Merge branch 'main' into issue-49

Author: mphoward

_toc.yml

@@ -6,12 +6,12 @@ chapters:
   - file: calculus/index
     sections:
       - file: calculus/functions
+      - file: calculus/function-identities
       - file: calculus/limits
       - file: calculus/derivatives
       - file: calculus/derivative-rules
-      - file: calculus/lhopitals-rule
       - file: calculus/series-expansion
-      - file: calculus/integration
+      # - file: calculus/integration
       - file: calculus/integration-substitution
       - file: calculus/integration-by-parts
       - file: calculus/integration-partial-fractions
@@ -23,9 +23,9 @@ chapters:
       - file: multivariable-calculus/derivatives
       - file: multivariable-calculus/total-differential
       - file: multivariable-calculus/manipulating-derivatives
-      - file: multivariable-calculus/series-expansion
-      - file: multivariable-calculus/integration
-      - file: multivariable-calculus/line-integral
+      # - file: multivariable-calculus/series-expansion
+      # - file: multivariable-calculus/integration
+      # - file: multivariable-calculus/line-integral
       # - file: multivariable-calculus/surface-integral
       # - file: multivariable-calculus/volume-integral
       # - file: multivariable-calculus/integration-theorems
@@ -42,14 +42,14 @@ chapters:
       - file: nonlinear-equations/definition
       - file: nonlinear-equations/quadratic-cubic
       - file: nonlinear-equations/root-finding
-      - file: nonlinear-equations/linearization
+      # - file: nonlinear-equations/linearization
   - file: first-order-odes/index
     sections:
       - file: first-order-odes/definition
       - file: first-order-odes/separable
       - file: first-order-odes/exact
-      - file: first-order-odes/linearity
-      - file: first-order-odes/undetermined-coefficients
+      # - file: first-order-odes/linearity
+      # - file: first-order-odes/undetermined-coefficients
       - file: first-order-odes/laplace-transform
       - file: first-order-odes/integrating-factor
       - file: first-order-odes/numerical-solution

calculus/derivative-rules.md

@@ -40,6 +40,7 @@ functions that are hard to expand!
 
 1. $f(x) = (x+1)(2x^2 + 5)(5x^3-4)$
 
+   ```{solution}
    Identify:
 
    \begin{equation}
@@ -67,9 +68,11 @@ functions that are hard to expand!
    &+ (2x^2+5)(5x^3-4)
    \end{align}
    <!--markdownlint-enable MD011 -->
+   ```
 
 2. $f(x) = \dfrac{1}{x} e^x$
 
+   ```{solution}
    Identify:
 
    \begin{align}
@@ -83,9 +86,11 @@ functions that are hard to expand!
    f'(x) &= \frac{1}{x}e^x + e^x(-\frac{1}{x^2}) \\
    &= e^x\left(\frac{1}{x} - \frac{1}{x^2}\right)
    \end{align}
+   ```
 
 3. $f(x) = (x^2+3)\ln x$
 
+   ```{solution}
    Identify:
 
    \begin{align}
@@ -99,6 +104,7 @@ functions that are hard to expand!
    f'(x) &= (x^2+3)\cdot\frac{1}{x} + (\ln x)(2x) \\
    &= \frac{x^2+3}{x} + 2x\ln x
    \end{align}
+   ```
 
 ## Quotient Rule
 
@@ -143,6 +149,7 @@ is helpful to do the quotient rule!
 
 1. $\displaystyle f(x) = \frac{x^2 -1}{x^4 + 2}$
 
+   ```{solution}
    \begin{align}
    u &= x^2 -1 & v &= x^4 +2 \\
    u' &= 2x & v' &= 4x^3
@@ -154,9 +161,11 @@ is helpful to do the quotient rule!
    f'(x) &= \frac{ (x^4 + 2) \cdot (2x) - (x^2 - 1) \cdot (4x^3)}{(x^4 +2)^2}\\
     &= \frac{2x^5 + 4x^2 - 4x^5 +4x^3}{x^8 + 2x^4 + 4}
    \end{align}
+   ```
 
 2. $\displaystyle f(x) = \frac{e^{x}}{1 + x}$
 
+   ```{solution}
    \begin{align}
    u &= e^{x} & v &= 1 + x \\
    u' &= e^{x} & v' &= 1
@@ -168,9 +177,11 @@ is helpful to do the quotient rule!
    f'(x) &= \frac{(1 + x) \cdot e^{x} - e^{x} \cdot 1}{(1 + x)^2} \\
    &= \frac{x e^{x}}{(1 + x)^2}
    \end{align}
+   ```
 
 3. $\displaystyle f(x) = \frac{(x - 1)(x^2 - 2x)}{x^4}$
 
+   ```{solution}
    \begin{align}
    u &=  & v &= x^4\\
    u' &= 3x^2 - 6x + 2 & v' &= 4x^3 \\
@@ -187,6 +198,7 @@ is helpful to do the quotient rule!
    Note, though, that in this case we could also have expanded the numerator,
    divided through by $x^8$, and differentiated term-by-term to arrive at the
    same answer. The faster route depends on the problem!
+   ```
 
 ## Chain rule
 
@@ -225,6 +237,7 @@ The results match! Some additional examples:
 
 1. $f(x) = e^{x^2}$
 
+   ```{solution}
    Make the replacement $u = x^2$:
 
    \begin{align}
@@ -237,9 +250,11 @@ The results match! Some additional examples:
    \begin{equation}
    f'(x) = \dd{}{f}{u} \dd{}{u}{x} = e^{u} \dd{}{u}{x} = e^{x^2} \cdot 2x
    \end{equation}
+   ```
 
