AP CALCULUS BC NEED-TO-KNOWS !

Euler's Method

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$, where $h$ is the step size

Logistic Growth Rate

$\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)$, where $M$ is carrying capacity

Logistic max growth rate

occurs at $P = \frac{M}{2}$; rate $= \frac{kM^2}{4}$

Logistic curve shape

S-shaped; concave up when $P < \frac{M}{2}$, concave down when $P > \frac{M}{2}$, inflection at $P = \frac{M}{2}$

Separable Differential Equation

rewrite as $g(y)\,dy = f(x)\,dx$, then integrate both sides independently

Taylor Series (centered at $c$)

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$$

Maclaurin Series

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$

Maclaurin Series for $e^x$

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

Maclaurin Series for $\sin x$

$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

Maclaurin Series for $\cos x$

$$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

Maclaurin Series for $\frac{1}{1-x}$

$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \cdots$, |x| < 1

Sum of infinite geometric series

$S = \frac{a}{1-r}$, valid when |r| < 1

nth Term Test (Divergence Test)

If $\lim_{n\to\infty} a_n \neq 0$, the series diverges. If $= 0$, inconclusive.

Geometric Series Test

$\sum ar^n$ converges to $\frac{a}{1-r}$ when |r|<1; diverges when $|r|\geq 1$

p-Series Test

$\sum \frac{1}{n^p}$ converges if p > 1; diverges if $p \leq 1$

Integral Test

If $f$ is positive, continuous, and decreasing, then $\sum a_n$ and $\int f(x)\,dx$ both converge or both diverge

Direct Comparison Test

If $0 \leq a_n \leq b_n$: $\sum b_n$ converges $\Rightarrow$ $\sum a_n$ converges; $\sum a_n$ diverges $\Rightarrow$ $\sum b_n$ diverges

Limit Comparison Test

If \lim \frac{a_n}{b_n} = L > 0 (finite), then $\sum a_n$ and $\sum b_n$ both converge or both diverge

Alternating Series Test

$\sum(-1)^n b_n$ converges if (1) $b_n$ is decreasing and (2) $\lim_{n\to\infty} b_n = 0$

Alternating Series Estimation Theorem

$$|\text{Error}| \leq b_{n+1}$$

Ratio Test

$\lim\left|\frac{a_{n+1}}{a_n}\right| < 1 \Rightarrow$ converges; $> 1 \Rightarrow$ diverges; $= 1 \Rightarrow$ inconclusive

Radius of Convergence

$R = \frac{1}{\lim|a_{n+1}/a_n|}$; series converges for |x-c| < R

Absolute vs. Conditional Convergence

Absolutely convergent: $\sum|a_n|$ converges. Conditionally convergent: $\sum a_n$ converges but $\sum|a_n|$ diverges.

Lagrange Error Bound

$|f(x)-P_n(x)| \leq \frac{M|x-c|^{n+1}}{(n+1)!}$, where $M \geq |f^{(n+1)}(t)|$ for all $t$ between $x$ and $c$

First Derivative (parametric)

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$

Second Derivative (parametric)

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

Arc Length (parametric)

$$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$$

Position vector

$$\langle x(t), y(t) \rangle$$

Velocity vector

$$\langle x'(t), y'(t) \rangle$$

Acceleration vector

$$\langle x''(t), y''(t) \rangle$$

Speed (parametric)

$$\sqrt{(x'(t))^2 + (y'(t))^2}$$

Polar Conversions

$x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2+y^2$, $\tan\theta = \frac{y}{x}$

Polar Area

$$A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$$

Polar Slope

$$\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$$

Position

$$x(t) = \int v(t)\,dt$$

Velocity

$$v(t) = x'(t) = \int a(t)\,dt$$

Acceleration

$$a(t) = v'(t) = x''(t)$$

Speed

$|v(t)|$, always non-negative

Displacement

$\int_a^b v(t)\,dt$, net change in position (can be negative)

Total Distance

$\int_a^b |v(t)|\,dt$, always non-negative

New Position

$$s(a) + \int_a^b v(t)\,dt$$

Average Velocity

$$\frac{x(t_2)-x(t_1)}{t_2-t_1}$$

Speed increasing

when $a(t)$ and $v(t)$ have the same sign

Speed decreasing

when $a(t)$ and $v(t)$ have opposite signs

Particle at rest

$$v(t) = 0$$

Particle moves right/up

v(t) > 0

Particle moves left/down

v(t) < 0

Integration by Parts

$$\int u\,dv = uv - \int v\,du$$

Average Value of a Function

$$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$$

Reverse Power Rule (Integration)

