Precalculus Toolkit

I thought I'd start by compiling a list of some of the useful algebraic and trigonometric rules that come up while solving differential equations. Many times, when students get stuck on a problem, it's not because they were unable to do the calculus. It's because they had trouble with the algebraic acrobatics that is sometimes necessary to get an answer into the proper form.

Sometimes, performing an operation on one representation of an equation would be very difficult or impossible using the methods you learn in this class. However, if you represent the equation in a different, equivalent way, the task might become much easier.

The list below is meant to refresh you on some of these methods, so that you won't spend hours beating your head against the wall to solve a problem that should be able to be solved easily. Also keep in mind that you might have to use the rules backwards, too!

Exponents:

1. Multiplying same base, different exponent: add the exponents
   (x^u)(x^v) = x^u+v
   Example: (x^4y)(x^y+6) = x^{4y + y +6} = x^{5y + 6}


2. Multiplying different base, same exponent: multiply the bases
   (x^u)(y^u) = (xy)^u
   Example: (x^3z+4)(y^3z+4) = (xy)^3z+4
    Please note that this rule DOES NOT apply if you are not multiplying! You cannot "distribute" an exponent over addition.


3. Raising an exponent to an exponent: multiply the exponents
   (x^u)^v = x^u*v
   Example:  (x^3z+4)² = x^(3z+4)(2) = x^6z+8


4. The Nth root is the same as having an exponent of 1/N
   _n√(u) = u^1/n
   Example: √(x+1) = (x+1)^1/2

Logarithms: 

1. Log of a product: add the log of each factor
   ln(u*v) = ln(u) + ln(v)
   Example: ln((x+1)(x-3)) = ln(x+1) + ln(x-3)
    
2. Log of a quotient: subtract the log of the denominator from the log of the numerator
   ln(u/v) = ln(u) - ln(v)
   Example: ln((x+1)/(x-3)) = ln(x+1) - ln(x-3)
    
3. Log of an exponential: bring the exponent out front
   ln(u^v)  =  u*ln(v)
   Example: ln((x+4)^x/5) = x/5*ln(x+4) 

Trigonometry:

1. Trig identities equal to 1
   sin²(u) + cos²(u) = 1
   sec²(u) - tan²(u) = 1
   csc²(u) - cot²(u) = 1
   Example:  sin²(2x + 3) + cos²(2x + 3) = 1
    
2. Multiple angle and power reduction trig formulas
   There are way too many to list here, but Wikipedia has a great table. These are to be used when you don't know what to do with that sin(2x) or sin³(x), and other similar situations.

Feel free to leave a comment if you think of more I should have included. You don't need to have a blogspot account to leave a comment. And as always, feel free to stop by the ARC in the lobby of Daniel's hall during my sessions to get help in person, or to email me at rachel.wigell (at) wpi (dot) edu.

Happy calculating,
Rachel