Formula Sheet
Every formula of the course on one searchable page — algebra, geometry, trigonometry, limits, derivatives and integrals.
Every formula used in the course, grouped by topic. Type in the box to filter (try chain, volume, sec or parts). Each card links to the section where it is taught. Use your browser's Print for a clean paper copy.
Algebra
Quadratic formula§ 0.1 →
$ax^2+bx+c=0\ \Rightarrow\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
Difference of squares§ 0.1 →
$a^2-b^2=(a-b)(a+b)$
Cubes§ 0.1 →
$a^3\mp b^3=(a\mp b)(a^2\pm ab+b^2)$
Binomial squares
$(a\pm b)^2=a^2\pm2ab+b^2$
Binomial cube§ 2.2 →
$(a+b)^3=a^3+3a^2b+3ab^2+b^3$
Exponent laws§ 0.1 →
$a^xa^y=a^{x+y}$$\dfrac{a^x}{a^y}=a^{x-y}$$(a^x)^y=a^{xy}$$a^{-x}=\dfrac1{a^x}$$a^{m/n}=\sqrt[n]{a^m}$
Logarithm laws§ 0.1 →
$\ln(xy)=\ln x+\ln y$$\ln\dfrac xy=\ln x-\ln y$$\ln x^r=r\ln x$
Log ↔ exponential§ 0.1 →
$y=\log_bx\iff b^y=x;$$ e^{\ln x}=x$$\ln e^x=x;$$ \log_bx=\dfrac{\ln x}{\ln b}$
Absolute value
$|x|\lt a\iff -a\lt x\lt a;$$ |x|=\sqrt{x^2}$
Lines§ 0.1 →
$m=\dfrac{y_2-y_1}{x_2-x_1};$$ y-y_1=m(x-x_1);$$ y=mx+b$
Parallel / perpendicular§ 3.1 →
parallel: $m_1=m_2$; perpendicular: $m_1m_2=-1$
Vertex of a parabola§ 0.1 →
$y=ax^2+bx+c$ has vertex at $x=-\dfrac{b}{2a}$
Distance
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Geometry (for related rates & optimization)
Pythagoras§ 3.3 →
$a^2+b^2=c^2$
Similar triangles§ 3.3 →
$\dfrac{r}{h}=\dfrac{R}{H}$ (corresponding sides in proportion)
Circle§ 3.3 →
$A=\pi r^2$$ C=2\pi r$
Rectangle / square§ 3.6 →
$A=lw,\ P=2l+2w;\quad$ square $A=s^2$
Triangle
$A=\tfrac12bh$
Sphere§ 3.3 →
$V=\tfrac43\pi r^3$$ S=4\pi r^2$
Cylinder§ 3.3 →
$V=\pi r^2h$$ S=2\pi r^2+2\pi rh$
Cone§ 3.3 →
$V=\tfrac13\pi r^2h$
Box / cube§ 3.6 →
$V=lwh;\quad$ cube $V=s^3,\ S=6s^2$
Distance–rate–time§ 3.6 →
$D=R\,T$
Trigonometry
Radians§ 0.1 →
$\pi\ \text{rad}=180^\circ;$$ 1^\circ=\tfrac{\pi}{180}\ \text{rad}$
Definitions§ 0.1 →
$\tan x=\dfrac{\sin x}{\cos x}$$\cot x=\dfrac{\cos x}{\sin x}$$\sec x=\dfrac1{\cos x}$$\csc x=\dfrac1{\sin x}$
Pythagorean identities§ 0.1 →
$\sin^2x+\cos^2x=1$$ 1+\tan^2x=\sec^2x$$ 1+\cot^2x=\csc^2x$
Double angle§ 0.1 →
$\sin2x=2\sin x\cos x;$$ \cos2x=\cos^2x-\sin^2x$$ =1-2\sin^2x$$ =2\cos^2x-1$
Half-angle (power reduction)
$\sin^2x=\dfrac{1-\cos2x}{2}$$ \cos^2x=\dfrac{1+\cos2x}{2}$
Even / odd
$\sin(-x)=-\sin x$$\cos(-x)=\cos x$$\tan(-x)=-\tan x$
Special values§ 0.1 →
$\sin\tfrac\pi6=\tfrac12$$\sin\tfrac\pi4=\tfrac{\sqrt2}2$$\sin\tfrac\pi3=\tfrac{\sqrt3}2$$\cos\tfrac\pi6=\tfrac{\sqrt3}2$$\cos\tfrac\pi3=\tfrac12$
Inverse trig ranges§ 0.1 →
$\sin^{-1}:[-\tfrac\pi2,\tfrac\pi2]$$\cos^{-1}:[0,\pi]$$\tan^{-1}:(-\tfrac\pi2,\tfrac\pi2)$
Limits and Continuity
Limit laws§ 1.2 →
