diff --git a/ptx/appendix_back_reference.ptx b/ptx/appendix_back_reference.ptx index 016a48550..01d4d4f74 100644 --- a/ptx/appendix_back_reference.ptx +++ b/ptx/appendix_back_reference.ptx @@ -61,51 +61,51 @@
  • - \lzo{x}(\sin x)=\cos x + \lzo{x}(\sin(x))=\cos(x)
  • - \lzo{x}(\cos x)=-\sin x + \lzo{x}(\cos(x))=-\sin(x)
  • - \lzo{x}(\csc x)=-\csc x\cot x + \lzo{x}(\csc(x))=-\csc(x)\cot(x)
  • - \lzo{x}(\sec x)=\sec x\tan x + \lzo{x}(\sec(x))=\sec(x)\tan(x)
  • - \lzo{x}(\tan x)=\sec^2 x + \lzo{x}(\tan(x))=\sec^2(x)
  • - \lzo{x}(\cot x)=-\csc^2 x + \lzo{x}(\cot(x))=-\csc^2(x)
  • - \lzo{x}(\cosh x)=\sinh x + \lzo{x}(\cosh=(x))=\sinh(x)
  • - \lzo{x}(\sinh x)=\cosh x + \lzo{x}(\sinh(x))=\cosh(x)
  • - \lzo{x}(\sech x)=-\sech x\tanh x + \lzo{x}(\sech(x))=-\sech(x)\tanh(x)
  • - \lzo{x}(\tanh x)=\sech^2 x + \lzo{x}(\tanh(x))=\sech^2(x)
  • - \lzo{x}(\csch x)=-\csch x\coth x + \lzo{x}(\csch(x))=-\csch(x)\coth(x)
  • - \lzo{x}(\coth x)=-\csch^2 x + \lzo{x}(\coth(x))=-\csch^2(x)
    • @@ -114,51 +114,51 @@ Derivatives of Inverse Functions
    1. - \lzo{x}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}} + \lzo{x}(\sin^{-1}(x))=\frac{1}{\sqrt{1-x^2}}
    2. - \lzo{x}(\cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}} + \lzo{x}(\cos^{-1}(x))=\frac{-1}{\sqrt{1-x^2}}
    3. - \lzo{x}(\csc^{-1}x)=\frac{-1}{\abs{x}\sqrt{x^2-1}} + \lzo{x}(\csc^{-1}(x))=\frac{-1}{\abs{x}\sqrt{x^2-1}}
    4. - \lzo{x}(\sec^{-1}x)=\frac{1}{\abs{x}\sqrt{x^2-1}} + \lzo{x}(\sec^{-1}(x))=\frac{1}{\abs{x}\sqrt{x^2-1}}
    5. - \lzo{x}(\tan^{-1}x)=\frac{1}{1+x^2} + \lzo{x}(\tan^{-1}(x))=\frac{1}{1+x^2}
    6. - \lzo{x}(\cot^{-1}x)=\frac{-1}{1+x^2} + \lzo{x}(\cot^{-1}(x))=\frac{-1}{1+x^2}
    7. - \lzo{x}(\cosh^{-1}x)=\frac1{\sqrt{x^2-1}} + \lzo{x}(\cosh^{-1}(x))=\frac1{\sqrt{x^2-1}}
    8. - \lzo{x}(\sinh^{-1}x)=\frac1{\sqrt{x^2+1}} + \lzo{x}(\sinh^{-1}(x))=\frac1{\sqrt{x^2+1}}
    9. - \lzo{x}(\sech^{-1}x)=\frac{-1}{x\sqrt{1-x^2}} + \lzo{x}(\sech^{-1}(x))=\frac{-1}{x\sqrt{1-x^2}}
    10. - \lzo{x}(\csch^{-1}x)=\frac{-1}{\abs{x}\sqrt{1+x^2}} + \lzo{x}(\csch^{-1}(x))=\frac{-1}{\abs{x}\sqrt{1+x^2}}
    11. - \lzo{x}(\tanh^{-1}x)=\frac1{1-x^2} + \lzo{x}(\tanh^{-1}(x))=\frac1{1-x^2}
    12. - \lzo{x}(\coth^{-1}x)=\frac1{1-x^2} + \lzo{x}(\coth^{-1}(x))=\frac1{1-x^2}
    @@ -196,11 +196,11 @@
  • - \int \ln x\,dx=x\ln x -x +C + \int \ln(x)\,dx=x\ln(x) -x +C
  • - \int a^x\,dx=\frac{1}{\ln a}\cdot a^x+C + \int a^x\,dx=\frac{1}{\ln(a)}\cdot a^x+C
  • @@ -217,51 +217,51 @@ Integrals Involving Trigonometric Functions
    1. - \int \cos x\,dx=\sin x+C + \int \cos(x)\,dx=\sin(x)+C
    2. - \int \sin x\,dx=-\cos x+C + \int \sin(x)\,dx=-\cos(x)+C
