- (cos(x))^2
- (1/2)(1+cos(2x)) OR 1-(sin(x))^2
- (sin(x))^2
(1/2)(1-cos(2x)) OR 1-(cos(x))^2- 8Mk=0 ar^k
- a/(1-r) if |r| (lt) 1
- Alternating Series Test for Convergence
- Suppose 8Mn=1 (-1)^(n+1)an is an alternating series. If 1)0(lt)a[n+1](lt)a[n] for every n, and lim a[n] = 0, then (-1)^(n+1)converges
- arc length formula
- L = ~sqrt(1+(f\'(x))^2) dx
- ay\'\' + by\' + cy = 0; r[1], r[2] equal
- y(x) = c[1]e^(r[1]x) + c[2]xe^(r[1]x)
- ay\'\' + by\' + cy = 0; r[1], r[2] imaginary
- y(x) = e^(px)*(c[1]cos(qx) + c[2]sin(qx));
p = -b/2a; q = sqrt(4ac-b^2)/2a; r = p +/- qi - ay\'\' + by\' + cy = 0; r[1], r[2] unequal
- y(x) = c[1]e^(r[1]x) + c[2]e^(r[2]x)
- a^2 - x^2
- u = sin^(-1)(x/a)
- Comparison Test - convergence
- M an and M bn are positive-term series, if Mbn converges and an(lt)bn for every n, then an converges
- Comparison Test - divergence
- M an and M bn are positive-term series, if bn diverges and an > bn for every n, then an diverges
- Comparison Test examples
- p/(q+k) < p/q < (p+k)/q, try to simplify to geometric or p-series
- conditions for integrating ~(sin(x))^m(cos(x))^n dx
- 1. Either n or m (or both) is an odd, positive interger
2. Both m and n are non-negative even intergers - conditions for integrating ~(tan(x))^m(sec(x))^n dx
- 1. m is an odd, positive interger
2. n is an even, non-negative interger - Dot Product of v and u
- u1v1+u2v2+...
- Dx cos^(-1)(x)
- -1/(sqrt(1-x^2))
- Dx csc^(-1)(x)
- -1/(|x|sqrt(x^2)-1)
- Dx sec^(-1)(x)
- 1/(|x|sqrt(x^2)-1)
- Dx sin^(-1)(x)
- 1/(sqrt(1-x^2))
- Euler\'s method/Step problem
- x[n+1] = x[n] + h;
y[n+1] = y[n] + h*y\';
h = step - Integral Test
- 8Mn=1 an, 1~8 f(x)dx = lim(t->8) 1~t f(x)dx, if 8Mn=1 an is a positive-term series and f is a continous positive-valued, decreasing (eventually) function for eacn n = 1, 2, 3, etc.
- Integration by parts
- ~udv = uv - ~vdu
- Limit Comparison Test
- Suppose An and Bn are positive-term series;
1)If lim An/Bn = L with 0(lt)L(lt)8, then either both converge or both diverge
2)if lim An/Bn = 0 and Bn converges, An converges
3) If lim An/Bn = 8 and Bn diverges, then An diverges
- nth degree Taylor Polynomial of f(x) at x = a
- Pn(x) = nMk=0 f(k)(a) * 1/k! * (x-a)k
- nth Term Test for Divergence
- if lim an # 0, then 8Mk=1 an diverges
- Population Model
- P\' = aP - bP^2 = kP(M-P);
M is limiting population - what it approaches - Ratio Test
- p = lim|An+1/An| Then An converges if p(lt)1, An diverges if p>1, and test is inconclusive if p=1
- Remainder of nth degree Taylor Polynomial
- Rn(x) = fn+1(z)/(n+1)! * (x-a)n+1
- Root Test
- p = lim |An|1\\\\n Then An converges if p(lt)1, An diverges if p>1, and test is inconclusive if p=1
- Separate (x^2+2)/(x^3)(x-2)^2(x^2+1) into partial fractions
- (a/(x))+(b/(x^2))+(c/(x^3))+(d/(x-2))+(e/(x-2)^2)+((fx+g)/(x^2+1))
- set-up for tank problem
- x(t) is the amount of salt in tank 1 after time t;
x\' = (gal/min input)-(gal/min output);
output = (v[2])(x(t)/(V-(v[1]-v[2])) - simplified integration by parts
- ~(polynomial)(e^x, sin(x), cos(x))
differentiate each separately, then combine crosswise, alternating between positive and negative starting with negative
- Surface area of f(x) around x-axis
- SA = ~2(pi)f(x)sqrt(1+(f\'(x))^2) dx (a, b on x-axis)
- Surface area of f(x) around y-axis
- SA = ~2(pi)(x)sqrt(1+(f\'(x))^2) dx (a, b, on x-axis)
- Surface area of g(y) around x-axis
- SA = ~2(pi)(y)sqrt(1+(g\'(y))^2) dy (a, b, on y-axis)
- Surface area of g(y) rotated around y-axis
- SA = ~2(pi)g(y)sqrt(1+(g\'(y))^2) dy (a, b on y-axis)
- Unit Vector
- v is vector. u = v/|v| = v1/|v|, v2/|v|
- Vector Cross Product
- uxv=
|i j k|
|u1 u2 u3|
|v1 v2 v3|
- volume generated by revolving f(x) and g(x) around x-axis
- v = ~(pi)[f(x)^2-g(x)^2] dx (a, b on x-axis)
- volume generated by revolving f(x) around y-axis
- v = ~2(pi)x(f(x)) dx (a, b on x-axis)
- volume generated by revolving g(y) around x-axis
- v = ~2(pi)y(g(y)) dx (a, b on y-axis)
- volume generated by rotating f(x)around x-axis
- v = ~(pi)f(x)^2 dx (a, b on x-axis)
- volume generated by rotating g(y) around y-axis (dish)
- v = ~(pi)g(y)^2 dy (a, b on y-axis)
- Volume of vectors
- V = |a.(bxc)|
- x^2 + a^2
- u = tan^(-1)(x/a)
- x^2 - a^2
- u = sec^(-1)(x/a)
- ~cos(kx) dx
- (1/k)sin(kx) + C
- ~cot(x) dx
- ln|sin(x)| + C OR -ln|csc(x)| + C
- ~csc(x) dx
- -ln|csc(x) + cot(x)| + C
- ~sec(x) dx
- ln|sec(x) + tan(x)| + C
- ~sin(kx) dx
- (-1/k)cos(kx) + C
- ~tan(x) dx
- -ln|cos(x)| + C OR ln|sec(x)| + C