$\lim(f\pm g)=L\pm M$$\lim(cf)=cL$$\lim(fg)=LM$$\lim\dfrac fg=\dfrac LM\ (M\ne0)$$\lim f^n=L^n$
Polynomials / rational§ 1.2 →
$\displaystyle \lim_{x\to a}p(x)=p(a);$$\displaystyle \lim_{x\to a}\frac{p(x)}{q(x)}=\frac{p(a)}{q(a)}\ (q(a)\ne0)$
One-sided ⇔ two-sided§ 1.1 →
$\displaystyle \lim_{x\to a}f=L\iff$$\displaystyle \lim_{x\to a^-}f=L\text{ and }$$\displaystyle \lim_{x\to a^+}f=L$
Special trig limits§ 1.1 →
$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$$\displaystyle \lim_{x\to0}\frac{1-\cos x}{x}=0$
Limits at infinity§ 1.4 →
$\displaystyle \lim_{x\to\pm\infty}\frac{1}{x^k}=0\ (k\gt 0)$
Rational functions at $\pm\infty$§ 1.4 →
deg top $\lt $ deg bottom: $0$; equal: ratio of leading coefficients; top bigger: $\pm\infty$
Continuity at a§ 1.3 →
$f(a)$ defined, $\displaystyle\lim_{x\to a}f(x)$ exists, and $\displaystyle\lim_{x\to a}f(x)=f(a)$
Vertical asymptote§ 1.4 →
$x=a$ if $\displaystyle\lim_{x\to a^\pm}f(x)=\pm\infty$
0/0 strategies§ 1.2 →
factor & cancel · multiply by the conjugate · simplify the complex fraction · L’Hôpital
Derivatives
Definition§ 2.1 →
$\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Basic rules§ 2.2 →
$(c)'=0$$(x^n)'=nx^{n-1}$$(cf)'=cf'$$(f\pm g)'=f'\pm g'$
Product & quotient§ 2.2 →
$(fg)'=f'g+fg'$$\left(\dfrac fg\right)'=\dfrac{f'g-fg'}{g^2}$
Chain rule§ 2.3 →
$\dfrac{d}{dx}f\big(g(x)\big)=f'\big(g(x)\big)\,g'(x);$$\dfrac{dy}{dx}=\dfrac{dy}{du}\dfrac{du}{dx}$
Trigonometric§ 2.2 →
$(\sin x)'=\cos x$$(\cos x)'=-\sin x$$(\tan x)'=\sec^2x$
Trigonometric (2)§ 2.2 →
$(\cot x)'=-\csc^2x$$(\sec x)'=\sec x\tan x$$(\csc x)'=-\csc x\cot x$
Exponential & log§ 2.2 →
$(e^x)'=e^x$$(a^x)'=a^x\ln a$$(\ln x)'=\dfrac1x$$(\log_bx)'=\dfrac{1}{x\ln b}$
Inverse trig§ 2.2 →
$(\sin^{-1}x)'=\dfrac{1}{\sqrt{1-x^2}}$$(\cos^{-1}x)'=\dfrac{-1}{\sqrt{1-x^2}}$$(\tan^{-1}x)'=\dfrac{1}{1+x^2}$
Inverse trig (2)§ 2.2 →
$(\cot^{-1}x)'=\dfrac{-1}{1+x^2}$$(\sec^{-1}x)'=\dfrac{1}{|x|\sqrt{x^2-1}}$$(\csc^{-1}x)'=\dfrac{-1}{|x|\sqrt{x^2-1}}$
Chain-rule forms§ 2.3 →
$\big(e^{g}\big)'=e^{g}g'$$\big(\ln g\big)'=\dfrac{g'}{g}$$\big(g^n\big)'=ng^{n-1}g'$
Implicit§ 2.4 →
differentiate both sides; each $y$-term gains $\dfrac{dy}{dx}$: $\ \dfrac{d}{dx}y^n=ny^{n-1}\dfrac{dy}{dx}$
Parametric§ 2.5 →
$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$$\dfrac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$
Applications of Derivatives
Tangent line§ 3.1 →
$y=f(a)+f'(a)(x-a)$
Normal line§ 3.1 →
$y=f(a)-\dfrac{1}{f'(a)}(x-a)$, ($f'(a)\neq0$; if $f'(a)=0$ the normal is $x=a$)
Linear approximation§ 3.1 →
$f(x)\approx f(a)+f'(a)(x-a)$ for $x$ near $a$
Motion§ 3.2 →
$v(t)=s'(t)$$a(t)=v'(t)=s''(t)$$\text{speed}=|v(t)|$
Critical point§ 3.4 →
$c$ interior with $f'(c)=0$ or $f'(c)$ undefined
