pure

• cosine rule: a^2 = b^2 + c^2 - 2bccosA
• area of triangle (trig):

  area = 1/2 x a x b x sinC
• integration by parts priority:

  Logs

  Algebra

  Trigonometry

  Exponentials
• unit vector (looks like a^):

  a^ = a / (magnitude of a)
• d/dx (a^x) = a^xlna
• integral of a^x = a^x / lna + C
• functions can either be one to one or many to one. they can not be many to many
• a geometric series is convergent if the magnitude of r < 1
• a sequence is increasing if u(n+1) > u(n) for all n

  a sequence is decreasing if u(n+1) < u(n) for all n
• a sequence is periodic if the terms repeat in a cycle. the order of a sequence is how many terms until the term repeats
• the inverse function of sinx is arcsinx
• a function is concave for a given interval if f''(x) <= 0 for every value of x in that interval
• a function is convex for a given interval if f''(x) >= 0 for every value of x in that interval
• a point of inflection is a point at which f''(x) changes sign
• the intergral of e^x is e^x + c
• the integral of 1/x is lnx + c
• the integral of sec^2x is tanx + c
• to integrate expressions of the form f'(x)/f(x), try lnf(x) and differentiate to check, then adjust constants
• to integrate expressions of the form f'(x)(f(x))^n, try (f(x))^n+1 and differentiate to check, then adjust constants
• if the vector a = xi+yj+zk makes an angle C with the positive x axis, then cosC = x / magnitude of a
• functions are increasing if f'(x) >= 0 for all values of x
• if f''(x) = 0, it could be a minimum, a maximum, or a point of inflection. to determine this, look at points on either side of ti
• the graph of y = e^x is a reflection of the graph y = lnx in the line y=x