\(\F^+\)/\(\F^+_0\) 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)

\[y' = f(x) y + g(x)\]

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 \(I(x) = e^{-\int f(x) \dd{x}}\)

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 \(x \in [a,b]\)