公式表

極限與連續

$f, g$ = functions · $c, a$ = numbers

• $\displaystyle \lim_{x\to0}\frac{\sin x}{x} = 1 \qquad \lim_{x\to0}\frac{1-\cos x}{x} = 0$
  Two trigonometric limits1.8 The squeeze theorem
• $f \text{ is continuous at } c \iff \lim_{x\to c}f(x) = f(c)$
  Continuity at a point1.11 Continuity at a point1.12 Continuity over an interval
• $f(a) < N < f(b) \implies f(c) = N \text{ for some } c \in (a, b)$
  Intermediate Value Theorem (f continuous on [a, b])1.16 The Intermediate Value Theorem
• $\displaystyle \lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}$
  L’Hôpital’s rule (the form 0/0 or ∞/∞)4.12 L’Hôpital’s rule4.13 L’Hôpital at infinity

導數

$f, g, u$ = functions of x · $n, a, k$ = constants

• $\displaystyle f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
  The derivative2.2 The derivative2.1 Rates of change
• $\displaystyle f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}$
  The derivative at a point2.2 The derivative2.3 Estimating derivatives
• $\displaystyle \frac{d}{dx}x^n = nx^{n-1}$
  Power rule2.5 The power rule
• $\displaystyle \frac{d}{dx}\big(kf + g\big) = kf' + g'$
  Sums and constant multiples2.6 Sums and constant multiples
• $\displaystyle \frac{d}{dx}(fg) = f'g + fg'$
  Product rule2.8 The product rule
• $\displaystyle \frac{d}{dx}\frac{f}{g} = \frac{f'g - fg'}{g^2}$
  Quotient rule2.9 The quotient rule
• $\displaystyle \frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$
  Chain rule3.1 The chain rule3.2 The chain rule on a graph
• $\displaystyle \frac{d}{dx}\sin x = \cos x \qquad \frac{d}{dx}\cos x = -\sin x$
  Sine and cosine2.7 Sine, cosine, eˣ and ln x
• $\displaystyle \frac{d}{dx}e^x = e^x \qquad \frac{d}{dx}\ln x = \frac{1}{x}$
  eˣ and ln x2.7 Sine, cosine, eˣ and ln x
• $\displaystyle \frac{d}{dx}a^x = a^x\ln a \qquad \frac{d}{dx}\log_a x = \frac{1}{x\ln a}$
  Other basesnot shown yet
• $\displaystyle \frac{d}{dx}\tan x = \sec^2x \qquad \frac{d}{dx}\cot x = -\csc^2x$
  Tangent and cotangent2.10 Tangent, cotangent, secant, cosecant
• $\displaystyle \frac{d}{dx}\sec x = \sec x\tan x \qquad \frac{d}{dx}\csc x = -\csc x\cot x$
  Secant and cosecant2.10 Tangent, cotangent, secant, cosecant
• $\displaystyle \big(f^{-1}\big)'(a) = \frac{1}{f'\big(f^{-1}(a)\big)}$
  Derivative of an inverse function3.5 Derivatives of inverse functions
• $\displaystyle \frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}} \qquad \frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}$
  Inverse sine and cosine3.6 Inverse trigonometric functions
• $\displaystyle \frac{d}{dx}\arctan x = \frac{1}{1+x^2}$
  Inverse tangent3.6 Inverse trigonometric functions
• $L(x) = f(a) + f'(a)(x-a)$
  Tangent line and local linearization4.10 Local linearity4.11 Over or under?

Motion and theorems

$x$ = position · $v$ = velocity · $a$ = acceleration · $t$ = time

• $v(t) = x'(t) \qquad a(t) = v'(t) = x''(t)$
  Velocity and acceleration4.2 Position, velocity, acceleration
• $\text{speed} = |v(t)|$
  Speed; speeding up when v and a share a sign4.3 Speeding up or slowing down
• $\displaystyle f'(c) = \frac{f(b)-f(a)}{b-a} \text{ for some } c \in (a, b)$
  Mean Value Theorem (f continuous on [a, b], differentiable on (a, b))5.1 The Mean Value Theorem
• $f \text{ continuous on } [a, b] \implies \text{a maximum and a minimum exist}$
  Extreme Value Theorem5.2 Extreme values5.5 The candidates test
• $f'(c) = 0 \text{ or undefined}$
  Critical point5.2 Extreme values5.4 The first derivative test
• $f'' > 0 \text{: concave up} \qquad f'' < 0 \text{: concave down}$
  Concavity; inflection where f″ changes sign5.6 Concavity5.7 The second derivative test

