From 1092511943ddc1379b45f68b6958a81c6b9fc2b5 Mon Sep 17 00:00:00 2001 From: sean-fitzpatrick Date: Fri, 9 May 2025 11:13:19 -0600 Subject: [PATCH 1/9] change two dl to ol --- ptx/sec_deriv_basic_rules.ptx | 63 +++++++++++++++++------------------ 1 file changed, 30 insertions(+), 33 deletions(-) diff --git a/ptx/sec_deriv_basic_rules.ptx b/ptx/sec_deriv_basic_rules.ptx index e4c83231a..89fc36801 100644 --- a/ptx/sec_deriv_basic_rules.ptx +++ b/ptx/sec_deriv_basic_rules.ptx @@ -51,8 +51,8 @@ Derivatives of Common Functions

-

-
  • +
      +
    1. Constant Rule derivativeConstant Rule @@ -75,24 +75,30 @@
    2. - Other common functions +

      \lzoo{x}{\sin(x)} = \cos(x)

      +
      3. +

    3. \lzoo{x}{\cos(x)} = {-\sin(x)}

      +
      4. +

    4. \lzoo{x}{e^x} = e^x

      +
      5. +

    5. \lzoo{x}{\ln(x)} = \frac{1}{x}, for x \gt 0.

      6. -
    • + derivativebasic rules @@ -293,14 +299,13 @@

    Let f and g be differentiable on an open interval I and let c be a real number. Then: -

    +
    1. Sum/Difference Rule

      - - \lzoo{x}{f(x) \pm g(x)} \amp= \lzoo{x}{f(x)} \pm \lzoo{x}{g(x)} - \amp= \fp(x)\pm g'(x) - + + \lzoo{x}{f(x) \pm g(x)} \lzoo{x}{f(x)} \pm \lzoo{x}{g(x)} = \fp(x)\pm g'(x) + derivativeSum/Difference Rule Sum/Difference Ruleof derivatives @@ -311,17 +316,16 @@

    2. Constant Multiple Rule

      - - \lzoo{x}{c\cdot f(x)} \amp= c\cdot\lzoo{x}{f(x)} - \amp = c\cdot\fp(x) - . + + \lzoo{x}{c\cdot f(x)} = c\cdot\lzoo{x}{f(x)} = c\cdot\fp(x) + . derivativeConstant Multiple Rule Constant Multiple Ruleof derivatives

      3. -
    +

    @@ -521,21 +525,7 @@ Higher Order Derivatives - - +

    The derivative of a function f is itself a function, therefore we can take its derivative. @@ -583,7 +573,7 @@

    @@ -669,9 +668,7 @@ What do higher order derivatives mean? What is the practical interpretation? - - - derivativehigher order!interpretation + derivativehigher orderinterpretation

    From 12e0f3ad209371bbe59e1ca0cde9806590e83023 Mon Sep 17 00:00:00 2001 From: sean-fitzpatrick Date: Fri, 9 May 2025 12:25:20 -0600 Subject: [PATCH 2/9] Greg's edits - derivatives --- ptx/sec_deriv_chainrule.ptx | 162 ++++++++++++++--------------- ptx/sec_deriv_implicit.ptx | 23 ++-- ptx/sec_deriv_interpret.ptx | 4 +- ptx/sec_deriv_intro.ptx | 14 ++- ptx/sec_deriv_inverse_function.ptx | 12 +-- 5 files changed, 109 insertions(+), 106 deletions(-) diff --git a/ptx/sec_deriv_chainrule.ptx b/ptx/sec_deriv_chainrule.ptx index a7b3f39ab..1989cfc8e 100644 --- a/ptx/sec_deriv_chainrule.ptx +++ b/ptx/sec_deriv_chainrule.ptx @@ -166,7 +166,7 @@

    The statement of takes care to ensure - this problem does not arise, but our focus is more on the derivative result than + this problem does not arise. We will focus more on the derivative result than on the domain/range conditions.

    @@ -1034,90 +1034,86 @@ That is, the rate at which the u gear makes a revolution is twice as fast as the rate at which the x gear makes a revolution.

