Math

Sequence

A list of numbers with a pattern.

Series

The sum of numbers in a sequence. Take that list of numbers separated by commas and replace it with a +.

Infinite Sequence

Sequences that have infinite terms

Common Log

y=log₁₀x

Natural Log

y=lnx OR y=logₑx

Partial Fractions

1. Check the degree (highest exponent): The degree of the numerator must be strictly less than the degree of the denominator. If it isn't (e.g., \(x^{3}\) on bottom and \(x^{2}\) on top), you must use polynomial long division first to simplify it.

2. Factor the denominator: Break the denominator down into its simplest multiplied components (factors).

3. Set up the fraction template: Assign unknown constants (usually \(A, B, C\), etc.) for the numerators over each factor.

4. Solve for the constants: Multiply the entire equation by the common denominator to eliminate the fractions, then equate the coefficients or plug in smart values for \(x\) to solve for $A, B$, and \(C\).

• Repeated Linear Factors: If a factor repeats, like \((x-a)^2\), you must account for each power: \(\frac{A}{x-a} + \frac{B}{(x-a)^2}\).

1. For each irreducible quadratic factor 𝑎⁢𝑥2+𝑏⁢𝑥+𝑐 that 𝑄⁡(𝑥) contains, the decomposition must include
   

   𝐴⁢𝑥+𝐵𝑎⁢𝑥2+𝑏⁢𝑥+𝑐.

2. For each repeated irreducible quadratic factor (𝑎⁢𝑥2+𝑏⁢𝑥+𝑐)𝑛, the decomposition must include
   

   𝐴1⁢𝑥+𝐵1𝑎⁢𝑥2+𝑏⁢𝑥+𝑐+𝐴2⁢𝑥+𝐵2(𝑎⁢𝑥2+𝑏⁢𝑥+𝑐)2+⋯+𝐴𝑛⁢𝑥+𝐵𝑛(𝑎⁢𝑥2+𝑏⁢𝑥+𝑐)𝑛.

Ex: ∫x²+3x+1/(x+2)(x-3)²(x²+4)² → A/x+2 + B/(x-3)^1 + C/(x-3)² + Dx+E/(x²+4)^1 + Fx+G/(x²+4)²

Double angle identity for sin2x

2sinxcosx

Double angle identity for cos2x

cos²x-sin²x, 2cos²x-1, 1-2sin²x

Double angle identity for tan2x

2tanx/1-tan²x

Power Reduction for sin²x

1-cos2x/2

Power Reduction for cos²x

1+cos2x/2

Power Reduction for tan²x

1-cos2x/1+cos2x

Power Reduction for csc²x

2/1-cos2x

Power Reduction for sec²x

2/1+cos2x

Power Reduction for cot²x

1+cos2x/1-cos2x

Product to Sum Formula for sinAcosB

(1/2) [sin(A+B) + sin(A-B)]

Product to Sum Formula for cosAsinB

(1/2) [sin(A+B) - sin(A-B)]

Product to Sum Formula for cosAcosB

(1/2) [cos(A+B) + cos(A-B)]

Product to Sum Formula for sinAsinB

(1/2) [cos(A-B) - cos(A+B)]

When you integrate a product of cosine and sin, where the exponent for one of the functions is 1, and the other is >1

let u=the trig function with the higher power

If you have the integral of cos²x sin³x dx

Take the one with the odd exponent and split it up, and then use pythag identity on what you just split up

-cosx + C

∫sinx dx

sinx + C

∫cosx dx

tanx + C

∫sec²x dx

-cotx + C

∫csc²x dx

secx + C

∫secx tanx dx

-cscx + C

∫cscx cotx dx

ln|secx + tanx| +C

∫secx dx

ln|secx| +C

∫tanx dx

PSST Rule

Sec → Sec ← Tan
Derivative of secx = secxtanx
Integral of secxtanx = secx

Derivative of tanx = sec²x
Integral of sec²x = tanx

CCC Rule

Csc → -Csc ← Cot
Derivative of cscx = -cscxcotx
Integral of -cscxcotx = cscx

Derivative of cotx = -csc²x
Integral of -csc²x = cotx

Derivate of ln|x|

= 1/x

Product Rule for Log

logₙ(xy)=logₙx+logₙy

Quotient Rule for Log

logₙ(x/y) = logₙx-logₙy

Power Rule for Log

logₙ(x^b) = b logₙx

Change of Base Rule for Log

logₙx = log(sub c) x / log (sub x)b

Zero Rule for Log

logₙb(1) = 0

Pythagorean Identities

csc²x + sec²x = 1

sin²x + cos²x = 1

tan²x + 1 = sec²x

Arithmetic Sequences

add a d(difference) to each term

formula: a₁+(n-1)d

find d: a(sub)n+1 - a(sub)n

Geometric Sequence

multiply a common ratio

find r: r=a(sub)n+1/an

Alternating Sequences

goes from positive term to negative, pos, neg, pos, neg, …

Recursive Formula

formula: an=asub(n-1) +d

Explicit Formula for all Geometric

an=a1(r^n-1)

Recursive Formula for Geometric

an=a(sub)n-1 * r

Convergent Sequence

Given a sequence {an} if the terms an becomes artbitrarily close to a finite number L, as n becomes sufficiently lathe, we say {an} is a convergent sequence and L is the lmit of the sequence, In this case we write the lim as n goes to inifnity of an = L.

if the sequence is not convergent we say it is divergent

L’Hopital Rule

Used to evaluate limit of a function is indeterminate (0/0 or oo/oo or any combination), then you can use this rule to find the derivative of the numerator by itself, and the derivative of the denominator by itself. do not use quotient rule!!!

Finding horizontal asymptope of px^a/qx^b

if a>b, no HA

if b>a, HA @ 0

if a=b, HA @ p/q