Analysis 2 Notes 5

Question 1

(5T-N39-5.1) Rolle’s Theorem

Answer 1

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Question 2

(5T-N39-5.2) MVT

Answer 2

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Question 3

(5T-N39-5.3) Cauchy’s generalisation of the MVT

Answer 3

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Question 4

(5T-N40-5.4) Let f, g : (a, b) → R be differentiable and let c ∈ (a, b) be such that f(c)=g(c)=0 and g(-)(x) /=0 for x/=c. Then lim as x goes to c of f(x)/g(x) = lim as x goes to c of f(-)(x) / g(-)(x)

Answer 4

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Question 5

(5E-N41-5.1) Lim as x goes to 0 of sin(x) / x

Answer 5

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Question 6

(5E-N41-5.2) Lim as x goes to 0 of 1 - cos (x) / x^2

Answer 6

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Question 7

(5R-N42-) Taylor’s formula, all of it in the notes

Answer 7

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Question 8

(5T-N42-5.5) Let f : (a,b) → R be n + 1 times differentiable and let x0 ∈ (a,b). Then for any x ∈ (a, b), x ̸= x0, there exists some ξ strictly between x0 and x such that …

Answer 8

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Question 9

(5T-N44-5.6) Let f : (a,b) → R be n + 1 times differentiable and let x0 ∈ (a,b). Then for any x ∈ (a, b), x ̸= x0, there exists some ξ strictly between x0 and x such that … (Cauchy’s form of the remainder term)

Answer 9

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Question 10

(5R-N45-) Maclaurin’s formula

Answer 10

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Question 11

(5E-N45-5.3) Maclaurin’s formula with Lagrange’s remainder for exp: R to R

Answer 11

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Question 12

(5R-N46-) How do you obtain infinite PS of exp from Maclaurins formula?

Answer 12

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Question 13

(5E-N46-5.4) Maclaurin’s formula with Lagrange’s remainder for f : (−1, +∞) → R, f (x) = ln(1 + x)

Answer 13

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Question 14

(5E-N47-5.5) Maclaurin’s formula for f : (−1, +∞) → R, f (x) = (1 + x)^a

Answer 14

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