    - - - -

    - Using the terminology of calculus, - the rate of u-change, with respect to x, - is \lz{u}{x} = 2. -

    - -

    - Likewise, every revolution of u causes 3 revolutions of y: - \lz{y}{u} = 3. - How does y change with respect to x? - For each revolution of x, - y revolves 6 times; that is, - - \frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx} = 2\cdot 3 = 6 - . -

    - -

    - We can then extend the with more variables by adding more gears to the picture. -

    -     + +
    + A series of gears to demonstrate the Chain Rule. Note how \lz{y}{x} = \lz{y}{u}\cdot\lz{u}{x} + + + + 3 gears of various sizes demonstrating the chain rule. + + + Three gears, connected in the order x,u,y. + x is the largest gear, having 36 teeth. It is rotating counter-clockwise. + u is connected to x, and it has 18 teeth. To the left of the connection is \frac{du}{dx} = 2. + y is connected to u, and it has 6 teeth. Below the connection is \frac{dy}{du}=3. + To the right of the gears is the expression \frac{dy}{dx} = 6. + + + + \begin{tikzpicture}[>=latex] + + \begin{scope}[shift={(0,-200pt)}] + \begin{scope} + \foreach \x in {0,1,2,...,35} + {% + \draw [rotate around={{\x*10}:(0,0)}] (60pt,0)--(65pt,0) arc (0:{4.}:65pt); + \draw [rotate around={{\x*10+4.}:(0,0)}] (65pt,0) -- (60pt,0) arc (0:6:60pt); + } + \draw [->] (40pt,0) arc (0:170:40pt); + \draw (0,0) node {$x$}; + \end{scope} + + \begin{scope}[shift={(4.5pt,-99pt)}] + \foreach \x in {0,1,2,...,17} + {% + \draw [rotate around={{\x*20}:(0,0)}] (30pt,0)--(35pt,0) arc (0:{9}:35pt); + \draw [rotate around={{\x*20+9}:(0,0)}] (35pt,0) -- (30pt,0) arc (0:11:30pt); + } + \draw [->] (0,25pt) arc (90:-80:25pt); + \draw (0,0) node {$u$}; + \draw (45pt,-30pt) node {\small $\ds \frac{dy}{du} = 3$}; + \draw (-50pt,30pt) node {\small $\ds \frac{du}{dx} = 2$}; + \draw (60pt,40pt) node {\small $\ds \frac{dy}{dx} = 6$}; + \end{scope} + + \begin{scope}[shift={(53.5pt,-100pt)}] + \foreach \x in {0,1,2,...,5} + {% + \draw [rotate around={{\x*60}:(0,0)}] (10pt,0)--(15pt,0) arc (0:{29}:15pt); + \draw [rotate around={{\x*60+29}:(0,0)}] (15pt,0) -- (10pt,0) arc (0:31:10pt); + } + \draw [->] (0,-20pt) arc (-90:70:20pt); + \draw (0,0) node {$y$}; + \end{scope} + \end{scope} + \end{tikzpicture} + + + + +
    + +

    + Using the terminology of calculus, + the rate of u-change, with respect to x, + is \lz{u}{x} = 2. +

    -
    - A series of gears to demonstrate the Chain Rule. Note how \lz{y}{x} = \lz{y}{u}\cdot\lz{u}{x} - - - - 3 gears of various sizes demonstrating the chain rule. - - - Three gears, connected in the order x,u,y. - x is the largest gear, having 36 teeth. It is rotating counter-clockwise. - u is connected to x, and it has 18 teeth. To the left of the connection is \frac{du}{dx} = 2. - y is connected to u, and it has 6 teeth. Below the connection is \frac{dy}{du}=3. - To the right of the gears is the expression \frac{dy}{dx} = 6. - - - - \begin{tikzpicture}[>=latex] - - \begin{scope}[shift={(0,-200pt)}] - \begin{scope} - \foreach \x in {0,1,2,...,35} - {% - \draw [rotate around={{\x*10}:(0,0)}] (60pt,0)--(65pt,0) arc (0:{4.}:65pt); - \draw [rotate around={{\x*10+4.}:(0,0)}] (65pt,0) -- (60pt,0) arc (0:6:60pt); - } - \draw [->] (40pt,0) arc (0:170:40pt); - \draw (0,0) node {$x$}; - \end{scope} - - \begin{scope}[shift={(4.5pt,-99pt)}] - \foreach \x in {0,1,2,...,17} - {% - \draw [rotate around={{\x*20}:(0,0)}] (30pt,0)--(35pt,0) arc (0:{9}:35pt); - \draw [rotate around={{\x*20+9}:(0,0)}] (35pt,0) -- (30pt,0) arc (0:11:30pt); - } - \draw [->] (0,25pt) arc (90:-80:25pt); - \draw (0,0) node {$u$}; - \draw (45pt,-30pt) node {\small $\ds \frac{dy}{du} = 3$}; - \draw (-50pt,30pt) node {\small $\ds \frac{du}{dx} = 2$}; - \draw (60pt,40pt) node {\small $\ds \frac{dy}{dx} = 6$}; - \end{scope} - - \begin{scope}[shift={(53.5pt,-100pt)}] - \foreach \x in {0,1,2,...,5} - {% - \draw [rotate around={{\x*60}:(0,0)}] (10pt,0)--(15pt,0) arc (0:{29}:15pt); - \draw [rotate around={{\x*60+29}:(0,0)}] (15pt,0) -- (10pt,0) arc (0:31:10pt); - } - \draw [->] (0,-20pt) arc (-90:70:20pt); - \draw (0,0) node {$y$}; - \end{scope} - \end{scope} - \end{tikzpicture} +

