Useful Formulas (Math 10C)
1. Fraction of population with characteristic a≤x≤b=Rb
ap(x)dx =P(b)−P(a).
2. Fraction of population with characteristic x≤t=P(t).
3. The median of a characteristic xis the value x=Tsuch that RT
−∞ p(x)dx = 0.5.
4. The mean of a characteristic xis equal to R∞
−∞ xp(x)dx.
5. a+ax +ax2+... +axn−1=a(1−xn)
1−xwhen x6= 1.
6. a+ax +ax2+... +axn−1+axn+... =a
1−xwhen |x| ≤ 1.
7. Near x= 0, f(x)≈Pn(x) = f(0) + f0(0)x+f00(0)
2! x2+f000(0)
3! x3+f(4)(0)
4! x4+... +f(n)(0)
n!xn.
8. Near x=a,f(x)≈Pn(x) = f(a) + f0(a)(x−a) + f00(a)
2! (x−a)2+f000(a)
3! (x−a)3+
f(4)(a)
4! (x−a)4+... +f(n)(a)
n!(x−a)n.
9. A plane through the point (x0, y0, z0), with slope min the xdirection and slope nin
the ydirection, has the equation
z=z0+m(x−x0) + n(y−y0).
10. The displacement vector from the point P1= (x1, y1, z1) to the point P2= (x2, y2, z2)
is given in components by
−→
P1P2= (x2−x1)
~
i+ (y2−y1)~
j+ (z2−z1)~
k.
11. If ~v =v1
~
i+v2~
j+v3~
k, then ||~v|| =pv2
1+v2
2+v2
3. The analogous formula works for
vectors in 2 dimensions.
12. ~v ·~w =||~v|| ||~w|| cos θ=v1w1+v2w2+v3w3. The analogous formula works for vectors
in 2 dimensions.
13. The equation of a plane with normal vector ~n =a
~
i+b~
j+c~
kand containing the point
(x0, y0, z0) is
a(x−x0) + b(y−y0) + c(z−z0)=0.
14. If ~vpar allel and ~vperp are the components of ~v which are parallel and perpendicular,
respectively, to the unit vector ~u, then ~vparallel = (~v ·~u)~u (this is the projection of ~v on
~u), and ~v =~vparallel +~vperp .
15. ~v ×~w = (||~v|| ||~w|| sin θ)~n = (v2w3−v3w2)
~
i+ (v3w1−v1w3)~
j+ (v1w2−v2w1)~
k.
16. A parallelogram with edges ~v and ~w has area ||~v ×~w||.
17. A parallelepiped with edges ~a,~
b, and ~c has volume |(~
b×~c)·a|.
