Useful Mathematics Formulas (Math 10C), Study notes of Calculus

Download Useful Mathematics Formulas (Math 10C) and more Study notes Calculus in PDF only on Docsity! Useful Formulas (Math 10C) 1. Fraction of population with characteristic a ≤ x ≤ b = ∫ b a p(x)dx = P (b)−P (a). 2. Fraction of population with characteristic x ≤ t = P (t). 3. The median of a characteristic x is the value x = T such that ∫ T −∞ p(x)dx = 0.5. 4. The mean of a characteristic x is equal to ∫∞ −∞ xp(x)dx. 5. a+ ax+ ax2 + ...+ axn−1 = a(1−x n) 1−x when x 6= 1. 6. a+ ax+ ax2 + ...+ axn−1 + axn + ... = a 1−x when |x| ≤ 1. 7. Near x = 0, f(x) ≈ Pn(x) = f(0) +f ′(0)x+ f ′′(0) 2! x2 + f ′′′(0) 3! x3 + f (4)(0) 4! x4 + ...+ f (n)(0) n! xn. 8. Near x = a, f(x) ≈ Pn(x) = f(a) + f ′(a)(x − a) + f ′′(a) 2! (x − a)2 + f ′′′(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 m in the x direction and slope n in the y direction, 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|| = √ v21 + v 2 2 + v 2 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~k and containing the point (x0, y0, z0) is a(x− x0) + b(y − y0) + c(z − z0) = 0. 14. If ~vparallel 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|.