Integrals

$F$ = an antiderivative of f · $C$ = constant of integration · $\Delta x$ = width of a subinterval

• $\sum_{i=1}^{n}f(x_i)\,\Delta x \qquad \tfrac{\Delta x}{2}\big(f(x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) + f(x_n)\big)$
  Riemann sum; trapezoidal sum6.2 Riemann sums6.3 The definite integral
• $\displaystyle \frac{d}{dx}\int_a^x f(t)\,dt = f(x)$
  Fundamental Theorem, part one6.4 Accumulation functions6.5 Reading an accumulation function
• $\displaystyle \frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\,g'(x)$
  Fundamental Theorem with a chain6.4 Accumulation functions
• $\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$
  Fundamental Theorem, part two6.7 The Fundamental Theorem, part two
• $\displaystyle \int_a^b f = -\int_b^a f \qquad \int_a^b f + \int_b^c f = \int_a^c f$
  Properties of definite integrals6.6 Properties of definite integrals
• $\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\quad n \ne -1 \qquad \int\frac{1}{x}\,dx = \ln|x| + C$
  Powers6.8 Antiderivatives
• $\displaystyle \int e^x\,dx = e^x + C \qquad \int a^x\,dx = \frac{a^x}{\ln a} + C$
  Exponentials6.8 Antiderivatives
• $\displaystyle \int\sin x\,dx = -\cos x + C \qquad \int\cos x\,dx = \sin x + C$
  Sine and cosine6.8 Antiderivatives
• $\displaystyle \int\sec^2x\,dx = \tan x + C \qquad \int\sec x\tan x\,dx = \sec x + C$
  Secant6.8 Antiderivatives
• $\displaystyle \int\frac{dx}{\sqrt{1-x^2}} = \arcsin x + C \qquad \int\frac{dx}{1+x^2} = \arctan x + C$
  Inverse trigonometric6.8 Antiderivatives6.10 Division and completing the square
• $\displaystyle \int f(g(x))\,g'(x)\,dx = \int f(u)\,du,\quad u = g(x)$
  Substitution6.9 Substitution

Applications of integrals and differential equations

$R, r$ = outer and inner radius · $A(x)$ = area of a cross section · $k$ = constant · $y_0$ = initial value

• $\displaystyle f_\text{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx$
  Average value8.1 Average value
• $\displaystyle x(b) = x(a) + \int_a^b v(t)\,dt \qquad \text{distance} = \int_a^b |v(t)|\,dt$
  Position, displacement and distance8.2 Motion along a line8.3 Accumulation in context
• $\displaystyle A = \int_a^b \big(f(x) - g(x)\big)\,dx$
  Area between curves (f above g)8.4 Area between curves8.5 Area in y8.6 More than two crossings
• $\displaystyle V = \int_a^b A(x)\,dx$
  Volume by cross sections8.7 Cross sections: squares and rectangles8.8 Cross sections: triangles and semicircles
• $\displaystyle V = \pi\int_a^b R(x)^2\,dx \qquad V = \pi\int_a^b \big(R(x)^2 - r(x)^2\big)\,dx$
  Disc and washer methods8.9 The disc method8.10 Discs about other axes8.11 The washer method8.12 Washers about other axes
• $\displaystyle \frac{dy}{dx} = f(x)g(y) \implies \int\frac{dy}{g(y)} = \int f(x)\,dx$
  Separation of variables7.6 Separation of variables7.7 Particular solutions
• $\displaystyle \frac{dy}{dt} = ky \implies y = y_0e^{kt}$
  Exponential growth and decay7.8 Exponential models

