$$\F^+$$/$$\F^+_0$$ is the positive/non-negative subset of $$\F$$. $$\N = \Z^+ = \\{1,2,\dotsc\\}$$. $$f(I)$$ is the range of $$f$$ on $$I$$. $$C^n([a,b])$$ is the set of functions with continuous $$n$$-th derivatives on $$[a,b]$$.

## Definitions

Name (Reference) Statement

Partition (§1.2, p. 13)

$[a,b]\subset\R$

Increasing sequence $$P=\\{t_n\\}$$ where $$a=t_0$$ and $$b=t_n$$

Partition Norm (§1.2, p. 13)

Partition $$P$$

$\norm{P} = \max\\{ \Delta t_1, \dotsc, \Delta t_n \\}$

Regular $$n$$-Partition (§1.2, p. 14)

$$[a,b]\subset\R$$, $$n\in\N$$

$$P^{(n)} = \\{t_n\\}$$ where $$\Delta t_i = \frac{b-a}{n}$$

Riemann Sum (§1.2, p. 13)

• Partition $$\\{t_n\\}$$ of $$I$$
• Bounded $$f:I\to\R$$
• $$c_i \in [t_{i-1},t_i]$$ for $$i \geq 1$$
$S = \sum\limits_{i=1}^n f(c_i) \Delta t_i$

Right-Hand Riemann Sum (§1.2, p. 15)

Riemann Sum requirements

$S = \sum\limits_{i=1}^n f(t_i) \Delta t_i$

Left-Hand Riemann Sum (§1.2, p. 15)

Riemann Sum requirements

$S = \sum\limits_{i=1}^n f(t_{i-1}) \Delta t_i$

Definite Integral/Integrability (§1.2, p. 16)

• Sequences $$\\{P_n\\}$$ with $$\norm{P_n} \to 0$$
• Riemann sums $$S_i$$ of $$P_i$$
• $$\exists$$ limit $$S_i \to I$$
$I = \displaystyle\int_a^b f(t) \dd{t}$

Identical Limits of Integration (§1.3.1, p. 20)

$$f(a)$$ exists

$\int_a^a f(t) \dd{t} = 0$

Switching the Limits of Integration (§1.3.1, p. 20)

$$f$$ integrable on $$[a,b]$$

$\int_b^a f(t) \dd{t} = -\int_a^b f(t) \dd{t}$

Average Value of $$f$$ (§1.4, p. 28)

$f \in C^0([a,b])$ $f_{avg}([a,b]) = \frac{1}{b-a}\int_a^b f(t) \dd{t}$

Indefinite Integral (§1.6.1, p. 42)

$$f$$ integrable

$\int f(x) \dd{x} = \\{ F : F' = f \\} = \\{ F(x) + C : C \in \R \\}$

Type I Improper Integral (§2.4, p. 82)

$$f$$ integrable

1. $\int_a^\infty f(x) \dd{x} = \lim\limits_{b\to\infty} \int_a^b f(x) \dd{x}$
2. $\int_{-\infty}^a f(x) \dd{x} = \lim\limits_{b\to-\infty} \int_b^a f(x) \dd{x}$
3. $\int_{-\infty}^\infty f(x) \dd{x} = \int_{-\infty}^c f(x) \dd{x} + \int_c^\infty f(x) \dd{x}$

Type II Improper Integral (§2.4, p. 97)

$$f$$ integrable except at asymptote $$c\in(a,b)$$

1. $\int_c^b f(x) \dd{x} = \lim\limits_{t\to c^+} \int_t^b f(x) \dd{x}$
2. $\int_a^c f(x) \dd{x} = \lim\limits_{t\to c^-} \int_a^t f(x) \dd{x}$
3. $\int_a^b f(x) \dd{x} = \int_a^c f(x) \dd{x} + \int_c^b f(x) \dd{x}$

Area Between Curves (§3.1)