    3. - \int \tan x\,dx=-\ln \abs{\cos x}+C + \int \tan(x)\,dx=-\ln \abs{\cos(x)}+C
    4. - \int \sec x\,dx=\ln \abs{\sec x+\tan x}+C + \int \sec(x)\,dx=\ln \abs{\sec(x)+\tan(x)}+C
    5. - \int \csc x\,dx=-\ln \abs{\csc x+\cot x}+C + \int \csc(x)\,dx=-\ln \abs{\csc(x)+\cot(x)}+C
    6. - \int \cot x\,dx=\ln \abs{\sin x}+C + \int \cot(x)\,dx=\ln \abs{\sin(x)}+C
    7. - \int \sec^2 x\,dx=\tan x+C + \int \sec^2(x)\,dx=\tan(x)+C
    8. - \int \csc^2x\,dx=-\cot x+C + \int \csc^2(x)\,dx=-\cot(x)+C
    9. - \int \sec x\tan x\,dx=\sec x+C + \int \sec(x)\tan(x)\,dx=\sec(x)+C
    10. - \int \csc x\cot x\,dx=-\csc x+C + \int \csc(x)\cot(x)\,dx=-\csc(x)+C
    11. - \int \cos^2x\,dx=\frac12x+\frac14\sin\big(2x\big)+C + \int \cos^2(x)\,dx=\frac12x+\frac14\sin\big(2x\big)+C
    12. - \int \sin^2x\,dx=\frac12x-\frac14\sin\big(2x\big)+C + \int \sin^2(x)\,dx=\frac12x-\frac14\sin\big(2x\big)+C
    13. @@ -282,19 +282,19 @@ Integrals Involving Hyperbolic Functions
      1. - \int \cosh x\,dx=\sinh x+C + \int \cosh(x)\,dx=\sinh(x)+C
      2. - \int \sinh x\,dx=\cosh x+C + \int \sinh(x)\,dx=\cosh(x)+C
      3. - \int \tanh x\,dx=\ln(\cosh x)+C + \int \tanh(x)\,dx=\ln(\cosh(x))+C
      4. - \int \coth x\,dx=\ln \abs{\sinh x}+C + \int \coth(x)\,dx=\ln \abs{\sinh(x)}+C
      5. @@ -397,22 +397,22 @@ - \sin \theta = y - \cos \theta = x + \sin(\theta) = y + \cos(\theta) = x - \ds\csc \theta = \frac1y - \ds\sec \theta = \frac1x + \ds\csc(\theta) = \frac1y + \ds\sec(\theta) = \frac1x - \ds\tan \theta = \frac yx - \ds\cot \theta = \frac xy + \ds\tan(\theta) = \frac yx + \ds\cot(\theta) = \frac xy @@ -439,22 +439,22 @@ - \ds\sin \theta = \frac{\text{O} }{\text{H} } - \ds\csc \theta = \frac{\text{H} }{\text{O} } + \ds\sin(\theta) = \frac{\text{O} }{\text{H} } + \ds\csc(\theta) = \frac{\text{H} }{\text{O} } - \ds\cos \theta = \frac{\text{A} }{\text{H} } - \ds\sec \theta = \frac{\text{H} }{\text{A} } + \ds\cos(\theta) = \frac{\text{A} }{\text{H} } + \ds\sec(\theta) = \frac{\text{H} }{\text{A} } - \ds\tan \theta = \frac{\text{O} }{\text{A} } - \ds\cot \theta = \frac{\text{A} }{\text{O} } + \ds\tan(\theta) = \frac{\text{O} }{\text{A} } + \ds\cot(\theta) = \frac{\text{A} }{\text{O} } @@ -467,28 +467,28 @@ Pythagorean Identities
          -
        1. \sin ^2x+\cos ^2x= 1
          2. -
        2. \tan^2x+ 1 = \sec^2 x
          3. -
        3. 1 + \cot^2x=\csc^2 x
          4. +
        4. \sin^2(x)+\cos^2(x)= 1
          5. +
        5. \tan^2(x)+ 1 = \sec^2(x)
          6. +
        6. 1 + \cot^2(x)=\csc^2(x)
         Double Angle Formulas
          -
        1. \sin 2x = 2\sin x\cos x
          2. +
        2. \sin(2x) = 2\sin(x)\cos(x)

