daviddoji/CalculusWithJuliaNotes.jl
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edits; simplify caching

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jverzani
2022-06-06 11:43:19 -04:00
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@@ -128,21 +128,16 @@ segments, and we could have derived the graph of `x` from that of
`speed`, just using the simple formula relating distance, rate, and
time.
```julia;echo=false
note("""
We were pretty loose with some key terms. There is a distinction
between "speed" and "velocity", this being the speed is the absolute
value of velocity. Velocity incorporates a direction as well as a
magnitude. Similarly, distance traveled and change in position are not
the same thing when there is back tracking involved. The total
distance traveled is computed with the speed, the change in position
is computed with the velocity. When there is no change of sign, it is
a bit more natural, perhaps, to use the language of speed and
distance.
""")
```
!!! note
We were pretty loose with some key terms. There is a
distinction between "speed" and "velocity", this being the speed
is the absolute value of velocity. Velocity incorporates a
direction as well as a magnitude. Similarly, distance traveled and
change in position are not the same thing when there is back
tracking involved. The total distance traveled is computed with
the speed, the change in position is computed with the
velocity. When there is no change of sign, it is a bit more
natural, perhaps, to use the language of speed and distance.
##### Example: Galileo's ball and ramp experiment
@@ -326,15 +321,13 @@ ImageFile(imgfile, caption)
The tangent line is not just a line that intersects the graph in one
point, nor does it need only intersect the line in just one point.
```julia; echo=false
note("""
This last point was certainly not obvious at
first. [Barrow](http://www.maa.org/sites/default/files/0746834234133.di020795.02p0640b.pdf),
who had Newton as a pupil, and was the first to sketch a proof of part
of the Fundamental Theorem of Calculus, understood a tangent line to
be a line that intersects a curve at only one point.
""")
```
!!! note
This last point was certainly not obvious at
first. [Barrow](http://www.maa.org/sites/default/files/0746834234133.di020795.02p0640b.pdf),
who had Newton as a pupil, and was the first to sketch a proof of
part of the Fundamental Theorem of Calculus, understood a tangent
line to be a line that intersects a curve at only one point.
##### Example
@@ -791,7 +784,7 @@ Find the derivative of ``f(x) = \sqrt{1 - x^2}``. We identify the composition of
```math
\begin{align*}
f(x) &=\sqrt{x} & g(x) &= 1 - x^2 \\
f(x) &=\sqrt{x} = x^{1/2} & g(x) &= 1 - x^2 \\
f'(\square) &=(1/2)(\square)^{-1/2} & g'(x) &= -2x
\end{align*}
```
@@ -864,8 +857,34 @@ f(g(a+h)) - f(g(a)) - f'(g(a)) g'(a) h = f'(g(a))\epsilon_g(h))h + \epsilon_f(h'
where $\epsilon(h)$ combines the above terms which go to zero as $h\rightarrow 0$ into one. This is
the alternative definition of the derivative, showing $(f\circ g)'(a) = f'(g(a)) g'(a)$ when $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$.
##### The "chain" rule
##### More examples
The chain rule name could also be simply the "composition rule," as that is the operation the rule works for. However, in practice, there are usually *multiple* compositions, and the "chain" rule is used to chain together the different pieces. To get a sense, consider a triple composition ``u(v(w(x())))``. This will have derivative:
```math
\begin{align*}
[u(v(w(x)))]' &= u'(v(w(x))) \cdot [v(w(x))]' \\
&= u'(v(w(x))) \cdot v'(w(x)) \cdot w'(x)
\end{align*}
```
The answer can be viewed as a repeated peeling off of the outer
function, a view with immediate application to many compositions. To
see that in action with an expression, consider this derivative
problem, shown in steps:
```math
\begin{align*}
[\sin(e^{\cos(x^2-x)})]'
&= \cos(e^{\cos(x^2-x)}) \cdot [e^{\cos(x^2-x)}]'\\
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot [\cos(x^2-x)]'\\
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot (-\sin(x^2-x)) \cdot [x^2-x]'\\
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot (-\sin(x^2-x)) \cdot (2x-1)\\
\end{align*}
```
##### More examples of differentiation
Find the derivative of $x^5 \cdot \sin(x)$.
@@ -928,14 +947,10 @@ and finally,
diff(sin(x)^5)
```
```julia; echo=false
note("""
The `diff` function can be called as `diff(ex)` when there is just one
free variable, as in the above examples; as `diff(ex, var)` when there are parameters in the
expression.
"""
)
```
!!! note
The `diff` function can be called as `diff(ex)` when there is
just one free variable, as in the above examples; as `diff(ex,
var)` when there are parameters in the expression.
----