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

$$\int \cos u\,du$$

$$\sin u + C$$

$$\int \sin u\,du$$

$$-\cos u + C$$

$$\int e^u\,du$$

$$e^u + C$$

$$\int u^{-1}\,du$$

$$\ln|u| + C$$

Disk Method

$$V = \pi\int_a^b [f(x)]^2\,dx$$

Washer Method

$V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx$, where $R$ = outer radius, $r$ = inner radius

General Volume (cross-sections)

$V = \int_a^b A(x)\,dx$, where $A(x)$ is the cross-sectional area

Area Between Two Curves

$A = \int_a^b [f(x) - g(x)]\,dx$, where $f(x) \geq g(x)$

Arc Length (rectangular)

$$L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$$

Mean Value Theorem (MVT)

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists\, c \in (a,b)$ where $f'(c) = \frac{f(b)-f(a)}{b-a}$

Fundamental Theorem of Calculus (FTC 1)

$\int_a^b f(x)\,dx = F(b) - F(a)$, where $F' = f$

Fundamental Theorem of Calculus (FTC 2)

$$\frac{d}{dx}\int_0^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$$

Rolle's Theorem

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then $\exists\, c$ where $f'(c)=0$

Critical Points

Where $f'=0$ or $f'$ is undefined

Relative Minimum

Where $f'$ changes from negative to positive; f''>0 confirms concave up

Relative Maximum

Where $f'$ changes from positive to negative; f''<0 confirms concave down

Point of Inflection

Where $f''$ changes sign; verify sign change, not just $f''=0$

Absolute Extrema

Occur at critical points or endpoints; evaluate $f$ at all and compare

Definition of Derivative

$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

Alternate Definition of Derivative

$$f'(x)\big|<em>{x=a} = \lim</em>{x \to a} \frac{f(x)-f(a)}{x-a}$$

Chain Rule

$$\frac{d}{dx}[f(u)] = f'(u) \cdot u'$$

Product Rule

$$\frac{d}{dx}[fg] = f'g + gf'$$

Quotient Rule

$$\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{gf' - fg'}{g^2}$$

Power Rule

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

$$\frac{d}{dx}(\sin u)$$

$$\cos u \cdot u'$$

$$\frac{d}{dx}(\cos u)$$

$$-\sin u \cdot u'$$

$$\frac{d}{dx}(\tan u)$$

$$\sec^2 u \cdot u'$$

$$\frac{d}{dx}(\cot u)$$

$$-\csc^2 u \cdot u'$$

$$\frac{d}{dx}(\sec u)$$

$$\sec u \tan u \cdot u'$$

$$\frac{d}{dx}(\csc u)$$

$$-\csc u \cot u \cdot u'$$

$$\frac{d}{dx}(\ln u)$$

$$\frac{u'}{u}$$

$$\frac{d}{dx}(e^u)$$

$$e^u \cdot u'$$

$$\frac{d}{dx}(\sqrt{u})$$

$$\frac{u'}{2\sqrt{u}}$$

$$\frac{d}{dx}(\sin^{-1} u)$$

$$\frac{u'}{\sqrt{1-u^2}}$$

$$\frac{d}{dx}(\cos^{-1} u)$$

$$\frac{-u'}{\sqrt{1-u^2}}$$

$$\frac{d}{dx}(\tan^{-1} u)$$

$$\frac{u'}{1+u^2}$$

$$\frac{d}{dx}(\cot^{-1} u)$$

$$\frac{-u'}{1+u^2}$$

Limit from the left

$\lim_{x \to c^-} f(x)$: the value $f(x)$ approaches as $x$ approaches $c$ from below.

Limit from the right

$\lim_{x \to c^+} f(x)$: the value $f(x)$ approaches as $x$ approaches $c$ from above.

Two-sided limit exists when

$$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$$

Limits fail to exist when

(1) left $\neq$ right limit, (2) vertical asymptote, (3) oscillation (e.g., $\sin(1/x)$ near 0)

Definition of Continuity at $x = c$

$f$ is continuous iff: (1) $f(c)$ exists, (2) $\lim_{x \to c} f(x)$ exists, (3) $\lim_{x \to c} f(x) = f(c)$

Derivatives fail to exist when

(1) not continuous, (2) sharp corner/cusp, (3) vertical tangent line

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then $\exists\, c \in (a,b)$ such that $f(c) = k$

Extreme Value Theorem (EVT)

If $f$ is continuous on closed $[a,b]$, then $f$ has both an absolute maximum and minimum on that interval

L'Hopital's Rule

If $\lim \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$

$$\lim_{x \to 0} \frac{\sin x}{x}$$

$$1$$

$$\lim_{x \to 0} \frac{1 - \cos x}{x}$$

$$0$$

$$\lim_{x \to 0} \frac{\tan x}{x}$$

$$1$$

$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$$

$$e$$