$f,g\in C^0([a,b])$ $A = \int_a^b \abs{f(x)-g(x)} \dd{x}$

Volume of Revolution (Disk I, §3.2)

$$f \in C^0([a,b])$$, $$f \geq 0$$

$V = \int_a^b \pi f(x)^2 \dd{x}$

Volume of Revolution (Disk II, §3.2)

$$f,g\in C^0([a,b])$$, $$0 \leq f \leq g$$

$V = \int_a^b \pi(g(x)^2 - f(x)^2) \dd{x}$

Volume of Revolution (Shell, §3.3)

$$a \geq 0$$, $$f,g\in C^0([a,b])$$, $$f \leq g$$

$V = \int_a^b 2 \pi x (g(x) - f(x)) \dd{x}$

Arc Length (§3.4)

$f \in C^1([a,b])$ $S = \int_a^b \sqrt{1+f'(x)^2} \dd{x}$

Separable Differential Equation (§4.2)

$y' = f(x) g(y)$

First-Order Linear Differentiable Equations (FOLDE, §4.3)

$y' = f(x) y + g(x)$

Series (§5.1, p. 165)

Sequence $$\{a_n\}$$

$\isum a_n$

Convergence of a Series (§5.1, p. 165)

Partial sums $$S_k = \sum_{n=1}^k a_n$$

$$S_k \to L$$, then $$\isum a_n = L$$

Geometric Series (§5.2, p. 167)

Ratio $$r$$

$\sum_{n=0}^\infty r^n = 1 + r + r^2 + \dotsb$

Positive Series (§5.5, p. 178)

Series $$\isum a_n$$

$$a_n \geq 0$$ for all $$n$$

Alternating Series (§5.7, p. 202)

$$\isum (-1)^{n-1} a_n = a_1 - a_2 + a_3 - a_4 + \dotsb$$ or $$\isum (-1)^{n} a_n = - a_1 + a_2 - a_3 + \dotsb$$ with $$a_n$$ positive

Interval and Radius of Convergence (§6.1, p. 234)

$\isum[n=0] a_n(x-a)^n$
• $I = \{ x_0 : \isum[n=0] a_n(x_0-a)^n \text{ converges } \}$
• $R = \begin{cases} \operatorname{lub}(\{\abs{x_0 - a} : x_0 \in I\}) & \text{I bounded} \\ \infty & \text{I unbounded} \end{cases}$

## Theorems

Name (Reference) Statement

Integrability Theorem for Continuous Functions (thm. 1.1, p. 17)

• $f \in C^0([a,b])$
• Any regular $$n$$-partition Riemann sum $$S_n$$
$$\implies$$ $\displaystyle\int_a^b f(t) \dd{t} = \lim_{n\to\infty}S_n$

Properties of Integrals (thm. 1.2, p. 18)

• $\int_a^b f(t) \dd{t}$
• $\int_a^b g(t) \dd{t}$
$$\implies$$
1. $$\forall c\in\R$$, $$\int_a^b c f(t) \dd{t} = c\int_a^b f(t) \dd{t}$$
2. $\int_a^b(f+g)(t)\dd{t} = \int_a^b f(t) \dd{t} + \int_a^b g(t) \dd{t}$
3. $$f([a,b])\subseteq[m,M]$$ then $$m(b-a) \leq \int_a^b f(t) \dd{t} \leq M(b-a)$$
4. $$f([a,b])$$ positive then $$\int_a^b f(t) \dd{t}$$ positive
5. $$g \leq f$$ on $$[a,b]$$ then $$\int_a^b g(t) \dd{t} \leq \int_a^b f(t) \dd{t}$$
6. $\abs{\int_a^b f(t) \dd{t}} \leq \int_a^b\abs{f(t)}\dd{t}$

Integrals over Subintervals (thm. 1.2, p. 21)

$$f$$ integrable on $$I$$ and $$a,b,c\in I$$

$$\implies$$ $\int_a^b f(t) \dd{t} = \int_a^c f(t) \dd{t} + \int_c^b f(t) \dd{t}$

Average Value Theorem (thm. 1.4, p. 29)

