NoteNextra-origin/content/Math4121/Exam_reviews/Math4121_E1.md

Math4121 Exam 1 Review

Range: Chapter 5 and 6 of Rudin. We skipped (and so you will not be tested on)

• Differentiation of Vector Valued Functions (pp. 111-113)
• Integration of Vector-Valued Function and Rectifiable Curves (pp.135-137)

You will also not be tested on Uniform Convergence and Integration, which we cover in class on Monday 2/10.

Chapter 5: Differentiation

Definition of the Derivative

Let f be a real function defined on an closed interval [a,b]. We say that f is differentiable at a point x \in [a,b] if the following limit exists:

f'(x) = \lim_{t\to x} \frac{f(t) - f(x)}{t - x}

If the limit exists, we call it the derivative of f at x and denote it by f'(x).

Theorem 5.2

Every differentiable function is continuous.

The converse is not true, consider f(x) = |x|.

Theorem 5.3

If f,g are differentiable at x, then

1. (f+g)'(x) = f'(x) + g'(x)
2. (fg)'(x) = f'(x)g(x) + f(x)g'(x)
3. If g(x) \neq 0, then (f/g)'(x) = (f'(x)g(x) - f(x)g'(x))/g(x)^2

Theorem 5.4

Constant function is differentiable and its derivative is 0.

Theorem 5.5

Chain rule: If f is differentiable at x and g is differentiable at f(x), then the composite function g\circ f is differentiable at x and

(g\circ f)'(x) = g'(f(x))f'(x)

Theorem 5.8

The derivative of local extremum (\exists \delta > 0 s.t. f(x)\geq f(y) or f(x)\leq f(y) for all y\in (x-\delta,x+\delta)) is 0.

Theorem 5.9

Generalized mean value theorem: If f,g are differentiable on (a,b), then there exists a point x\in (a,b) such that

(f(b)-f(a))g'(x) = (g(b)-g(a))f'(x)

If we put g(x) = x, we get the mean value theorem.

f(b)-f(a) = f'(x)(b-a)

for some x\in (a,b).

Theorem 5.12

Intermediate value theorem:

If f is differentiable on [a,b], for all \lambda between f'(a) and f'(b), there exists a c\in (a,b) such that f'(x) = \lambda.

Theorem 5.13

L'Hôpital's rule: If f,g are differentiable in (a,b) and g'(x) \neq 0 for all x\in (a,b), where -\infty \leq a < b \leq \infty,

Suppose

\frac{f'(x)}{g'(x)} \to A \text{ as } x\to a

If

f(x) \to 0, g(x) \to 0 \text{ as } x\to a

or if

g(x) \to \infty \text{ as } x\to a

then

\lim_{x\to a} \frac{f(x)}{g(x)} = A

Theorem 5.15

Taylor's theorem: If f is n times differentiable on [a,b], f^{(n-1)} is continuous on [a,b], and f^{(n)} exists on (a,b), for any distinct points \alpha, \beta \in [a,b], there exists a point x\in (\alpha, \beta) such that

f(\beta) =\left(\sum_{k=0}^{n-1} \frac{f^{(k)}(\alpha)}{k!}(\beta-\alpha)^k\right) + \frac{f^{(n)}(x)}{n!}(\beta-\alpha)^n

Chapter 6: Riemann-Stieltjes Integration

Definition of the Integral

Let \alpha be a monotonically increasing function on [a,b].

A partition of [a,b] is a set of points P = \{x_0, x_1, \cdots, x_n\} such that

a = x_0 < x_1 < \cdots < x_n = b

Let \Delta \alpha_i = \alpha(x_{i}) - \alpha(x_{i-1}) for i = 1, \cdots, n.

Let m_i = \inf \{f(x) : x_{i-1} \leq x \leq x_{i}\} and M_i = \sup \{f(x) : x_{i-1} \leq x \leq x_{i}\} for i = 1, \cdots, n.

The lower sum of f with respect to \alpha is

L(f,P,\alpha) = \sum_{i=1}^{n} m_i \Delta \alpha_i

The upper sum of f with respect to \alpha is

U(f,P,\alpha) = \sum_{i=1}^{n} M_i \Delta \alpha_i

Let \overline{\int_a^b} f(x) d\alpha(x)=\sup_P L(f,P,\alpha) and \underline{\int_a^b} f(x) d\alpha(x)=\inf_P U(f,P,\alpha).

If \overline{\int_a^b} f(x) d\alpha(x) = \underline{\int_a^b} f(x) d\alpha(x), we say that f is Riemann-Stieltjes integrable with respect to \alpha on [a,b] and we write

\int_a^b f(x) d\alpha(x) = \overline{\int_a^b} f(x) d\alpha(x) = \underline{\int_a^b} f(x) d\alpha(x)

Theorem 6.4

Refinement of partition will never make the lower sum smaller or the upper sum larger.

