Formulas



Formulas - Math 216

Trigonometric Identities

sin(x+y) = sinx cosy + cosx siny cos(x+y) = cosx cosy - sinx siny

sinx siny = ½ [ cos(x-y) - cos(x+y) ] cosx cosy = ½ [ cos(x-y) + cos(x+y) ]

sinx cosy = ½ [ sin(x+y) + sin(x-y) ]

sin2x = ½ [ 1 - cos(2x) ] cos2x = ½ [ 1 + cos(2x) ]

sinh x = ½ [ e x - e-x ] cosh x = ½ [ e x + e-x ]

eiy = cos y + i sin y e-iy = cos y - i sin y

sin x = [ eix - e-ix ] cos x = ½ [ eix + e-ix ]

Geometry

Circles: Area = (r2 Circumference = 2(r

Spheres: Volume = (r3 Surface Area = 4(r2

Cylinders: Volume = (Area of base) ( (Height)

Cones: Volume = (Area of base) ( (Height)

Summation Formulas

1 + 2 + 3 + ( + n =

12 + 22 + 32 + ( + n2 =

13 + 23 + 33 + ( + n3 =

1 + x + x2 + ( + xn =

Derivatives and Integrals:

ax = ax ln(a) =

tanx = sec2x

secx = tanx secx

sin-1x = ) ) dx) = sin-1x

cos-1x = ) ) dx) = - cos-1x

tan-1x = dx) = tan-1x

= uv -

= xex - ex

= - x cos x + sin x

= x sin x + cos x

= -x2 cos x + 2x sin x + 2cos x

= x2 sin x + 2x cos x - 2sin x

= x ln(x) - x

= (a sin(bx) - b cos(bx) )

= (a cos(bx) + b sin(bx) )

Ei(x) = dt)

Ei(x) = dx) = Ei(x)

erf(x) = )

erf(x) = ) e-x2 = ,2) erf(x)

Differential Equations

First order linear

+ p(t)u = f(t) Multiply by e ( p(t) dt

Second order linear homogeneous

p(t) + q(t) + r(t)u = 0

u = C1u1 + C2u2 where u1 and u2 are solutions (superposition principle)

u2 = u1 (second solution formula)

a + b + cu = 0 (constant coefficient)

Try u = ert ( Solve ar2 +br + c = 0 to get roots r1, r2

u = Aer1t + Ber2t if roots are real and unequal

u = (A + Bt)ert if r1 = r2 = r

u = Ae(tcos((t)+ Be(tsin((t) if roots are complex with r1 = ( + (i

ax2 + bx + cu = 0 (Euler)

Try u = xr ( Solve ar(r-1) +br + c = 0 to get roots r1, r2

u = Axr1 + Bxr2 if roots are real and unequal

u = (A + B ln(x))xr if r1 = r2 = r

u = Ax(cos(( ln(x))+ Bx(sin(( ln(x)) if roots are complex with r1 = ( + (i

Second order linear inhomogeneous

p(t) + q(t) + r(t)u = f(t)

u = up + uh up = a particular solution (superposition principle)

uh = general solution to the homogeneous equation

up = - u1 + u2 W =

Approximations to the solution - Euler's method

= f(t,u) u(to) = uo

Choose a stepsize h and let

tj = to + jh uj+1 = uj + h f(tj,uj)

Then u(tj) = uj + error where

error ( Ch where C is a constant that depends on t but not h.

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