Formulas
Geometry:
Circles:Area = r2Circumference = 2r
Spheres:Volume = r3Surface Area = 4r2
Cylinders:Volume = (Area of base)(Height)
Cones:Volume = (Area of base)(Height)
Trigonometric Identities:
sin2x + cos2x = 1tan2x + 1 = sec2xctn2x + 1 = csc2x
sin(x+y) = sinx cosy + cosx sinycos(x+y) = cosx cosy - sinx siny
sin 2x = 2 sinx cosxcos 2x = cos2x - sin2x
sinx siny = cosx cosy =
sinx cosy = sin2x = cos2x =
y = is that angle y such that x = and
sin-1(- x) = - sin-1xcos-1(- x) = - cos-1xcos-1x = - sin-1x
Hyperbolic Functions:
sinh x = cosh x = tanh x = =
coth x = = sech x = = csch x = =
cosh2x - sinh2x = 1tanh2x + sech2x = 1coth2x = 1 + csch2x
sinh(x+y) = sinh x cosh y + cosh x sinh ycosh(x+y) = cosh x cosh y + sinh x sinh y
sinh 2x = 2 sinh x cosh xcosh 2x = cosh2x + sinh2x
sinh2x = cosh2x =
y = = is that y such that x =
Derivatives
= f’(x) =
Linear approximation (3 ways of writing the same thing)
f(x) f(xo) + f’(xo)(x-xo)f(x+x) f(x) + f’(x)xy x
Quadratic approximation
f(x) f(xo) + f’(xo)(x-xo) + (x-xo)2f(x+x) f(x) + f’(x)x + (x)2
y x + (x)2
Mean Value Theorem (Slope of secant line = slope of tangent line at intermediate point. Same as linear approximation but uncertainty is where derivative is evaluated.)
f(x) = f(xo) + f’(c)(x-xo)f(x+x) = f(x) + f’(c)x (c)
L’Hospital’s Rule = provided f(x) and g(x) are either both 0 or both
Summation Formulas
1 + 2 + 3 + ... + n = 12 + 22 + 32 + ... + n2 =
13 + 23 + 33 + ... + n3 = 1 + x + x2 + ... + xn =
Integrals
=
Area = L(x) = Length of cross section through x.
Volume= A(x) = Area of cross section through x.
Volume= For the solid created by rotating a region about the x axis. The region is bounded above by y = f(x), below by y = g(x), on the left by x = a and on the right by x = b.
Volume= 2For the solid created by rotating the same region about the y axis.
Work= F(x) = Force on object at position x.
Table of Derivatives and Integrals
tan x = sec2x / = tan xcot x = - csc2x / = - cot x
sec x = tan x sec x / = secx
csc x = - cot x csc x / = - cscx
ax = ax ln(a) / =
sin-1x = / = sin-1x
cos-1x = / = - cos-1x
tan-1x = / = tan-1x
cot-1x = / = - cot-1x
sec-1x = / = sec-1x
csc-1x = / = - csc-1x
sinh x = cosh x / = sinh x
cosh x = sinh x / = cosh x
tanh x = sech2x / = tanh x
coth x = - csch2x / = - coth x
sech x = - tanh x sech x / = - sechx
csch x = - coth x csch x / = - cschx
sinh-1x = / = sinh-1x
cosh-1x = / = cosh-1x
tanh-1x = / = tanh-1x
coth-1x = / = coth-1x
sech-1x = / = - sech-1x
csch-1x = / = - csch-1x
More Integrals
= uv -
= ln( tan x + sec x)
= ln( csc x - cot x)
= - sinn-1x cos x +
= cosn-1x sin x +
= tann-1x -
= - cotn-1x -
= tan x secn-2x +
= - cot x cscn-2x +
= after substituting u = sin x and using cos2x = 1 – sin2x
Similarly when sin is to an odd power
Use sin2x = and cos2x =
= after substituting u = tan x and using sec2x = 1 + tan2x
= after substituting u = sec x and using tan2x = sec2x - 1
= after tan2x = sec2x – 1. Now expand out and use reduction formula for
Integral involving Let x = a sin u
Integral involving Let x = a tan u
Integral involving Let x = a sec u
Integral involving Complete the square: = . Then substitute u = x + which reduces the integral to one of the three previous
Integral of a rational function . First, divide denominator into numerator to get a polynomial plus a rational function where the numerator has degree less than the denominator. Integrate the polynomial as usual. For the new rational function, factor the denominator into the product of linear and quadratic factors and split it up using partial fractions: = + + + + + . To find the numerators, cross multiply to get anxn + an-1xn-1 + + a1x + a0 = A1(x-r2)(x-s)2(x2+bx+c) + A2(xr1)(xr3)(xs)2(x2+bx+c) + + B(xr1)(xr2)(xs)(x2+bx+c) + C(xr1)(xr2)(x2+bx+c) + (Dx+E)(xr1)(xr2)(xs)2. You will need to plug in additional values to find B, D and E. Example: = + 1 = A1(x-5) + A2(x4). x = 4 A1 = -1. The resulting fractions with linear denominators are easy to integrate. Complete the square, if necessary, to integrate the ones with quadratic denominators.
= f(x) dx = f(x) dx = f(x) dx +
= if f(x) is unbounded as xa
Numerical Integration
Midpoint method
2(x) [ f(x1) + f(x3) + f(x5) + + f(xn-1) ] where x = and xj = a + j(x) and n is even
Simpson’s method
[ f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + + 2f(xn-2) + 4f(xn-1) + f(xn) ] with x,xj as above, n is even
Arc Length
L = length of curve y = f(x) where axb =
Center of Mass
(,) = center of mass of thin plate of uniform density occupying the region bounded above by y=f(x) and below y = g(x) and between x = a and x = b. Let A = area. Then
= =
Parametric Equations of Curves: x = f(t) and y = g(t)
To write as y = F(x): Solve x = f(t) for t in terms of x and substitute in y = g(t). Similarly to write as x=G(y)
Slopes: = Second derivatives: =
Areas: = where a = f() and b = f()
Arc Lengths: L = = length of the part corresponding to t
Polar Coordinates r and
Curve defined by a polar equation r = f(): x = f() cos and y = f() sin
Slopes: =
Arc Lengths: L = = length of the part corresponding to t
Areas: A = = area of region where g() rf() and
Conic Sections
Ellipse: + = 1
ab foci at (uc, v)
c2 = a2 – b2
Hyperbola: - = 1
foci at (uc, v) c2 = a2 + b2
Sequences and Series
Geometric series: if -1 < x < 1
nth term test: 0 diverges
Absolute convergence: < converges
Integral test: f(x) positive & decreasing < <
and, more generally < <
p - series: Converges if p > 1 and diverges if p 1
Limit comparison test: Assume an > 0 & bn > 0 for all n
< & < <
> 0 = =
Ratio test:
If an > 0 then < 1 < & > 1 =
< 1 converges & > 1 diverges
Root test:
If an > 0 then < 1 < & > 1 =
< 1 converges & > 1 diverges
Alternating series test: an decreasing with n = 0 converges
Taylor series: f(x) =
where M = maximum of | f(n+1)(t) | for t between a and x
1