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sDifferential CalculusThe mathematics of teeny little amounts

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The Slope as a Limit

Suppose we have measured the position versus time of a bicyclist using ultra-fancy laser equipment, etc.

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Cyclist Position versus Time

• Plotting the data gives

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.50.0050.00100.00150.00200.00250.00300.00

position

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Cyclist versus Time

• From inspection we can immediately see that

• Now we ask the question we’re really interested in: what is her speed versus time?

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Cyclist versus Time

Let’s be specific: what is her speed at the time 2.7 seconds?

Speed is

How big should the be?

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Change in Time

Let’s compute some values from our data.

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change in time pseudo-speed1 31.450000

0.5 35.2000000.25 36.137500

0.125 36.3718750.0625 36.430469

0.01 36.4495000.001 36.449995

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.50.0050.00100.00150.00200.00250.00300.00

position

slope here

We Want the Slope

What we really need is the (exact) slope of the position curve at t = 2.7 seconds.

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Limits

We were computing

where stands for some (small) amount of time.

The true speed is the limit of as goes to zero. We write this as

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Limits: How to Actually Compute One

Compute:

Note that )2

2)

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Taking It to the Limit

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And The Answer Is

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𝑠 (2.7 )=27

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What Have We Done?

We’ve computed

or

the (first) derivative of .

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What a Lot of Work, Eh?

• We would be crazy don’t have to do this every time.

• Mostly we use a small set of standard results from either memory (or a book) plus a few composition rules.

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Specific Derivatives

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Exponentials and Logarithms

is the transcendental number

raised to the real power .

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Exponentials and Logarithms

It is effectively defined by the differential equation

the solution to which is

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Exponentials and Logarithms

0 0.2 0.4 0.6 0.8 11.25 1.5 1.7

5 2

2.3333

2.6666

0

2

4

6

8

10

12

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e**t

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Exponentials and Logarithms

The (natural) logarithm, or , is the inverse of the exponential.

It’s defined by

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Exponentials and Logarithms

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0.5 0.75 1

1.25 1.5 1.7

5 22.3332.666 3 3.5 4 4.5 5 5.5 6

-1

-0.5

0

0.5

1

1.5

2

log(t)

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Composition Rules

Composition rules are ways to break certain forms of complicated derivatives into expressions containing simpler derivatives.

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Sum and Product Composition Rules

• Sum rule

• Product rule

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The Chain Composition Rule

• Suppose . We sometimes call the composition of f and g.• Then

• looks confusing. Because it is.

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More Chain Rule

To compute for some fixed t, we compute

1. the number and2. the number

Then compute the number

The result is

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Takeaway

• Derivatives are the (exact) slope of a curve.

• They are computed from a difference as the change in the arguments goes to zero.

• They are almost always computed by applying composition rules to the derivatives of some simple functions, or by using a reference book.

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sAnd That’s Differential Calculus

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