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Calculus Final
| Question | Answer |
|---|---|
| n-th term / divergence test | 1. check if the limit of the sequence goes to zero 2. if it does, the series is inconclusive -> use another test if it does not, the series diverges |
| p-test | if p is less than or equal to 1, the series diverges if p > 1, the series converges |
| integral test | 1. change the series into a function 2. if f is eventually continuous, positive, and decreasing, then the series can be an integral and will either converge or diverge |
| geometric series test | in geometric series form, if abs value of r < 1, the series converges if abs value of r is greater than or equal to 1, the series diverges Sum = a0/1-r |
| direct comparison test | check if an < bn 1. if bn converges, an converges 2. if an diverges, bn diverges |
| limit comparison test | if the limit of an/bn is a finite number, then an and bn both converge or diverge if the limit of an/bn = 0, then bn converges and an converges if the limit of an/bn equals infinity, bn diverges and an diverges |
| alternating series test | if an (without the alternating part) are positive and decreasing and the limit as n goes to infinity is 0, then the series converges |
| ratio test | p=limit as n goes to infinity |(an+1)/(an)| if p is from 0 to 1, the series converges ABSOLUTELY if p > 1 or infinity, the series diverges if p = 1, the test is inconclusive |
| root test | p=limit as n goes to infinity |an|^1/n (root) if p is from 0 to 1, the series converges ABSOLUTELY if p > 1 or infinity, the series diverges if p = 1, the test is inconclusive |
| absolute convergence test | if the absolute value of the terms of a series form a convergent series, then the original series will converge absolutely |
| three pythagorean identities | sin^(2)x + cos^(2)x = 1 1 + tan^(2)x = sec^(2)x 1 + cot^(2)x = csc^(2)x |
| double angle formulas | sin(2x)=2sinxcosx cos(2x) =cos^(2)x-sin^(2)x =1-2sin^(2)x =2cos^(2)x-1 |
| power reduction formulas | sin^(2)x=1/2[1-cos2x] cos^(2)x=1/2[1+cos2x] |
| trig substitution formulas | integral of sqrt(a^(2)-x^(2)) -> x = a sin(theta) integral of sqrt(a^(2)+x^(2)) -> x = atan(theta) integral of sqrt(x^(2)-a^(2)) -> x = asec(theta) |
| hooke's law | F(x)=-kx -k is the spring constant x is the displacement |
| checking irreducible quality with discriminant | D=b^(2)-4ac 1. if D < 0 -> no real roots, irreducible over R (real roots) 2. if D is not a perfect square -> irreducible over Q (rational roots) ...just check for roots? |
| area of a triangle | 1/2(magnitude of the sides that form the largest angle) |
| vector length | magnitude of vector |
| cross product | 1. form 3x3 matrix (i,j,k on top) 2. i[]-j[]+k[] |
| how to find vectors | subtract from each other...if it was AC then you subtract C-A |
| dot product | 1. multiply vector 2. add up the numbers DOT PRODUCT IS A SCALAR |
| partial fraction decompositon case 1: distinct linear factors | examples of distinct linear factors (x, x+1,4x) just do it normally...easiest way |
| partial fraction decomposition case 2: distinct quadratic factors w/ irreducible quadratic | examples of irreducible quadratic (x^(2)+2) use Bx+C |
| partial fraction decomposition case 3: distinct quadratic factors | examples of distinct quadratic factors (x^(2)+2,x^(2)+x+1) it repeats...it will go up so B is over x and C is over x^(2) |
| partial fraction decomposition case 4: distinct quadratic factors with a high exponent | it will go up again... if x^(4) was in the denominator, it will be broken up to be A/x + B/x^(2) + C/x^(3) + D/x^(4) |
| partial fraction decomposition case 5: repeated linear factors | it will be the same as going up... if (x+2)^(2) was in the denominator, it will be broken up to be |
Created by:
ephemeral