Theorem: The less you Theorem: The less you know, the more money know, the more money you makeyou make ProofProof::

We know thatWe know thata) Time is Money (T=M)a) Time is Money (T=M)b) Knowledge is Power (K=P)b) Knowledge is Power (K=P)and from Physicsand from Physicsc) Power = Work / Time (P=W/T)c) Power = Work / Time (P=W/T)

By substitution, K = W/MBy substitution, K = W/M Rearrange the equation and M = W/K, or Rearrange the equation and M = W/K, or

Money = Work/KnowledgeMoney = Work/Knowledge

From this equation, it follows that as knowledge goes From this equation, it follows that as knowledge goes to 0, money goes to infinity.to 0, money goes to infinity.

2

0

2

Consider the function siny

We could make a graph of the slope: slope

1

0

1

0

1Now we connect the dots!

The resulting curve is a cosine curve.

sin cosd

x xdx

2

0

2

We can do the same thing for cosy slope

0

1

0

1

0The resulting curve is a sine curve that has been reflected about the x-axis.

cos sind

x xdx

Derivative of y=sinxDerivative of y=sinx

Use the definition of the derivative Use the definition of the derivative

To prove the derivative of y=sinx is y’=cosx.To prove the derivative of y=sinx is y’=cosx.

lim ( ) ( )'

0

f x h f xy

h h

Hints:

sin( ) sin cosh cos sinh

lim cosh 10

0

x h x x

h h

Derivative of y=sinxDerivative of y=sinx

lim ( ) ( )'

0

f x h f xy

h h

lim sin( ) sin

0

x h x

h h

lim sin cosh cos sinh sin

0

x x x

h h

lim sin (cosh 1) cos sinh

0

x x

h h

lim lim lim limcosh 1 sinhsin cos0 0 0 0

x xh h h hh h

lim cosh 1 sinhsin cos

0x x

h h h

sin 0 cos 1x x

Shortcut: y’=cosx

The proof of the d(cosx) = -sinx is almost identical

Derivative of y=tanxDerivative of y=tanx

Use the quotient rule to show the Use the quotient rule to show the derivative is y’=secderivative is y’=sec22xx

sin

cos

xy

x sin cos

cos sin

HI x dHI x

HO x dHO x

2 2

2

cos sin'

cos

x xy

x

2

1

cos x

Shortcut: y’=sec2x

The proof of the d(cotx) = -csc2x is almost identical

Derivative of y=secxDerivative of y=secx

Use the quotient rule to show the Use the quotient rule to show the derivative is y’=secxtanxderivative is y’=secxtanx

1

cosy

x 1 0

cos sin

HI dHI

HO x dHO x

2

cos 0 1 ( sin )'

cos

x xy

x

1 sin

cos cos

x

x x

Shortcut: y’=secxtanx

The proof of the d(cscx) = -cscxcotx is almost identical

Summary of trig derivatives

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

Now for a worksheet