Formulas - University of Michigan
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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