L(f,P,\alpha) \leq L(f,P^*,\alpha) \leq U(f,P^*,\alpha) \leq U(f,P,\alpha)

Theorem 6.5

\underline{\int_a^b} f(x) d\alpha(x) \leq \overline{\int_a^b} f(x) d\alpha(x)

Theorem 6.6

f\in \mathscr{R}(\alpha) on [a,b] if and only if for every \epsilon > 0, there exists a partition P of [a,b] such that

U(f,P,\alpha) - L(f,P,\alpha) < \epsilon

Theorem 6.8

Every continuous function on a closed interval is Riemann-Stieltjes integrable with respect to any monotonically increasing function.

Theorem 6.9

If f is monotonically increasing on [a,b] and \alpha is continuous on $[a,b]$, then f\in \mathscr{R}(\alpha) on [a,b].

Key: We can repartition the interval [a,b] using f.

Theorem 6.10

If f is bounded on [a,b] and has only finitely many discontinuities on [a,b], then f\in \mathscr{R}(\alpha) on [a,b].

Key: We can use the bound and partition around the points of discontinuity to make the error arbitrary small.

Theorem 6.11

If f\in \mathscr{R}(\alpha) on [a,b], and m\leq f(x) \leq M for all x\in [a,b], and \phi is a continuous function on [m,M], then \phi\circ f\in \mathscr{R}(\alpha) on [a,b].

Composition of bounded integrable functions and continuous functions is integrable.

Theorem 6.12

Properties of the integral:

Let f,g\in \mathscr{R}(\alpha) on [a,b], and c be a constant. Then

1. f+g\in \mathscr{R}(\alpha) on [a,b] and \int_a^b (f(x) + g(x)) d\alpha(x) = \int_a^b f(x) d\alpha(x) + \int_a^b g(x) d\alpha(x)
2. cf\in \mathscr{R}(\alpha) on [a,b] and \int_a^b cf(x) d\alpha(x) = c\int_a^b f(x) d\alpha(x)
3. f\in \mathscr{R}(\alpha) on [a,b] and c\in [a,b], then \int_a^b f(x) d\alpha(x) = \int_a^c f(x) d\alpha(x) + \int_c^b f(x) d\alpha(x).
4. Favorite Estimate: If |f(x)| \leq M for all x\in [a,b], then \left|\int_a^b f(x) d\alpha(x)\right| \leq M(\alpha(b)-\alpha(a)).
5. If f\in \mathscr{R}(\beta) on [a,b], then \int_a^b f(x) d(\alpha+\beta) = \int_a^b f(x) d\alpha + \int_a^b f(x) d\beta.

Theorem 6.13

If f,g\in \mathscr{R}(\alpha) on [a,b], then

1. fg\in \mathscr{R}(\alpha) on [a,b]
2. |f|\in \mathscr{R}(\alpha) on [a,b] and \left|\int_a^b f(x) d\alpha(x)\right| \leq \int_a^b |f(x)| d\alpha(x)

Key: (1), use Theorem 6.12, 6.11 to build up fg from (f+g)^2-f^2-g^2. (2), take \phi(x) = |x| in Theorem 6.11.

Theorem 6.14

Integration over indicator functions:

If a<s<b, f is bounded on [a,b], and f is continuous at s, and \alpha(x)=I(x-s), then

\int_a^b f(x) d\alpha(x) = f(s)

Key: Note the max difference can be made only occurs at s.

Theorem 6.15

Integration over step functions:

If \alpha(x) = \sum_{i=1}^{n} c_i I(x-x_i) for x\in [a,b], then

\int_a^b f(x) d\alpha(x) = \sum_{i=1}^{n} c_i f(x_i)

Theorem 6.21

Fundamental theorem of calculus:

Let f\in \mathscr{R}(\alpha) on [a,b], and F(x) = \int_a^x f(t) d\alpha(t). Then

1. F is continuous on [a,b]
2. If f is continuous at x\in [a,b], then F is differentiable at x and F'(x) = f(x)

Chapter 7: Sequence and Series of Functions

Example of non-Riemann integrable function

\lim_{m\to \infty} \lim_{n\to \infty} (\cos(m!\pi x))^{2n}=\begin{cases} 1 & x\in \mathbb{Q} \\ 0 & x\notin \mathbb{Q} \end{cases}

This function is everywhere discontinuous and not Riemann integrable.

Uniform Convergence

Definition 7.7

A sequence of functions \{f_n\} converges uniformly to f on E if for every \epsilon > 0, there exists a positive integer N such that

|f_n(x) - f(x)| < \epsilon \text{ for all } x\in E \text{ and } n\geq N

If E is a point, then that's the common definition of convergence.

If we have uniform convergence, then we can swap the order of limits.

Theorem 7.16

If \{f_n\}\in \mathscr{R}(\alpha) on [a,b], and \{f_n\} converges uniformly to f on [a,b], then

\int_a^b f(x) d\alpha(x) = \lim_{n\to \infty} \int_a^b f_n(x) d\alpha(x)

Key: Use the definition of uniform convergence to bound the difference between the integral of the limit and the limit of the integral. \int_a^b (f-f_n)d\alpha \leq |f-f_n| \int_a^b d\alpha = |f-f_n| (\alpha(b)-\alpha(a)).