        3. - \cos 2x \amp = \cos^2x - \sin^2 x \amp \amp - \amp = 2\cos^2x-1 \amp \amp - \amp = 1-2\sin^2x \amp \amp + \cos(2x) \amp = \cos^2(x) - \sin^2(x) \amp \amp + \amp = 2\cos^2(x)-1 \amp \amp + \amp = 1-2\sin^2(x) \amp \amp

          4. -
        4. \tan 2x = \frac{2\tan x}{1-\tan^2 x}
          5. +
        5. \tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}
         @@ -497,33 +497,33 @@ Cofunction Identities
          -
        1. \sin\left(\frac{\pi}{2}-x\right) = \cos x
          2. +
        2. \sin\left(\frac{\pi}{2}-x\right) = \cos(x)
          3. -
        3. \cos\left(\frac{\pi}{2}-x\right) = \sin x
          4. +
        4. \cos\left(\frac{\pi}{2}-x\right) = \sin(x)
          5. -
        5. \tan\left(\frac{\pi}{2}-x\right) = \cot x
          6. -
        6. \csc\left(\frac{\pi}{2}-x\right) = \sec x
          7. +
        7. \tan\left(\frac{\pi}{2}-x\right) = \cot(x)
          8. +
        8. \csc\left(\frac{\pi}{2}-x\right) = \sec(x)
          9. -
        9. \sec\left(\frac{\pi}{2}-x\right) = \csc x
          10. +
        10. \sec\left(\frac{\pi}{2}-x\right) = \csc(x)
          11. -
        11. \cot\left(\frac{\pi}{2}-x\right) = \tan x
          12. +
        12. \cot\left(\frac{\pi}{2}-x\right) = \tan(x)
         Even/Odd Identities
          -
        1. \sin(-x) = -\sin x
          2. +
        2. \sin(-x) = -\sin(x)
          3. -
        3. \cos (-x) = \cos x
          4. +
        4. \cos (-x) = \cos(x)
          5. -
        5. \tan (-x) = -\tan x
          6. +
        6. \tan (-x) = -\tan(x)
          7. -
        7. \csc(-x) = -\csc x
          8. +
        8. \csc(-x) = -\csc(x)
          9. -
        9. \sec (-x) = \sec x
          10. +
        10. \sec (-x) = \sec(x)
          11. -
        11. \cot (-x) = -\cot x
          12. +
        12. \cot (-x) = -\cot(x)
         @@ -532,24 +532,24 @@ Power-Reducing Formulas
          -
        1. \sin^2 x = \frac{1-\cos 2x}{2}
          2. +
        2. \sin^2(x) = \frac{1-\cos(2x)}{2}
          3. -
        3. \cos^2 x = \frac{1+\cos 2x}{2}
          4. +
        4. \cos^2(x) = \frac{1+\cos(2x)}{2}
          5. -
        5. \tan^2x = \frac{1-\cos 2x}{1+\cos 2x}
          6. +
        6. \tan^2(x) = \frac{1-\cos(2x)}{1+\cos(2x)}
         Sum to Product Formulas
          -
        1. \sin x+\sin y = 2\sin \left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)
          2. +
        2. \sin(x)+\sin(y) = 2\sin\left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)
          3. -
        3. \sin x-\sin y = 2\sin \left(\frac{x-y}2\right)\cos\left(\frac{x+y}2\right)
          4. +
        4. \sin(x)-\sin(y) = 2\sin\left(\frac{x-y}2\right)\cos\left(\frac{x+y}2\right)
          5. -
        5. \cos x+\cos y = 2\cos \left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)
          6. +
        6. \cos(x)+\cos(y) = 2\cos\left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)
          7. -
        7. \cos x-\cos y = -2\sin \left(\frac{x+y}2\right)\sin\left(\frac{x-y}2\right)
          8. +
        8. \cos(x)-\cos(y) = -2\sin\left(\frac{x+y}2\right)\sin\left(\frac{x-y}2\right)
         @@ -557,22 +557,22 @@ Product to Sum Formulas
          -
        1. \sin x\sin y = \frac12 \big(\cos(x-y) - \cos (x+y)\big)
          2. +
        2. \sin(x)\sin(y) = \frac12 \big(\cos(x-y) - \cos (x+y)\big)
          3. -
        3. \cos x\cos y = \frac12\big(\cos (x-y) +\cos (x+y)\big)
          4. +
        4. \cos(x)\cos(y) = \frac12\big(\cos (x-y) +\cos (x+y)\big)
          5. -
        5. \sin x\cos y = \frac12 \big(\sin(x+y) + \sin (x-y)\big)
          6. +
        6. \sin(x)\cos(y) = \frac12 \big(\sin(x+y) + \sin (x-y)\big)
         Angle Sum/Difference Formulas
          -
        1. \sin (x\pm y) = \sin x\cos y \pm \cos x\sin y
          2. +
        2. \sin (x\pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)
          3. -
        3. \cos (x\pm y) = \cos x\cos y \mp \sin x\sin y
          4. +
        4. \cos (x\pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)
          5. -
        5. \tan (x\pm y) = \frac{\tan x\pm \tan y}{1\mp \tan x\tan y}
          6. +
        6. \tan (x\pm y) = \frac{\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}
         @@ -588,7 +588,7 @@

