Theorem: The less you know, the more money you make • Proof:We know thata) Time is Money (T=M)b) Knowledge is Power (K=P)and from Physicsc) Power = Work / Time (P=W/T) • By substitution, K = W/M • Rearrange the equation and M = W/K, or Money = Work/Knowledge • From this equation, it follows that as knowledge goes to 0, money goes to infinity.
slope Consider the function We could make a graph of the slope: Now we connect the dots! The resulting curve is a cosine curve.
slope We can do the same thing for The resulting curve is a sine curve that has been reflected about the x-axis.
Derivative of y=sinx • Use the definition of the derivative To prove the derivative of y=sinx is y’=cosx.
Derivative of y=sinx Shortcut: y’=cosx The proof of the d(cosx) = -sinx is almost identical
Derivative of y=tanx • Use the quotient rule to show the derivative is y’=sec2x The proof of the d(cotx) = -csc2x is almost identical Shortcut: y’=sec2x
Derivative of y=secx • Use the quotient rule to show the derivative is y’=secxtanx The proof of the d(cscx) = -cscxcotx is almost identical Shortcut: y’=secxtanx
Practice Time Now for a worksheet