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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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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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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
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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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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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