        • - h=a\sin \theta + h=a\sin(\theta)

        • @@ -599,7 +599,7 @@

        • Law of Cosines: - c^2=a^2+b^2-2ab\cos \theta + c^2=a^2+b^2-2ab\cos(\theta)

        diff --git a/ptx/sec_ABC.ptx b/ptx/sec_ABC.ptx index 91461afb4..15461061b 100644 --- a/ptx/sec_ABC.ptx +++ b/ptx/sec_ABC.ptx @@ -1558,7 +1558,7 @@

        The functions f(x) = \cos (x) and - g(x) = \sin x intersect infinitely many times, + g(x) = \sin(x) intersect infinitely many times, forming an infinite number of repeated, enclosed regions. Find the areas of these regions.

        diff --git a/ptx/sec_FTC.ptx b/ptx/sec_FTC.ptx index 71b574d1e..b5b6b7011 100644 --- a/ptx/sec_FTC.ptx +++ b/ptx/sec_FTC.ptx @@ -2548,7 +2548,7 @@

        - Explain why \ds\int_{a}^{a+2\pi} \sin t\, dt = 0 for all values of a. + Explain why \ds\int_{a}^{a+2\pi} \sin(t)\, dt = 0 for all values of a.

        diff --git a/ptx/sec_Separable.ptx b/ptx/sec_Separable.ptx index 2ab03775e..4d90a399b 100644 --- a/ptx/sec_Separable.ptx +++ b/ptx/sec_Separable.ptx @@ -424,7 +424,7 @@

        - \displaystyle (y + 3)\yp + (\ln(x)) \yp - x\sin y = (y+3)\ln(x) + \displaystyle (y + 3)\yp + (\ln(x)) \yp - x\sin(y) = (y+3)\ln(x)

        @@ -436,13 +436,13 @@

        - \displaystyle \yp -x^2\cos y + y = \cos y - x^2 y + \displaystyle \yp -x^2\cos(y) + y = \cos(y) - x^2 y

        Separable. - \displaystyle \frac{1}{\cos y - y}\,dy = (x^2+1)\,dx + \displaystyle \frac{1}{\cos(y) - y}\,dy = (x^2+1)\,dx

        @@ -562,13 +562,13 @@

        - \displaystyle \yp = \frac{\sin(x)}{\cos y}, + \displaystyle \yp = \frac{\sin(x)}{\cos(y)}, with y(0) = \displaystyle \frac{\pi}{2}

        - \sin y + \cos(x) = 2 + \sin(y) + \cos(x) = 2

        @@ -636,12 +636,12 @@

        - \displaystyle \yp = (\cos^2x)(\cos^2 2y), with y(0) = 0 + \displaystyle \yp = (\cos^2(x))(\cos^2 (2y)), with y(0) = 0

        - 2\tan 2y = 2x + \sin 2x + 2\tan(2y) = 2x + \sin(2x)

        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

        -

        -
      6. +
          +
        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. -
        7. +
      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

    diff --git a/ptx/sec_deriv_chainrule.ptx b/ptx/sec_deriv_chainrule.ptx index 0e6f985a0..a1b10aebc 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.

    @@ -1048,90 +1048,86 @@ 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.