Day 29 · Cheat Sheet

Final Cheat Sheet

Every formula, rule, and theorem for AP Calculus AB in one place. Read it through the night before; do not try to learn anything new from it.

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A limit exists: $\lim_{x \to c} f(x) = L$ iff $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$
Continuity at x = c: $f(c)$ defined, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$
Limits at infinity (rational): Compare degrees: top $<$ bottom $\Rightarrow 0$ ; equal $\Rightarrow$ ratio of leading coefficients; top $>$ bottom $\Rightarrow \pm\infty$
Special trig limits: $\lim_{x \to 0} \dfrac{\sin x}{x} = 1, \qquad \lim_{x \to 0} \dfrac{1 - \cos x}{x} = 0$
L'Hopital's Rule: If $\lim \dfrac{f}{g}$ is $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ , then $\lim \dfrac{f}{g} = \lim \dfrac{f'}{g'}$

Definition: $f'(a) = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}$
Power, product, quotient: $(x^n)' = nx^{n-1}, \quad (fg)' = f'g + fg', \quad \left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$
Chain rule: $\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
Trig: $(\sin x)' = \cos x, \ (\cos x)' = -\sin x, \ (\tan x)' = \sec^2 x$
$(\cot x)' = -\csc^2 x, \ (\sec x)' = \sec x \tan x, \ (\csc x)' = -\csc x \cot x$
Exponential & log: $(e^x)' = e^x, \quad (a^x)' = a^x \ln a, \quad (\ln x)' = \dfrac{1}{x}, \quad (\log_a x)' = \dfrac{1}{x \ln a}$
Inverse trig: $(\sin^{-1} x)' = \dfrac{1}{\sqrt{1 - x^2}}, \quad (\tan^{-1} x)' = \dfrac{1}{1 + x^2}, \quad (\sec^{-1} x)' = \dfrac{1}{|x|\sqrt{x^2 - 1}}$
Inverse function: $(f^{-1})'(b) = \dfrac{1}{f'(f^{-1}(b))}$ Find the input $a$ with $f(a) = b$ first, then take $1/f'(a)$ .

Tangent / normal line: Tangent: $y = f(a) + f'(a)(x - a)$ ; normal slope $= -\dfrac{1}{f'(a)}$
Motion: $v(t) = s'(t), \ a(t) = v'(t)$ . Speed $= |v(t)|$ . Speeding up when $v$ and $a$ have the same sign.
Increasing / decreasing: $f$ increasing where $f' > 0$ ; decreasing where $f' < 0$
1st derivative test: Local max where $f'$ goes $+ \to -$ ; local min where $f'$ goes $- \to +$
Concavity & inflection: Concave up where $f'' > 0$ ; concave down where $f'' < 0$ ; inflection where $f''$ changes sign
2nd derivative test: $f'(c) = 0$ and $f''(c) > 0 \Rightarrow$ local min; $f''(c) < 0 \Rightarrow$ local max
Absolute extrema (closed interval): Compare $f$ at every critical number in $(a, b)$ and at $a$ and $b$
Related rates: Differentiate the relating equation with respect to $t$ ; substitute known values last
Optimization: Write the quantity in one variable using a constraint, set its derivative to $0$ , check it is the extremum
Linear approximation: $f(x) \approx f(a) + f'(a)(x - a)$ near $x = a$

Power / basic: $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\ (n \ne -1), \quad \int \dfrac{1}{x}\,dx = \ln|x| + C$
Exponential / trig: $\displaystyle\int e^x\,dx = e^x + C, \ \int \sin x\,dx = -\cos x + C, \ \int \cos x\,dx = \sin x + C$
$\displaystyle\int \sec^2 x\,dx = \tan x + C, \ \int \dfrac{dx}{1 + x^2} = \tan^{-1} x + C, \ \int \dfrac{dx}{\sqrt{1 - x^2}} = \sin^{-1} x + C$
u-substitution: $\displaystyle\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ ; for a definite integral, change the limits to $u$ -values
FTC Part 1: $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$ where $F' = f$
FTC Part 2: $\dfrac{d}{dx}\displaystyle\int_a^{u(x)} f(t)\,dt = f(u(x)) \cdot u'(x)$
Riemann sums / trapezoid: Left/right/midpoint: $\sum f(x_i^*)\Delta x$ . Trapezoid: $\dfrac{\Delta x}{2}\big[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)\big]$
Integral properties: $\displaystyle\int_a^b (cf \pm g) = c\int_a^b f \pm \int_a^b g, \quad \int_a^c f + \int_c^b f = \int_a^b f, \quad \int_a^b f = -\int_b^a f$

Accumulation: $F(b) = F(a) + \displaystyle\int_a^b F'(x)\,dx$ (final = initial + total change)
Displacement vs. distance: Displacement $= \displaystyle\int_a^b v(t)\,dt$ ; total distance $= \int_a^b |v(t)|\,dt$ (split at $v = 0$ )
Average value: $f_{\text{avg}} = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$
Area between curves: $\displaystyle\int_a^b (\text{top} - \text{bottom})\,dx$ or $\int_c^d (\text{right} - \text{left})\,dy$
Volume — disk / washer: Disk: $\pi\displaystyle\int R^2\,dx$ . Washer: $\pi\int (R^2 - r^2)\,dx$ (square each radius first)
Volume — known cross sections: $\displaystyle\int_a^b A(x)\,dx$ with $A$ from $s = \text{top} - \text{bottom}$ : square $s^2$ ; semicircle $\tfrac{\pi}{8}s^2$ ; equilateral triangle $\tfrac{\sqrt{3}}{4}s^2$
Separable differential equation: $\dfrac{dy}{dx} = f(x)g(y) \Rightarrow \displaystyle\int \dfrac{dy}{g(y)} = \int f(x)\,dx$ , then use the initial condition for $C$
Exponential model: $\dfrac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}$ ; half-life $= \dfrac{\ln 2}{|k|}$

Intermediate Value Theorem: $f$ continuous on $[a, b]$ , $N$ between $f(a)$ and $f(b)$ $\Rightarrow$ some $c \in (a, b)$ has $f(c) = N$
Extreme Value Theorem: $f$ continuous on $[a, b]$ $\Rightarrow$ $f$ attains an absolute max and min on $[a, b]$
Mean Value Theorem: $f$ continuous on $[a, b]$ , differentiable on $(a, b)$ $\Rightarrow$ some $c$ has $f'(c) = \dfrac{f(b) - f(a)}{b - a}$
MVT for Integrals: $f$ continuous on $[a, b]$ $\Rightarrow$ some $c$ has $f(c) = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$
Differentiable ⟹ continuous: The converse is false: $|x|$ is continuous but not differentiable at $0$

Unit circle sine / cosine: $\sin: 0, \tfrac{1}{2}, \tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{3}}{2}, 1$ at $0, \tfrac{\pi}{6}, \tfrac{\pi}{4}, \tfrac{\pi}{3}, \tfrac{\pi}{2}$ ; cosine is the same list reversed
Logs: $\ln 1 = 0, \ \ln e = 1, \ e^0 = 1, \ \ln(ab) = \ln a + \ln b, \ \ln(a^p) = p\ln a$
Pythagorean identity: $\sin^2 x + \cos^2 x = 1, \quad 1 + \tan^2 x = \sec^2 x$
Double angle: $\sin 2x = 2\sin x \cos x, \quad \cos 2x = 1 - 2\sin^2 x = 2\cos^2 x - 1$