\(\F^+\)/\(\F^+_0\) 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) |
|
\[S = \sum\limits_{i=1}^n f(c_i) \Delta t_i\] |
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) |
|
\[I = \displaystyle\int_a^b f(t) \dd{t}\] |
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 |
|
Type II Improper Integral (§2.4, p. 97) |
\(f\) integrable except at asymptote \(c\in(a,b)\) |
|
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) |
|
\( \implies \) | \[\displaystyle\int_a^b f(t) \dd{t} = \lim_{n\to\infty}S_n\] |
Properties of Integrals (thm. 1.2, p. 18) |
|
\( \implies \) |
|
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) |
|
\( \implies \) |
|
\(p\)-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 \(x \geq a\) |
\( \implies \) |
|
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 \(x_1 < x_2\):
|
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 |
|
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) |
|
\( \implies \) | \[f'(c) = 0\] |
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 |
\( \implies \) |
|
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 \(x \in [a,b]\) |