$f \in C^0([a,b])$ $$\implies$$ $\exists c, f(c) = f_{avg}([a,b])$

Fundamental Theorem of Calculus, Part 1 (thm. 1.5 [FTC1], p. 36)

$$f \in C^0(I)$$, $$a\in I$$, $$G(x) = \int_a^x f(t) \dd{t}$$

$$\implies$$ $G'(x) = f(x)$

Extended Version of the Fundamental Theorem of Calculus (thm. 1.6, p. 40)

$$f \in C^0$$, $$g'$$, $$h'$$, $$H(x) = \int_{g(x)}^{h(x)} f(x) \dd{t}$$

$$\implies$$ $H'(x) = f(h(x))h'(x) - f(g(x))g'(x)$

Fundamental Theorem of Calculus, Part 2 (thm. 1.8 [FTC2], p. 45)

$$f \in C^0$$, $$F'(x) = f(x)$$

$$\implies$$ $\int_a^b f(t) \dd{t} = F(b) - F(a)$

Change of Variables (thm 1.9, p. 53)

$$g \in C^1([a,b])$$, $$f \in C^0(g([a,b]))$$

$$\implies$$ $\int_{x=a}^{x=b} f(g(x)) g'(x) \dd{x} = \int_{u=g(a)}^{u=g(b)} f(u) \dd{u}$

Integration by Parts (thm. 2.1, p. 71)

$f,g \in C^1([a,b])$ $$\implies$$ $\int_a^b f(x) g'(x) \dd{x} = f(x)g(x)\Big\vert_a^b - \int_a^b f'(x)g(x) \dd{x}$

Monotone Convergence Theorem for Functions (thm. 2.5 [MCTF], p. 89)

• $$f$$ non-decreasing on $$[a,\infty)$$
• $S = f([a,\infty))$
$$\implies$$
• If $$S$$ bounded above, $$f(x) \to \operatorname{lub} S$$
• Else, $$f(x) \to \infty$$

$$p$$-Test for Type I Improper Integrals (thm. 2.3, p. 85)

$p > 1$ $$\iff$$ $\int_1^\infty \frac{1}{x^p} \dd{x} = \frac{1}{p-1}$

Comparison Test for Type I Improper Integrals (thm. 2.6, p. 90)

$$0 \leq f(x) \leq g(x)$$ for all $$x \geq a$$

$$\implies$$
1. If $$\int_a^\infty g(x)$$ converges, so does $$\int_a^\infty f(x)$$
2. If $$\int_a^\infty f(x)$$ diverges, so does $$\int_a^\infty g(x)$$

Absolute Convergence Theorem for Improper Integrals (thm. 2.7, p. 93)

$$\int_a^\infty \abs{f(x)} \dd{x}$$ converges

$$\implies$$

$$\int_a^\infty f(x) \dd{x}$$ converges

$$p$$-Test for Type II Improper Integrals (thm. 2.3, p. 85)

$p < 1$ $$\iff$$ $\int_0^1 \frac{1}{x^p} \dd{x} = \frac{1}{1-p}$

Solving First-Order Linear Differential Equations (thm 4.1, p. 135)

$$f,g\in C^0$$, $$y' = f(x)y + g(y)$$

$$\implies$$

$$y = \frac{1}{I(x)}\int g(x)I(x) \dd{x}$$ where $$I(x) = e^{-\int f(x) \dd{x}}$$

Geometric Series Test (thm. 5.1, p. 169)

Geometric series $$\sum_{n=0}^\infty r^n$$ converges

$$\impliedby$$ $\abs{r} < 1$

Divergence Test (thm 5.2, p. 169)

Series $$\isum a_n$$ converges

$$\implies$$ $\ilim{n} a_n = 0$

Arithmetic for Series I (thm. 5.3, p. 173)

$$\isum a_n$$ and $$\isum b_n$$ converge


1. $$\isum ca_n = c\isum a_n$$ converges
2. $$\isum (a_n + b_n) = \isum a_n + \isum b_n$$ converges

Arithmetic for Series II (thm. 5.4, p. 174)

1. $$\isum a_n$$ converges $$\implies$$ $$\isum{j=1} a_n$$ converges for all $$j$$
2. $$\isum{j=1} a_n$$ converges for some $$j$$ $$\implies$$ $$\isum a_n$$ converges

Monotonic Convergence Theorem (thm. 5.5, p. 177)

Sequence $$\{a_n\}$$ non-decreasing and converges

$$\iff$$

$$\{a_n\}$$ is bounded above

Comparison Test for Series (thm. 5.6, p. 180)