積分

$u, v$ = functions of x · $p$ = a constant

• $\displaystyle \int u\,dv = uv - \int v\,du$
  Integration by parts6.11 Integration by parts
• $\displaystyle \frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
  Partial fractions (distinct linear factors)6.12 Partial fractions
• $\displaystyle \int_a^\infty f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx$
  Improper integral6.13 Improper integrals
• $\displaystyle L = \int_a^b\sqrt{1 + \big(f'(x)\big)^2}\,dx$
  Arc length of a graph8.13 Arc length

微分方程

$\Delta x$ = step size · $k$ = growth constant · $L$ = carrying capacity

• $\displaystyle y_{n+1} = y_n + \frac{dy}{dx}\Big|_{(x_n,\,y_n)}\Delta x$
  Euler’s method7.5 Euler’s method
• $\displaystyle \frac{dP}{dt} = kP\Big(1 - \frac{P}{L}\Big)$
  Logistic growth: P → L, fastest at P = L/27.9 Logistic models

參數式、極座標與向量函數

$t$ = parameter · $r$ = polar radius · $\theta$ = polar angle

• $\displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
  Slope of a parametric curve9.2 The slope of a parametric curve
• $\displaystyle \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\big(\frac{dy}{dx}\big)}{dx/dt}$
  Second derivative of a parametric curve9.3 The second derivative
• $\displaystyle L = \int_a^b\sqrt{\Big(\frac{dx}{dt}\Big)^2 + \Big(\frac{dy}{dt}\Big)^2}\,dt$
  Arc length of a parametric curve9.4 Arc length
• $\text{speed} = |\vec{v}(t)| = \sqrt{x'(t)^2 + y'(t)^2}$
  Speed in the plane9.5 Vector-valued functions9.7 Motion in the plane
• $\displaystyle \vec{r}(b) = \vec{r}(a) + \int_a^b\vec{v}(t)\,dt$
  Position from velocity9.6 Integrating vector functions9.7 Motion in the plane
• $x = r\cos\theta \qquad y = r\sin\theta$
  Polar to rectangular9.8 Polar coordinates
• $\displaystyle \frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$
  Slope of a polar curve9.9 Slopes of polar curves
• $\displaystyle A = \frac12\int_\alpha^\beta r^2\,d\theta \qquad A = \frac12\int_\alpha^\beta\big(R^2 - r^2\big)\,d\theta$
  Area in polar; between polar curves9.10 Area in polar9.11 Area between polar curves

Series

$a_n$ = the nth term · $r$ = common ratio · $c$ = centre · $R_n$ = remainder after the nth Taylor polynomial

• $\displaystyle \sum_{n=0}^\infty ar^n = \frac{a}{1-r},\quad |r| < 1$
  Geometric series10.2 Geometric series
• $\lim_{n\to\infty}a_n \ne 0 \implies \sum a_n \text{ diverges}$
  nth term test10.3 The nth term test
• $\displaystyle \sum\frac{1}{n^p} \text{ converges} \iff p > 1$
  p-series (harmonic: p = 1)10.5 Harmonic series and p-series10.4 The integral test
• $\displaystyle \lim_{n\to\infty}\Big|\frac{a_{n+1}}{a_n}\Big| < 1 \implies \sum a_n \text{ converges absolutely}$
  Ratio test10.8 The ratio test10.13 Radius and interval of convergence
• $|S - S_n| \le |a_{n+1}|$
  Alternating series error bound10.10 Alternating series error bound10.7 The alternating series test
• $\displaystyle P_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(c)}{k!}(x-c)^k$
  Taylor polynomial10.11 Taylor polynomials
• $\displaystyle |R_n(x)| \le \frac{\max|f^{(n+1)}|}{(n+1)!}|x-c|^{n+1}$
  Lagrange error bound10.12 Lagrange error bound
• $\displaystyle e^x = \sum_{n=0}^\infty\frac{x^n}{n!} \qquad \frac{1}{1-x} = \sum_{n=0}^\infty x^n,\ |x| < 1$
  Maclaurin series: eˣ and 1/(1 − x)10.14 Taylor and Maclaurin series10.15 Representing functions as power series
• $\displaystyle \sin x = \sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!} \qquad \cos x = \sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!}$
  Maclaurin series: sine and cosine10.14 Taylor and Maclaurin series