    + Likewise, every revolution of u causes 3 revolutions of y: + \lz{y}{u} = 3. + How does y change with respect to x? + For each revolution of x, + y revolves 6 times; that is, + + \frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx} = 2\cdot 3 = 6 + . +

    -     - - -
    -     +

    + We can then extend the with more variables by adding more gears to the picture. +

    It is difficult to overstate the importance of the . diff --git a/ptx/sec_deriv_implicit.ptx b/ptx/sec_deriv_implicit.ptx index eabb5b2a7..7ad209131 100644 --- a/ptx/sec_deriv_implicit.ptx +++ b/ptx/sec_deriv_implicit.ptx @@ -504,9 +504,9 @@ \begin{tikzpicture} - \begin{axis}[xmin=-1.95,xmax=1.99, + \begin{axis}[xmin=-2.05,xmax=2.05, ymin=-1.95,ymax=1.95,] - \addplot+[smooth,infinite] coordinates {(-1.91,-1.57) (-1.82,-1.61) (-1.68,-1.61) (-1.7,-1.49) (-1.74,-1.38)(-1.76,-1.28) (-1.64,-1.26) (-1.54,-1.29) (-1.36,-1.35) (-1.2,-1.42)(-1.09,-1.46) (-0.989,-1.49) (-0.86,-1.5) (-0.714,-1.45)(-0.566,-1.35) (-0.327,-1.18) (-0.15,-1.08)(0.107,-0.947) (0.216,-0.895) (0.332,-0.832) (0.429,-0.755) (0.464,-0.623) (0.429,-0.52) (0.308,-0.335)(0.142,-0.144) (-0.071,0.0714) (-0.239,0.261) (-0.336,0.45) (-0.336,0.622) (-0.278,0.757) (-0.17,0.884) (-0.046,0.975)(0.0714,1.03) (0.216,1.07) (0.48,1.09) (0.714,1.07) (0.929,1.05)(1.15,1.07) (1.24,1.13) (1.27,1.25) (1.28,1.37) (1.29,1.49)(1.36,1.58) (1.47,1.57) (1.59,1.52) (1.68,1.48) (1.79,1.43)(1.89,1.39) (1.98,1.36) (1.98,1.6)}; + \addplot+[smooth,infinite] coordinates {(-2.03,-1.513)(-1.988,-1.536)(-1.972,-1.543)(-1.879,-1.586)(-1.786,-1.621)(-1.68,-1.572)(-1.707,-1.464)(-1.749,-1.357)(-1.739,-1.261)(-1.621,-1.264)(-1.5,-1.299)(-1.348,-1.357)(-1.179,-1.428)(-1.074,-1.467)(-0.9643,-1.494)(-0.8524,-1.495)(-0.6789,-1.429)(-0.5,-1.303)(-0.3182,-1.179)(-0.1408,-1.073)(0,-1.)(0.1397,-0.9317)(0.2397,-0.8826)(0.3496,-0.8214)(0.4457,-0.7329)(0.4619,-0.6071)(0.4189,-0.5)(0.276,-0.2954)(0.07143,-0.07178)(-0.07143,0.07183)(-0.2684,0.3031)(-0.3426,0.4855)(-0.3214,0.6721)(-0.2591,0.7857)(-0.1429,0.9072)(0,1.)