$0 \leq a_n \leq b_n$ $$\implies$$
• $$\isum b_n$$ converges $$\implies$$ $$\isum a_n$$ converges
• $$\isum a_n$$ diverges $$\implies$$ $$\isum b_n$$ diverges

Limit Comparison Test (thm 5.7, p. 185)

$$a_n > 0$$, $$b_n > 0$$, $$\ilim{n} \frac{a_n}{b_n} = L$$

$$\implies$$
• $$0 < L < \infty$$: $$\isum a_n$$ converges $$\iff$$ $$\isum b_n$$ converges
• $$L = 0$$: $$\isum b_n$$ converges $$\implies$$ $$\isum a_n$$ converges (and contrapositive)
• $$L = \infty$$: $$\isum a_n$$ converges $$\implies$$ $$\isum b_n$$ converges (and contrapositive)

Integral Test for Convergence (thm. 5.8, p. 194)

$$S_n = \sum{k=1}^n a_k$$, and $$f$$ exists so

1. $f \in \C^0[1,\infty)$
2. $$f(x) > 0$$ on $$[1, \infty)$$
3. $$f$$ decreasing on $$[1, \infty)$$
4. $a_k = f(k)$
$$\implies$$
1. $\int_1^{n+1} f(x) \dd{x} \leq S_n \leq a_1 + \int_1^n f(x) \dd{x}$
2. $$\isum a_n$$ converges $$\iff$$ $$\int_1^\infty f(x) \dd{x}$$ converges
3. $$\int_1^\infty f(x) \dd{x} \leq \isum a_n \leq a_1 + \int_1^\infty f(x) \dd{x}$$ and $$\int_{n+1}^\infty f(x) \dd{x} \leq \isum a_n - S_n \leq \int_n^\infty f(x) \dd{x}$$

$$p$$-Series Test (thm. 5.9, p. 196)

$$\isum \frac{1}{n^p}$$ converges

$$\iff$$ $p > 1$

Alternate Series Test (thm 5.10, p. 208)

1. $$a_n > 0$$ for all $$n$$
2. $$a_{n+1} \leq a_n$$ for all $$n$$
3. $\ilim{n} a_n = 0$
$$\implies$$

$$\isum (-1)^{n-1}a_n$$ converges with $$\abs{S - S_k} \leq a_{k+1}$$

Absolute Convergence Theorem (thm 5.11, p. 214)

$$\isum \abs{a_n}$$ converges

$$\implies$$

$$\isum a_n$$ converges

Rearrangement Theorem (thm. 5.12, p. 218)

• $$\sum a_n$$ absolutely convergent
• $$\sum c_n$$ conditionally convergent
$$\implies$$
• $$\sum b_n$$ any rearrangement of $$a_n$$ convergent
• for any $$\alpha \in \R$$ or $$\alpha = \pm \infty$$, exists $$\sum d_n = \alpha$$ rearrangement of $$c_n$$

Ratio Test (thm. 5.13, p. 221)

$$\isum a_n$$ with $$\ilim{n}\abs{\frac{a_{n+1}}{a_n}}=L$$

$$\implies$$
• $$0 \leq L < 1$$, converges absolutely
• $$L > 1$$, diverges
• $$L = 1$$, indeterminate

Polynomials vs. Factorials (thm. 5.14, p. 224)

$x \in \R$ $$\implies$$ $\ilim{n} \frac{x^n}{n!} = 0$

Root Test (thm. 5.15, p. 228)

$$\isum a_n$$ with $$\ilim{n} \sqrt[n]{\abs{a_n}} = L$$

$$\implies$$
• $$0 \leq L < 1$$, converges absolutely
• $$L > 1$$, diverges
• $$L = 1$$, indeterminate

Fundamental Convergence Theorem for Power Series (thm. 6.1, p. 234)

$$\isum[n=0] a_n(x-a)^n$$ with radius of convergence $$R$$

$$\implies$$
• $$R = 0$$, converges for $$x = a$$ and diverges otherwise
• $$0 < R < \infty$$, converges absolutely if $$\abs{x-a} < R$$ and diverges if $$\abs{x-a}>R$$
• $$R = \infty$$, converges absolutely everywhere

