$\F^+$/$\F^+_0$ 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	\[\int_a^\infty f(x) \dd{x} = \lim\limits_{b\to\infty} \int_a^b f(x) \dd{x}\] \[\int_{-\infty}^a f(x) \dd{x} = \lim\limits_{b\to-\infty} \int_b^a f(x) \dd{x}\] \[\int_{-\infty}^\infty f(x) \dd{x} = \int_{-\infty}^c f(x) \dd{x} + \int_c^\infty f(x) \dd{x}\]
Type II Improper Integral (§2.4, p. 97)	$f$ integrable except at asymptote $c\in(a,b)$	\[\int_c^b f(x) \dd{x} = \lim\limits_{t\to c^+} \int_t^b f(x) \dd{x}\] \[\int_a^c f(x) \dd{x} = \lim\limits_{t\to c^-} \int_a^t f(x) \dd{x}\] \[\int_a^b f(x) \dd{x} = \int_a^c f(x) \dd{x} + \int_c^b f(x) \dd{x}\]
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 $	$\forall c\in\R$, $\int_a^b c f(t) \dd{t} = c\int_a^b f(t) \dd{t}$ \[\int_a^b(f+g)(t)\dd{t} = \int_a^b f(t) \dd{t} + \int_a^b g(t) \dd{t}\] $f([a,b])\subseteq[m,M]$ then $m(b-a) \leq \int_a^b f(t) \dd{t} \leq M(b-a)$ $f([a,b])$ positive then $\int_a^b f(t) \dd{t}$ positive $g \leq f$ on $[a,b]$ then $\int_a^b g(t) \dd{t} \leq \int_a^b f(t) \dd{t}$ \[\abs{\int_a^b f(t) \dd{t}} \leq \int_a^b\abs{f(t)}\dd{t}\]
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 $	If $\int_a^\infty g(x)$ converges, so does $\int_a^\infty f(x)$ 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		$\isum ca_n = c\isum a_n$ converges $\isum (a_n + b_n) = \isum a_n + \isum b_n$ converges
Arithmetic for Series II (thm. 5.4, p. 174)	$\isum a_n$ converges $\implies$ $\isum{j=1} a_n$ converges for all $j$ $\isum{j=1} a_n$ 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 \[f \in \C^0[1,\infty)\] $f(x) > 0$ on $[1, \infty)$ $f$ decreasing on $[1, \infty)$ \[a_k = f(k)\]	$ \implies $	\[\int_1^{n+1} f(x) \dd{x} \leq S_n \leq a_1 + \int_1^n f(x) \dd{x}\] $\isum a_n$ converges $\iff$ $\int_1^\infty f(x) \dd{x}$ converges $\int_1^\infty f(x) \dd{x} \leq \isum a_n \leq a_1 + \int_1^\infty f(x) \dd{x}$ 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)	$a_n > 0$ for all $n$ $a_{n+1} \leq a_n$ for all $n$ \[\ilim{n} a_n = 0\]	$ \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 $x_1 < x_2$: Increasing on $I$ if $f(x_1) < f(x_2)$ Decreasing on $I$ if $f(x_1) > f(x_2)$ Non-increasing on $I$ if $f(x_1) \leq f(x_2)$ Non-decreasing on $I$ if $f(x_1) \geq f(x_2)$ 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	\[f \in C^0(c)\] 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 $	$f'$ positive on $I$ $\implies$ $f$ increasing on $I$ $f'$ negative on $I$ $\implies$ $f$ decreasing on $I$ $f'$ non-positive on $I$ $\implies$ $f$ non-increasing on $I$ $f'$ 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 $x \in [a,b]$

