BC Calculus Memorization Sheet Notes

Derivatives

• Product Rule: $\frac{d}{dx}(uv) = u'v + uv'$

• Quotient Rule: $\frac{d}{dx}(\frac{t}{b}) = \frac{t'b - tb'}{b^2}$

• Chain Rule: $\frac{d}{dx}f(u) = f'(u) \cdot u'$ or $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

• Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$

• $\frac{d}{dx} ln(x) = \frac{1}{x}$, $\frac{d}{dx} ln(u) = \frac{u'}{u}$

• $\frac{d}{dx} log_a(x) = \frac{1}{x ln(a)}$

• $\frac{d}{dx} e^x = e^x$, $\frac{d}{dx} a^x = a^x ln(a)$

• $\frac{d}{dx} sin(x) = cos(x)$

• $\frac{d}{dx} cos(x) = -sin(x)$

• $\frac{d}{dx} tan(x) = sec^2(x)$

• $\frac{d}{dx} cot(x) = -csc^2(x)$

• $\frac{d}{dx} sec(x) = sec(x)tan(x)$

• $\frac{d}{dx} csc(x) = -csc(x)cot(x)$

• $\frac{d}{dx} arcsin(x) = \frac{1}{\sqrt{1-x^2}}$

• $\frac{d}{dx} arctan(x) = \frac{1}{1+x^2}$

• Definition of Derivative: $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$

• Alternative form: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Another alternative form: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Integrals

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$

• $\int \frac{1}{x} dx = ln|x| + C$

• $\int \frac{u'}{u} dx = ln|u| + C$

• $\int e^x dx = e^x + C$

• $\int a^x dx = \frac{a^x}{ln(a)} + C$

• $\int cos(x) dx = sin(x) + C$

• $\int sin(x) dx = -cos(x) + C$

• $\int sec^2(x) dx = tan(x) + C$

• $\int tan(x) dx = -ln|cos(x)| + C$

• $\int cot(x) dx = ln|sin(x)| + C$

• $\int sec(x)tan(x) dx = sec(x) + C$

• $\int csc(x)cot(x) dx = -csc(x) + C$

• $\int csc^2(x) dx = -cot(x) + C$

• $\int \frac{u'}{a^2 + u^2} du = \frac{1}{a}arctan(\frac{u}{a}) + C$

• $\int \frac{u'}{\sqrt{a^2 - u^2}} du = arcsin(\frac{u}{a}) + C$

• Integration by Parts: $\int u dv = uv - \int v du$

Trigonometric Identities

• $tan(x) = \frac{sin(x)}{cos(x)}$

• $sin^2(x) + cos^2(x) = 1$

• $tan^2(x) = sec^2(x) - 1$

Elementary Series

• $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + … + \frac{x^n}{n!} + …$

• $sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - … + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + …$

• $cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - … + (-1)^n \frac{x^{2n}}{(2n)!} + …$

Taylor and Maclaurin Series

• Taylor Series:
  $f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)(x-c)^2}{2!} + \frac{f'''(c)(x-c)^3}{3!} + … + \frac{f^n(c)(x-c)^n}{n!} + …$

• Maclaurin Series: Taylor series with $c = 0$

Logistic Growth

• $\frac{dP}{dt} = kP(M - P)$, where $M$ is the carrying capacity.

• Fastest growth occurs when $P = \frac{M}{2}$.

Euler's Method

• $\frac{dy}{dx} \approx \frac{\Delta y}{\Delta x}$

Second Fundamental Theorem of Calculus

• $\frac{d}{dx} \int_{u}^{v} f(t) dt = f(v) \cdot v' - f(u) \cdot u'$

Area Approximation

• Area of Trapezoid: $A = w(\frac{h<em>1 + h</em>2}{2})$

• Riemann Sum: Sum of areas of rectangles.