Closed-interval method§ 3.4 →
evaluate $f$ at end points and critical points; largest = abs. max, smallest = abs. min
Increasing / decreasing§ 3.4 →
$f'\gt 0\Rightarrow$ increasing; $f'\lt 0\Rightarrow$ decreasing
First derivative test§ 3.4 →
$f'$: $+\to-$ local max; $-\to+$ local min; no change: neither
Concavity§ 3.4 →
$f''\gt 0$: concave up $\cup$; $f''\lt 0$: concave down $\cap$; sign change: inflection point
Second derivative test§ 3.4 →
$f'(c)=0$: $f''(c)\gt 0$ min, $f''(c)\lt 0$ max, $f''(c)=0$ inconclusive
Rolle's theorem§ 3.5 →
If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f(a)=f(b)$, then $f'(c)=0$ for some $c\in(a,b)$
Mean Value Theorem§ 3.5 →
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$: $f'(c)=\dfrac{f(b)-f(a)}{b-a}$ for some $c\in(a,b)$
Related rates§ 3.3 →
draw · name variables · relate them · differentiate in $t$ · substitute last
Optimization§ 3.6 →
objective + constraint → one variable → domain → critical points & ends → answer
L'Hôpital's rule§ 3.7 →
$\dfrac00$ or $\dfrac\infty\infty$: $\displaystyle\lim\frac{f}{g}=\lim\frac{f'}{g'}$
Integrals
Definition§ 4.1 →
$\int f(x)\,dx=F(x)+C\iff F'(x)=f(x)$
Power rule§ 4.1 →
$\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\ (n\ne-1)$$\int\dfrac1x\,dx=\ln|x|+C$
Exponentials§ 4.1 →
$\int e^x\,dx=e^x+C$$\int a^x\,dx=\dfrac{a^x}{\ln a}+C$
Trigonometric§ 4.1 →
$\int\cos x\,dx=\sin x+C$$\int\sin x\,dx=-\cos x+C$$\int\sec^2x\,dx=\tan x+C$
Trigonometric (2)§ 4.1 →
$\int\csc^2x\,dx=-\cot x+C$$\int\sec x\tan x\,dx=\sec x+C$$\int\csc x\cot x\,dx=-\csc x+C$
Inverse-trig results§ 4.1 →
$\int\dfrac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+C$$\int\dfrac{dx}{1+x^2}=\tan^{-1}x+C$$\int\dfrac{dx}{x\sqrt{x^2-1}}=\sec^{-1}|x|+C$
Tangent / log§ 4.4 →
$\int\tan x\,dx=-\ln|\cos x|+C$$\int\ln x\,dx=x\ln x-x+C$
Linearity§ 4.1 →
$\int\big(f\pm g\big)dx=\int f\,dx\pm\int g\,dx$$\int kf\,dx=k\int f\,dx$
Definite-integral properties§ 4.2 →
$\int_a^af=0$$\int_a^bf=-\int_b^af$$\int_a^bf=\int_a^cf+\int_c^bf$
FTC Part 1§ 4.2 →
$\displaystyle \dfrac{d}{dx}\int_a^xf(t)\,dt=f(x)$
FTC Part 2§ 4.2 →
$\displaystyle \int_a^bf(x)\,dx=F(b)-F(a)=F(x)\Big|_a^b$
Average value§ 4.2 →
$f_{\text{ave}}=\dfrac{1}{b-a}\displaystyle\int_a^bf(x)\,dx$; MVT for integrals: $f(c)=f_{\text{ave}}$ for some $c\in[a,b]$
Substitution§ 4.3 →
$\int f\big(g(x)\big)g'(x)\,dx=\int f(u)\,du,\quad u=g(x)$; definite: change the limits to $g(a),g(b)$
Linear inside§ 4.3 →
$\int f(ax+b)\,dx=\dfrac1aF(ax+b)+C$, where $F'=f$ and $a\neq0$
Integration by parts§ 4.4 →
$\int u\,dv=uv-\int v\,du$; choose $u$ by LIATE (Log, Inverse trig, Algebraic, Trig, Exponential)
Initial-value problem§ 4.1 →
integrate, then use $y(x_0)=y_0$ to find $C$