theorem: the less you know, the more money you make

12

Theorem: The less you Theorem: The less you know, the more money you know, the more money you make make Proof Proof : : We know that We know that a) 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 Physics and from Physics c) Power = Work / Time (P=W/T) c) Power = Work / Time (P=W/T) By substitution, K = W/M By substitution, K = W/M Rearrange the equation and M = W/K, or Rearrange the equation and M = W/K, or Money = Work/Knowledge Money = Work/Knowledge From this equation, it follows that as From this equation, it follows that as knowledge goes to 0, money goes to infinity. knowledge goes to 0, money goes to infinity.

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Theorem: The less you know, the more money you make. Proof : We know that a) Time is Money (T=M) b) Knowledge is Power (K=P) and from Physics c) Power = Work / Time (P=W/T) By substitution, K = W/M Rearrange the equation and M = W/K, or Money = Work/Knowledge - PowerPoint PPT Presentation

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Page 1: Theorem: The less you know, the more money you make

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.

Page 2: Theorem: The less you know, the more money you make

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

Page 3: Theorem: The less you know, the more money you make

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

Page 4: Theorem: The less you know, the more money you make

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

Page 5: Theorem: The less you know, the more money you make

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

Page 6: Theorem: The less you know, the more money you make

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

Page 7: Theorem: The less you know, the more money you make

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

Page 8: Theorem: The less you know, the more money you make

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

Page 9: Theorem: The less you know, the more money you make

Page 10: Theorem: The less you know, the more money you make

Page 11: Theorem: The less you know, the more money you make

Page 12: Theorem: The less you know, the more money you make

Now for a worksheet

http://archives.math.utk.edu/visual.calculus/2/trig.1/index.html

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