Series Error Bounds

• Alternating Series Error: $|error| \le a_{n+1}$ (the next term)

• Lagrange Error Bound:
  $|error| \le \frac{f^{n+1}(z)(x-c)^{n+1}}{(n+1)!}$, where $f^{n+1}(z)$ is the maximum value of the (n+1)-th derivative between $x$ and $c$

Volume

• Disc Method: $V = \pi \int_{a}^{b} r^2 dx$

• Washer Method: $V = \pi \int_{a}^{b} (R^2 - r^2) dx$

• Cross-Sectional Area: $V = \int_{a}^{b} A(x) dx$

Start plus Accumulation

• $f(b) = f(a) + \int_{a}^{b} f'(x) dx$

Fundamental Theorem of Calculus

• $\int_{a}^{b} f'(x) dx = f(b) - f(a)$

Tests for Convergence/Divergence

• nth Term Test: If $\lim<em>{n \to \infty} a</em>n \neq 0$, then the series diverges (but cannot be used to show convergence).

• Geometric Series Test: $\sum_{n=1}^{\infty} ar^{n-1}$ converges if |r| < 1 and diverges if $|r| \ge 1$. If convergent, the sum $S = \frac{a}{1-r}$.

• p-Series Test: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if p > 1 and diverges if $p \le 1$.

• Alternating Series Test: For an alternating series with decreasing terms where $\lim<em>{n \to \infty} a</em>n = 0$, the series converges.

• Integral Test: If $f(x)$ is positive, continuous, and decreasing, then $\sum<em>{n=1}^{\infty} a</em>n$ converges if $\int_{1}^{\infty} f(x) dx$ converges, and diverges if the integral diverges.

• Ratio Test:

  • If \lim{n \to \infty} |\frac{a{n+1}}{a_n}| < 1, the series converges.

  • If \lim{n \to \infty} |\frac{a{n+1}}{a_n}| > 1, the series diverges.

  • If $\lim<em>{n \to \infty} |\frac{a</em>{n+1}}{a_n}| = 1$, the test is inconclusive (works well for factorials and exponentials).

• Direct Comparison Test:

  • If $0 \le a<em>n \le b</em>n$ and $\sum b<em>n$ converges, then $\sum a</em>n$ converges.

  • If $a<em>n \ge b</em>n \ge 0$ and $\sum b<em>n$ diverges, then $\sum a</em>n$ diverges.

• Limit Comparison Test: If $\lim<em>{n \to \infty} \frac{a</em>n}{b_n}$ is finite and positive, then both series converge or both diverge (use with "messy" algebraic series, usually compared to a p-series).

Rates of Change

• Average Rate of Change (AROC): $\frac{f(b) - f(a)}{b - a}$ (slope between two points)

• Instantaneous Rate of Change (IROC): $f'(c)$ (slope at a single point)

Mean Value Theorem

• $\frac{f(b) - f(a)}{b - a} = f'(c)$; find $c$ where $m<em>{sec} = m</em>{tan}$.

• Must be continuous on $[a, b]$ and differentiable on $(a, b)$.

Average Value of a Function

• $f<em>{avg} = \frac{1}{b-a} \int</em>{a}^{b} f(x) dx = \frac{area}{width}$

Intermediate Value Theorem

• If a function $f$ is continuous on $[a, b]$, it takes on every y-value between $f(a)$ and $f(b)$.

Extreme Value Theorem

• If a function $f$ is continuous on $[a, b]$, it has both an absolute minimum and an absolute maximum on the interval.

Definition of Continuity

• A function $f$ is continuous at $x = c$ if and only if $\lim<em>{x \to c^-} f(x) = \lim</em>{x \to c^+} f(x) = f(c)$.

Squeeze Theorem

• If $f(x) \le g(x) \le h(x)$ for all $x \neq c$ in some interval containing $c$, and if $\lim<em>{x \to c} f(x) = \lim</em>{x \to c} h(x) = L$, then $\lim_{x \to c} g(x) = L$.

Arc Length

• $L = \int<em>{a}^{b} \sqrt{1 + (f'(x))^2} dx = \int</em>{t<em>1}^{t</em>2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

Speed

• $v(t) = \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$

Total Distance

• $\int<em>{t</em>1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

Polar Area

• $A = \frac{1}{2} \int<em>{\theta</em>1}^{\theta_2} r^2 d\theta$

Parametric Derivatives

• $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, $\frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt}$

Polar Conversions

• $r^2 = x^2 + y^2$, $x = r cos(\theta)$, $y = r sin(\theta)$, $\theta = arctan(\frac{y}{x})$

• For parametric/vectors, speed is increasing when velocity and acceleration have the same sign.