Test for Radius of Convergence (thm. 6.2, p. 237)

$$\isum[n=0] a_n(x-a)^n$$ with $$\ilim{n} \abs{\frac{a_{n+1}}{a_n}} = L$$

$$\implies$$
• $$0 < L < \infty$$, then $$R = \frac1L$$
• $$L = 0$$, then $$R = \infty$$
• $$L = \infty$$, then $$R = 0$$

Equivalence of Radius of Convergence (thm. 6.3, p. 239)

$$\isum[n=0] a_n(x-a)^n$$ with $$R$$, $$p,q\in\R[x]$$

$$\implies$$

$$\isum[n=0] \frac{p(x)a_n(x-a)^n}{q(x)}$$ has same $$R$$

Abel’s Theorem (thm. 6.4, p. 241)

$$f(x) = \isum[n=0] a_n(x-a)^n$$ with $$I$$

$$\implies$$ $f \in C^0(I)$

Addition of Power Series (thm. 6.5, p. 242)

$$f(x) = \isum[n=0] a_n(x-a)^n$$, $$g(x) = \isum[n=0] b_n(x-a)^n$$

$$\implies$$

$$(f+g)(x) = \isum[n=0] (a_n + b_n)(x-a)^n$$ has $$R = \min\{R_f,R_g\}$$ and $$I=I_f\cap I_g$$

Multiplication of Power Series (thm. 6.6, p. 242)

$f(x) = \isum[n=0] a_n(x-a)^n$ $$\implies$$

$$(x-a)^m f(x) = \isum[n=0] a_n(x-a)^{m+n}$$ with same $$R$$ and $$I$$

Power Series of Composite Functions (thm. 6.7, p. 243)

$f(x) = \isum[n=0] a_n(x-a)^n$ $$\implies$$

$$f(c\cdot x^m) = \isum[n=0] (a_n\cdot c^n)x^{mn}$$ with $$I = \{ x\in\R : c\cdot x^m \in I_f \}$$ and $$R = \sqrt[m]{\frac{R_f}{\abs{c}}}$$

Term-by-Term Differentiation of Power Series (thm. 6.8, p. 246)

$$f(x) = \isum[n=0] a_n(x-a)^n$$ with $$R > 0$$

$$\implies$$

$$f'(x) = \isum[n=1] na_n(x-a)^{n-1}$$ on $$x \in (a-R,a+R)$$

Uniqueness of Power Series (thm. 6.9, p. 252)

$$f(x) = \isum[n=0] a_n(x-a)^n$$ with $$R > 0$$

$$\implies$$ $a_n = \frac{f^{(n)}(a)}{n!}$

Term-by-Term Integration of Power Series (thm. 6.10, p. 254)

$$f(x) = \isum[n=0] a_n(x-a)^n$$ with $$R > 0$$ and $$[c,b] \subset (a-R,a+R)$$

$$\implies$$ $\int_c^b f(x) \dd{x} = \isum[n=0] \frac{a_n}{n+1}((b-a)^{n+1} - (c-a)^{n+1})$

Taylor’s Theorem (thm. 6.11, p. 270)

$$f^{(n+1)}$$ exists on $$a \in I$$

$$\implies$$

$$f(x) - T_{n,a}(x) = R_{n,a}(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$$ for a $$c \in I$$

Taylor’s Approximation Theorem I (thm. 6.12, p. 274)

$f^{(k+1)} \in C^0[-1,1]$ $$\implies$$

Exists $$M$$ where $$\abs{f(x) - T_{k,0}} \leq M\abs{x}^{k+1}$$ on $$x \in [-1,1]$$

Convergence Theorem for Taylor Series (thm. 6.13, p. 283)

$$f \in C^\infty(I)$$ where $$f^{(k)} \leq M \in \R$$ on $$I$$

$$\implies$$

$$f(x) = \isum[n=0] \frac{f^{(n)}(a)}{n!}(x-a)^n$$ along $$I$$

Generalized Binomial Theorem (thm. 6.15, p. 288)

$$a \in \R$$ and $$x \in (-1,1)$$

 $(1+x)^\alpha = 1 + \isum[n=1]\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-n+1)}{n!}x^n$

