0 < |x - c | < δ ⟹ | f(x) - f(c) | < ε )
