UC Berkeley Math Prelim Workshop

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UC Berkeley Math Prelim Workshop

Real Analysis

James Rowan

August 12, 2020

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Roadmap

1. Real analysis

1.1 Intermediate and mean value theorems 1.2 Sequences of real numbers

1.3 Taylor’s formula (with remainder) 1.4 Multivariable generalizations

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Functions from R to R

There are two major new properties in real analysis beyond basic metric space topology that can come in handy

I The ordering of the real numbers

I Completeness: every Cauchy sequence converges The ordering gives a couple of useful theorems

1. Intermediate value theorem

2. Mean value theorem (and various other versions such as the integral or Cauchy versions)

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Functions from R to R

I Completeness: every Cauchy sequence converges

The ordering gives a couple of useful theorems 1. Intermediate value theorem

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Functions from R to R

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Functions from R to R

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1.1.10 – Fall 1982 11

1. Prove that there is no continuous map from the closed interval [0, 1] onto the open interval (0, 1).

2. Find a continuous surjective map from the open interval (0, 1) onto the closed interval [0, 1].

3. Prove that no map in Part 2 can be bijective.

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1.5.3 – Fall 1990 4 Suppose f is a continuous real valued function. Show that

Z 1 0

f (x )x²dx = 1 3f (ξ) for some ξ ∈ [0, 1].

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Inequalities

Some key inequalities to know I Cauchy-Schwarz:

If f and g are two continuous, complex-valued functions on [a, b], then

Z b a

f gdx ≤ Z b

a

|f |²dx

!1/2

Z b a

|g|²dx

!1/2

,

with equality if and only if f = λg for some λ.

Note that there is also a version for finite and infinite sums I Jensen’s inequality:

If f is a convex function, then f x₁+x₂

2

≤ f (x₁) +f (x₂)

2 .

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Inequalities

Z b a

f gdx ≤ Z b

a

|f |²dx

!1/2

Z b a

|g|²dx

!1/2

,

2

≤ f (x₁) +f (x₂)

2 .

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Inequalities

Z b a

f gdx ≤ Z b

a

|f |²dx

!1/2

Z b a

|g|²dx

!1/2

,

Note that there is also a version for finite and infinite sums

I Jensen’s inequality:

2

≤ f (x₁) +f (x₂)

2 .

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Inequalities

Z b a

f gdx ≤ Z b

a

|f |²dx

!1/2

Z b a

|g|²dx

!1/2

,

2

≤ f (x₁) +f (x₂)

2 .

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Inequalities

Z b a

f gdx ≤ Z b

a

|f |²dx

!1/2

Z b a

|g|²dx

!1/2

,

2

≤ f (x₁) +f (x₂)

2 .

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1.5.9 – Fall 1985 15 Let 0 ≤ a ≤ 1 be given. Determine all nonnegative continuous functions f on [0, 1] which satisfy the following three conditions:

Z 1 0

f (x ) dx = 1,

Z 1 0

xf (x ) dx = a,

Z 1 0

x²f (x ) dx = a².

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Sequences of real numbers

I Any monotonic, bounded sequence converges

I Any Cauchy sequence converges (the converse is true in any metric space)

I We can also look at the lim sup and lim inf of a sequence of real numbers; if these are equal, then the sequence

converges.

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Sequences of real numbers

converges.

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Sequences of real numbers

converges.

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1.3.8 – Spring 2003 6A Let x_nbe a sequence of real numbers so that lim_n→∞2x_n+1− xn=x . Show that lim_n→∞x_n=x .

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1.3.9 – Spring 2000 5 Let a and x₀be positive numbers, and define the sequence (x_n)^∞_n=1 recursively by

x_n= 1 2

x_n−1+ a x_n−1

.

Prove that this sequence converges, and find its limit.

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Taylor’s Theorem with Remainder

I How well a Taylor series approximates a smooth function can be understood quantitatively (useful for e.g. proving error bounds on approximations for integrals)

Taylor’s theorem with remainder: Let f be k times differentiable. The difference between f (x ) and its k th Taylor polynomial

f (a) + f⁰(a)(x − a) +1

2f⁰⁰(a)(x − a)²+ · · · + 1

k !f^{(k )}(a)(x − a)^k is given by

R_k(x ) = 1

(k + 1)!f^{(k +1)}(ξ)(x − a)^{k +1} for some ξ between a and x .

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Taylor’s Theorem with Remainder

I How well a Taylor series approximates a smooth function can be understood quantitatively (useful for e.g. proving error bounds on approximations for integrals)

Taylor’s theorem with remainder: Let f be k times differentiable. The difference between f (x ) and its k th Taylor polynomial

f (a) + f⁰(a)(x − a) +1

2f⁰⁰(a)(x − a)²+ · · · + 1

k !f^{(k )}(a)(x − a)^k is given by

R_k(x ) = 1

(k + 1)!f^{(k +1)}(ξ)(x − a)^{k +1} for some ξ between a and x .

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1.1.17 – Fall 1997 11 Let f be a C²function on the real line.

Assume f is bounded with bounded second derivative. Let A = sup

x ∈R

|f (x)|, B = sup

x ∈|R

|f⁰⁰(x )|.

Prove that

sup

x ∈R

|f⁰(x )| ≤ 2

√ AB.

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Multivariable Extensions

1. “The derivative the Jacobian” (e.g. in the inverse function theorem, ^dy_dx 6= 0 becomes det J(x) 6= 0)

2. “The second derivative the Hessian” (e.g. for convexity)

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Multivariable Extensions

1. “The derivative the Jacobian” (e.g. in the inverse function theorem, ^dy_dx 6= 0 becomes det J(x) 6= 0)

2. “The second derivative the Hessian” (e.g. for convexity)

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2.2.43 – Spring 1996 12 Let M_2×2be the space of 2 × 2 matrices over R, identified in the usual way with R⁴. Let the function F from M_2×2into M_2×2be defined by

F (X ) = X + X².

Prove that the range of F contains a neighborhood of the origin.

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Key takeaways

I If you know that a function is a function from R to R, you might be able to exploit ordering-related properties like the IVT

I Similarly, there is additional structure to sequences of real numbers coming from the ordering properties

I Using Taylor’s theorem with remainder can give you quantitative control of how good your approximations are I Many one-variable results generalize to several variables

with appropriate modifications

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Key takeaways

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Key takeaways

I Using Taylor’s theorem with remainder can give you quantitative control of how good your approximations are

I Many one-variable results generalize to several variables with appropriate modifications

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