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--- math104-s21:s:franceskim [2021/05/12 03:40]
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 D) E ⊂ X is dense iff (closure of E) = X
+**Lecture 19**
+A) If f is differentiable at p ∈ [a, b], then f is continuous at p.
+B) If f(x) is differentiable at p, then there exists u(x) s.t. f(x) = f(p) + (x-p)f'(p) + (x-p)u(x) where lim u(x) = 0 as x -> p
+u(x) = (f(x) - f(p))/(x - p) - f'(p) when x ≠ p, u(x) = 0 when x = p
+C) p can be a local maximum, but f'(p) is non existent (cusp).
+D) Local maximum and minimum can occur at endpoints.
+E) f: [a, b] -> R is a continuous function. Assume f'(x) exists for all x ∈ (a, b). If f(a) = f(b), then there exists c ∈ (a, b), s.t. f'(c) = 0. (Rolle's Theorem)
+F) If f, g: [a, b] -> R are continuous and differentiable on (a, b), then there exist c ∈ (a, b) s.t. [f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c).
+- Special case:
+f(b) - f(a) = (b-a)f'(c), c ∈ (a, b)
+**Lecture 20**
+. "If c is a local maximum of a function, and if derivative exists at c, and c is an interior point, derivative vanishes." How does this statement hold true?
+A) f: R -> R, f is continuous, f'(x) exists for all x ∈ R. Assume there exists M > 0 s.t. |f'(x)| <= M for all x. Then, f is uniformly continuous.
+B) f: [a, b] -> R is a differentiable function. f'(a) < f'(b). Then, for any c ∈ R w/ f'(a) < c < f'(b), there exists a d ∈ (a, b) s.t. f'(d) = c
+C) lim (f(x)/g(x)) = C as x -> a if:
+f, g: (a, b) -> R are differentiable, g(x), g'(x) ≠ 0 over (a, b) AND
+lim (f'(x)/g'(x)) = C AND
+i) lim f(x) = 0 as x -> a, lim g(x) = 0 as x -> a OR
+ii) lim g(x) = +inf as x -> a
+**Lecture 21**
+A) Smooth functions:
+Derivatives exist to all order.
+B) Taylor Theorem:
+f: [a, b] -> R is a function s.t. f^(n-1) (x) exists and is continuous on [a, b]. f^(n) (x) exists on (a, b).
+Then, for any α, β ∈ [a, b], we have:
+f(β) = f(α) + f'(α)(β-α) + f''(α)/2!(β-α)^2 + ... + f^(n-1)(α)/(n-1)!(β-α)^(n-1) + R_n(α, β)
+R_n(α, β) = 0 if α = β, R_n(α, β) = f^n(r)/n!(β-α)^n if α ≠ β for some r ∈ (α, β)
+**Lecture 22**
+A) Power series:
+Series of the form ∑ C_n(x-x_0)^n from n = 0 to n = +inf
+B) Radius of Convergence R:
+R = sup {r >= 0, s.t. if |x-x_0| <= r, the series converges}
+C) If R = 1/a, where a = limsup|C_n|^(1/n) as n -> +inf:
+If |x-x_0| < R, the series converges.
+If |x-x_0| > R, the series diverges.
+. Prove the diverging case of (C).
+D) Real Analytic function f:
+f: (a, b) -> R is smooth, for all x_0 ∈ (a, b), f(x) = ∑ C_n(x-x_0)^n for x in some neighborhood of x_0.
+. Regarding the definition of real analytic function, what does it mean for "f(x) = ∑ C_n(x-x_0)^n for x in some neighborhood of x_0"?
+E) Length of a function:
+∫sqrt(1 + (f'(x))^2)dx
+F) U(P, f) = ∑ M_i*Δx_i from i = 1 to i = n
+L(P, f) = ∑ m_i*Δx_i from i = 1 to i = n
+M_i = sup{f(x)|x ∈ [x_(i-1), x_i]}
+m_i = inf{f(x)|x ∈ [x_(i-1), x_i]}
+Δx_i = x_i - x_(i-1)
+G) U(f) = inf U(P, f)
+L(f) = sup L(P, f)
+H) f is integrable if U(f) = L(f)
+I) Generalization of Riemann-Stieltjes integrable:
+Let α: [a, b] -> R be a monotone increasing function, define partition P = {a = x_0 <= x_1 <= ... <= x_n = b}, define Δα_i = α(x_i) - α(x_(i-1))
+The remaining definitions are similar as in parts (F) and (G), except Δx_i -> Δα_i, and U(P, α) = L(P, α) implies that f is Riemann-Stieltjes integrable w.r.t. α.
+**Lecture 23**
+A) If a partition Q is a refinement of partition P on [a, b], then L_P <= L_Q <= U_Q <= U_P.
+B) L(f, α) <= U(f, α)
+C) f is integrable w.r.t. α iff for all ε > 0, there exists P partition s.t. U_P - L_P < ε
+.
 General Note:

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