Math 132 equations
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- Product Rule
- (fg)'=f'g+fg'
- Integration by Parts
- ∫udv=uv-∫vdu
- Equation of Ellipse
- (x^2/a^2)+(y^2/b^2)=1
- √(#) =
- l#l... the absolute value of the number
- ∫(sinx)^(n)*(cosx)^(m) = if n is odd...
- take out a sin and convert everything to cosines w/ (sinx)^2=1-(cosx)^2
- ∫(sinx)^(n)*(cosx)^(m) = if m is odd
- take out one cosine and convert everything to sine using (cosx)^2=1-(sinx)^2
- ∫(tanx)^n*(secx)^n = if n is odd
- take out one tangent and one secant make everything tangents u-sub w/ tan^2=sec^2-1
- ∫(sinx)^(n)*(cosx)^(m) = if n and m are even
- Use double angle formula for sine and/or halfangle formula
- ∫(tanx)^n*(secx)^n = if m is even
- take out 2 secants make remaining into tangent w/ sec^2=1+tan^2
- Trig Sub of: √(a^2-x^2)
- x=asinø
- Trig Sub of: √(a^2+x^2)
- x=tanø
- Trig Sub of: √(x^2-a^2)
- x=asecø
- (cosø)^2=
- 1-(sinø)^2
- (secø)^2=
- 1+(tanø)^2
- Divergence Test
- If lim(an)=DNE or ≠0, then ∑(an) is divergent If ∑(an) is convergent, then lim(an)=0
- Integral Test
- *must be positive, decreasing, continuous 1)∫f(x)dx is convergent, then ∑an is convergent 2)√f(x)dx is divergent, then ∑(an) is divergent
- P-test
- ∑1/(nP) is convergent if p>1 and divergent if p<1
- Comparison Test
- ∑an & ∑bn are series w/ positive terms If ∑bn is convergent & an≤bn for all n, then ∑an is convergent If ∑bn is divergent & an≥bn for all n, then ∑an is divergent
- Limit Comparison Test
- ∑an & ∑bn are positive terms lim (an/bn) = c, where c is a finite number & c>0, then both diverge or converge
- Alternating Series Test
- ∑(-1)bn must satisfy: 1) b(n+1)≤bn for all n 2) lim bn = 0 then the series is convergent
- Absolutely Convergent
- ∑an & ∑lanl are convergent
- Ratio Test
- Given a series ∑an, we have: lim l a(n+1)/an l 1) L<1, then ∑an is abs. conv. 2) L>1 or ∞, then ∑an is div. 3) 1, the test is inconclusive
- Root Test
- Given a series ∑an, we have: lim√l(an)l = 1) L<1, then ∑an is abs. conv. 2) L>1 or ∞, then ∑an is div. 3) 1, the test is inconclusive
- Geometric Series
- ∑ar(n-1)= a+ar+ar2+ar3+... is convergent if lrl<1 and its sum is ∑ar(n-1)= a/(1-r) If lrl ≥1, the series is divergent
- 0!=?
- 1
- 1! = ?
- 1
- Sequence
- a list of numbers in a definite order
- If lim f(x) = L, then...
- f(n) = L
- If l(an)l --> 0, then...
- an --> 0
- If an ≤bn≤cn for n≥n0, and lim= an = L = lim Cn Then...
- lim bn = L
- Fibinoci Sequence
- {fn} = {1,1,2,3,5,8,13,21,...}
- Maclaurin Series 1/(1-x) =?
- ∑x^n= 1+x+x2+x3+x4+... R=1
- Maclaurin Series e^x =
- ∑xn/n! = 1+(x/1!)+(x2/2!)+(x3/3!)+(x4/4!)+...
- Maclaurin Series sin(x) =
- ∑(-1)n* [x^(2n+1)]/(2n+1!) = x-(x3/x!)+(x5/5!)-(x7/7!)+... R=∞
- Maclaurin Series cos(x) =
- ∑(-1)n* (x^2x)/(2n)! = 1-(x2/2!)+(x4/4!)-(x6/6!)+...
- Maclaurin Series arctan(x) =
- ∑(-1)n* [x^(2n+1)]/(2n+1) = x - (x3/x)+(x5/5)-(x7/7)+...
- Maclaurin Series (1+x)^k =
- ∑(k/n)x^n = 1+kx+[k(k-1)/2!]x2+[k(k-1)(k-2)/3!]x3+...