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)	\[\int_a^b f(t) \dd{t}\] \[\int_a^b g(t) \dd{t}\]	\( \implies \)	\(\forall c\in\R\), \(\int_a^b c f(t) \dd{t} = c\int_a^b f(t) \dd{t}\) \[\int_a^b(f+g)(t)\dd{t} = \int_a^b f(t) \dd{t} + \int_a^b g(t) \dd{t}\] \(f([a,b])\subseteq[m,M]\) then \(m(b-a) \leq \int_a^b f(t) \dd{t} \leq M(b-a)\) \(f([a,b])\) positive then \(\int_a^b f(t) \dd{t}\) positive \(g \leq f\) on \([a,b]\) then \(\int_a^b g(t) \dd{t} \leq \int_a^b f(t) \dd{t}\) \[\abs{\int_a^b f(t) \dd{t}} \leq \int_a^b\abs{f(t)}\dd{t}\]
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 \(x \geq a\)	\( \implies \)	If \(\int_a^\infty g(x)\) converges, so does \(\int_a^\infty f(x)\) 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	\( \)	\(\isum ca_n = c\isum a_n\) converges \(\isum (a_n + b_n) = \isum a_n + \isum b_n\) converges
Arithmetic for Series II (thm. 5.4, p. 174)	\(\isum a_n\) converges \(\implies\) \(\isum{j=1} a_n\) converges for all \(j\) \(\isum{j=1} a_n\) 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 \[f \in \C^0[1,\infty)\] \(f(x) > 0\) on \([1, \infty)\) \(f\) decreasing on \([1, \infty)\) \[a_k = f(k)\]	\( \implies \)	\[\int_1^{n+1} f(x) \dd{x} \leq S_n \leq a_1 + \int_1^n f(x) \dd{x}\] \(\isum a_n\) converges \(\iff\) \(\int_1^\infty f(x) \dd{x}\) converges \(\int_1^\infty f(x) \dd{x} \leq \isum a_n \leq a_1 + \int_1^\infty f(x) \dd{x}\) 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)	\(a_n > 0\) for all \(n\) \(a_{n+1} \leq a_n\) for all \(n\) \[\ilim{n} a_n = 0\]	\( \implies \)	\(\isum (-1)^{n-1}a_n\) 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 \(x \in (-1,1)\)	\( \)	\[(1+x)^\alpha = 1 + \isum[n=1]\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-n+1)}{n!}x^n\]

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 \(x_1 < x_2\): Increasing on \(I\) if \(f(x_1) < f(x_2)\) Decreasing on \(I\) if \(f(x_1) > f(x_2)\) Non-increasing on \(I\) if \(f(x_1) \leq f(x_2)\) Non-decreasing on \(I\) if \(f(x_1) \geq f(x_2)\) 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	\[f \in C^0(c)\] Concavity changes at \(c\)

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	\( \implies \)	\(f'\) positive on \(I\) \(\implies\) \(f\) increasing on \(I\) \(f'\) negative on \(I\) \(\implies\) \(f\) decreasing on \(I\) \(f'\) non-positive on \(I\) \(\implies\) \(f\) non-increasing on \(I\) \(f'\) 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 \(x \in [a,b]\)

MATH 138 Reference Sheet

Collected from the Winter 2021 course notes

Definitions

Theorems

Series

MATH 137 Reference

Definitions

Theorems

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	\[\int_a^\infty f(x) \dd{x} = \lim\limits_{b\to\infty} \int_a^b f(x) \dd{x}\] \[\int_{-\infty}^a f(x) \dd{x} = \lim\limits_{b\to-\infty} \int_b^a f(x) \dd{x}\] \[\int_{-\infty}^\infty f(x) \dd{x} = \int_{-\infty}^c f(x) \dd{x} + \int_c^\infty f(x) \dd{x}\]
Type II Improper Integral (§2.4, p. 97)	\(f\) integrable except at asymptote \(c\in(a,b)\)	\[\int_c^b f(x) \dd{x} = \lim\limits_{t\to c^+} \int_t^b f(x) \dd{x}\] \[\int_a^c f(x) \dd{x} = \lim\limits_{t\to c^-} \int_a^t f(x) \dd{x}\] \[\int_a^b f(x) \dd{x} = \int_a^c f(x) \dd{x} + \int_c^b f(x) \dd{x}\]
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)	\[\isum[n=0] a_n(x-a)^n\]	\[I = \{ x_0 : \isum[n=0] a_n(x_0-a)^n \text{ converges } \}\] \[R = \begin{cases} \operatorname{lub}(\{\abs{x_0 - a} : x_0 \in I\}) & \text{$I$ bounded} \\ \infty & \text{$I$ unbounded} \end{cases}\]