## Series

Name (Reference) Statement

Geometric Series

$\isum[n=0] x^n = \frac{1}{1-x}$ $R = 1$

Exponential Series

$\isum[n=0] \frac{x^n}{n!} = e^x$ $R = \infty$

Trigonometric Series

$$\isum[n=0] (-1)^n\frac{x^{2n}}{(2n)!} = \cos(x)$$
$$\isum[n=0] (-1)^n\frac{x^{2n+1}}{(2n+1)!} = \sin(x)$$

$R = \infty$

Alternating Series

$\isum \frac{(-1)^{1-n}}{n}x^{n} = \ln(1+x)$ $R = 1$

Binomial Series

$\isum \binom{r}{n}x^n = 1 + \isum \frac{r(r-1)\cdots(r-n+1)}{n!}x^n = (1+x)^r$ $R = 1$

Arctangent (based on Geometric)

$\isum[n=0] \frac{(-1)^n}{(2n+1)}x^{2n+1} = \arctan(x)$ $R = 1$

# MATH 137 Reference

Because I know I’m gonna need this.

## Definitions

Name (Reference) Statement

Local Extrema (§3.13, p. 197)

$$f(c)$$ exists

$$I=(a,b)$$ exists so $$f(c) = \min f(I)$$ (or max)

Antiderivative (§4.2.1, p. 213)

$$f(I)$$ exists

Function $$F$$ where $$F'(x) = f(x)$$ for all $$x\in I$$.

Increasing/Decreasing (§4.2.2, p. 219)

$$f(I)$$ exists

For all $$x_1,x_2\in I$$ with $$x_1 < x_2$$:

1. Increasing on $$I$$ if $$f(x_1) < f(x_2)$$
2. Decreasing on $$I$$ if $$f(x_1) > f(x_2)$$
3. Non-increasing on $$I$$ if $$f(x_1) \leq f(x_2)$$
4. Non-decreasing on $$I$$ if $$f(x_1) \geq f(x_2)$$
5. Monotone if non-increasing or non-decreasing

Concavity

$$f(I)$$ exists

For all pairs of points $$a$$ and $$b$$ in $$I$$, the secant line from $$(a, f(a))$$ to $$(b, f(b))$$ is above (up) or below (down) the graph of $$f$$.

Inflection Point

$$f(c)$$ exists

1. $f \in C^0(c)$
2. Concavity changes at $$c$$

## Theorems

Name (Reference) Statement

Fundamental Trigonometric Limit (thm. 2.8, p. 84)

$\lim_{x\to\infty} \frac{\sin x}{x} = 0$

Fundamental Log Limit (thm. 2.10, p. 95)

$\lim_{x\to\infty} \frac{\ln x}{x} = 0$

Intermediate Value Theorem (thm. 2.16 [IVT], p. 116)

$$f \in C^0([a,b])$$, $$\alpha \in f((a,b))$$

$$\implies$$

$$\exists c\in(a,b)$$, $$f(c) = \alpha$$

Local Extrema Theorem (thm. 3.11, p. 201)

• $$c$$ local extrema for $$f$$
• $$f'(c)$$ exists
$$\implies$$ $f'(c) = 0$

Mean Value Theorem (thm. 4.1 [MVT], p. 210)

$$f \in C^0([a,b])$$, $$f'([a,b])$$

$$\implies$$

$$c\in(a,b)$$ where $$f'(c) = \frac{f(b)-f(a)}{b-a}$$

Increasing/Decreasing Function Theorem (thm. 4.6, p. 220)

$$f'(I)$$ exists

$$\implies$$
1. $$f'$$ positive on $$I$$ $$\implies$$ $$f$$ increasing on $$I$$
2. $$f'$$ negative on $$I$$ $$\implies$$ $$f$$ decreasing on $$I$$
3. $$f'$$ non-positive on $$I$$ $$\implies$$ $$f$$ non-increasing on $$I$$
4. $$f'$$ non-negative on $$I$$ $$\implies$$ $$f$$ non-decreasing on $$I$$

Bounded Derivative Theorem (thm. 4.7, p. 222)

$$f\in C^0([a,b])$$, $$f'((a,b)) \subseteq [m,M]$$

$$\implies$$

$$f(a) + m(x-a) \leq f(x) \leq f(a) + M(x-a)$$ for $$x \in [a,b]$$