(0.1009,1.042)(0.2857,1.083)(0.5178,1.089)(0.7793,1.065)(0.9789,1.05)(1.179,1.08)(1.248,1.145)(1.275,1.275)(1.279,1.393)(1.3,1.514)(1.393,1.582)(1.5,1.558)(1.607,1.514)(1.712,1.467)(1.804,1.426)(1.856,1.405)}; \end{axis} \end{tikzpicture} @@ -550,9 +550,9 @@ \begin{tikzpicture} - \begin{axis}[xmin=-1.95,xmax=1.99, + \begin{axis}[xmin=-2.05,xmax=2.05, ymin=-1.95,ymax=1.95,] - \addplot+[infinite,smooth] coordinates {(-1.91,-1.57) (-1.82,-1.61) (-1.68,-1.61) (-1.7,-1.49) (-1.74,-1.38)(-1.76,-1.28) (-1.64,-1.26) (-1.54,-1.29) (-1.36,-1.35) (-1.2,-1.42)(-1.09,-1.46) (-0.989,-1.49) (-0.86,-1.5) (-0.714,-1.45)(-0.566,-1.35) (-0.327,-1.18) (-0.15,-1.08)(0.107,-0.947) (0.216,-0.895) (0.332,-0.832) (0.429,-0.755) (0.464,-0.623) (0.429,-0.52) (0.308,-0.335)(0.142,-0.144) (-0.071,0.0714) (-0.239,0.261) (-0.336,0.45) (-0.336,0.622) (-0.278,0.757) (-0.17,0.884) (-0.046,0.975)(0.0714,1.03) (0.216,1.07) (0.48,1.09) (0.714,1.07) (0.929,1.05)(1.15,1.07) (1.24,1.13) (1.27,1.25) (1.28,1.37) (1.29,1.49)(1.36,1.58) (1.47,1.57) (1.59,1.52) (1.68,1.48) (1.79,1.43)(1.89,1.39) (1.98,1.36) (1.98,1.6)}; + \addplot+[infinite,smooth] coordinates {(-2.03,-1.513)(-1.988,-1.536)(-1.972,-1.543)(-1.879,-1.586)(-1.786,-1.621)(-1.68,-1.572)(-1.707,-1.464)(-1.749,-1.357)(-1.739,-1.261)(-1.621,-1.264)(-1.5,-1.299)(-1.348,-1.357)(-1.179,-1.428)(-1.074,-1.467)(-0.9643,-1.494)(-0.8524,-1.495)(-0.6789,-1.429)(-0.5,-1.303)(-0.3182,-1.179)(-0.1408,-1.073)(0,-1.)(0.1397,-0.9317)(0.2397,-0.8826)(0.3496,-0.8214)(0.4457,-0.7329)(0.4619,-0.6071)(0.4189,-0.5)(0.276,-0.2954)(0.07143,-0.07178)(-0.07143,0.07183)(-0.2684,0.3031)(-0.3426,0.4855)(-0.3214,0.6721)(-0.2591,0.7857)(-0.1429,0.9072)(0,1.)(0.1009,1.042)(0.2857,1.083)(0.5178,1.089)(0.7793,1.065)(0.9789,1.05)(1.179,1.08)(1.248,1.145)(1.275,1.275)(1.279,1.393)(1.3,1.514)(1.393,1.582)(1.5,1.558)(1.607,1.514)(1.712,1.467)(1.804,1.426)(1.856,1.405)}; \addplot [tangentline,domain=-.5:.5] {-x}; \addplot [tangentline,domain=-.5:.5] {0.5*x+1}; @@ -904,18 +904,21 @@ It is well-defined for x \gt 0 and we might be interested in finding equations of lines tangent and normal to its graph. How do we take its derivative? - logarithmic differentiation - derivative - logarithmic - + logarithmic differentiation + derivativelogarithmic

    diff --git a/ptx/sec_deriv_interpret.ptx b/ptx/sec_deriv_interpret.ptx index fb96b7543..3ea9245e1 100644 --- a/ptx/sec_deriv_interpret.ptx +++ b/ptx/sec_deriv_interpret.ptx @@ -285,7 +285,7 @@ s could measure the height of a projectile or the distance an object has traveled.