 2. $f(x) = \ln(1 + 2x)$
 
+   ```{solution}
    Make the replacement $u = 1+2x$:
 
    \begin{align}
@@ -252,9 +267,11 @@ The results match! Some additional examples:
    \begin{equation}
    f'(x) = \dd{}{f}{u} \dd{}{u}{x} = \frac{1}{u} \dd{}{u}{x} = \frac{2}{1 + 2x}
    \end{equation}
+   ```
 
 3. $f(x) = \dfrac{2}{1 + 2x}$
 
+   ```{solution}
    Make the replacement $u = 1+2x$:
 
    \begin{align}
@@ -268,6 +285,7 @@ The results match! Some additional examples:
    f'(x) = \dd{}{f}{u} \dd{}{u}{x} = -2u^{-2} \cdot \dd{}{u}{x} =
      \frac{-4}{(1 + 2x)^2}
    \end{equation}
+   ```
 
 ## Trigonometric functions
 
@@ -361,8 +379,6 @@ The roots occur at $t = T/4$ or $3T/4$, when $x = 0$ and the spring is no longer
 stretched. All potential energy has been converted to kinetic energy!
 ````
 
-## Skill builder problems
-
 1. $f(x) = 3 \cos x + \sin x$
 
    ```{solution}

calculus/derivatives.md

@@ -87,6 +87,7 @@ examples:
 
 1. $f(x) = (x - 1)^2 + 1$
 
+   ```{solution}
    \begin{align}
    f'(x) &= \lim_{h \to 0} \frac{[(x + h - 1)^2 + 1] - [(x - 1)^2 + 1]}{h} \\
        &= \lim_{h \to 0} \frac{(x - 1)^2 +
@@ -95,16 +96,19 @@ examples:
        &= \lim_{h \to 0} 2(x - 1) + h \\
        &= 2(x - 1)
    \end{align}
+   ```
 
 2. $f(x) = 1/x$
 
+   ```{solution}
    \begin{align}
    f'(x) &= \lim_{h \to 0} \frac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h} \\
        &= \lim_{h \to 0} \frac{\dfrac{x - (x + h)}{x(x+h)}}{h}  \\
        &= \lim_{h \to 0} \frac{\dfrac{- h}{x(x+h)}}{h}  \\
        &= \lim_{h \to 0} \frac{-1}{(x+h)x} \\
        &= \frac{-1}{x^2}
    \end{align}
+   ```
 
 ## Differentiability
 

calculus/function-identities.md

@@ -0,0 +1,79 @@
+# Function identities
+
+## Trigonmetric functions
+
+\begin{align}
+\sin \theta &= \frac{y}{r} & \csc \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\
+\cos \theta &= \frac{x}{r} & \sec \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\
+\tan \theta &= \frac{y}{x} = \frac{\sin \theta}{\cos \theta} &
+\cot \theta &= \frac{x}{y} = \frac{\cos \theta}{\sin \theta}
+\end{align}
+
+\begin{align}
+\sin^2 \theta + \cos^2 \theta = 1 \\
+1 + \tan^2 \theta = \sec^2 \theta \\
+1 + \cot^2 \theta = \csc^2 \theta
+\end{align}
+
+\begin{align}
+\cos(A+B) = \cos A \cos B - \sin A \sin B \\
+\sin(A+B) = \sin A \cos B - \cos A \sin B
+\end{align}
+
+\begin{align}
+\cos 2 \theta = \cos^2 \theta - \sin^2 \theta \\
+\sin 2 \theta = 2 \sin \theta \cos \theta
+\end{align}
+
+\begin{align}
+\cos^2 \theta = \frac{1 + \cos 2 \theta}{2} \\
+\sin^2 \theta = \frac{1 - \cos 2 \theta}{2}
+\end{align}
+
+\begin{align}
+\sin(\theta + 2 \pi) = \sin \theta \\
+\cos(\theta + 2 \pi) = \cos \theta
+\end{align}
+
+\begin{align}
+\sin(- \theta) = - \sin \theta \\
+\cos(- \theta) = cos \theta
+\end{align}
+
+## Exponential functions
+
+\begin{align}
+a^x a^y &= a^{x+y} \\
+\frac{a^x}{a^y} &= a^{x-y} \\
+(a^x)^y = (a^y)^x &= a^{xy} \\
+a^x b^x &= (ab)^x \\
+\frac{a^x}{b^x} &= \left (\frac{a}{b} \right)^x
+\end{align}
+
+## Logarithmic functions
+
+Definition:
+
+\begin{align}
+y &= \log_{a}x \\
+x &= a^y
+\end{align}
+
+Natural Log:
+
+\begin{equation}
+\ln x = \log_{e} x
+\end{equation}
+
+Common log:
+
+\begin{equation}
+\log x = \log_{10} x
+\end{equation}
+
+\begin{align}
+\ln(bx) &= \ln(b) + \ln(x) \\
+\ln\left(\frac{b}{x}\right) &= \ln(b) - \ln(x) \\
+\ln(x^r) &= r \ln(x) \\
+\log_{a}x &= \frac{\ln(x)}{\ln(a)} \\
+\end{align}

calculus/functions.md

@@ -266,83 +266,3 @@ y = A\sin\left[ \frac{2\pi}{L}(x + x_0) \right] + y_0
 
 where *A* is the amplitude of the wave, *L* is the period of the wave, $x_0$ is
 a horizontal (phase) shift, and $y_0$ is a vertical shift.
-
-## Identities
-
-### Trigonmetric functions
-
-\begin{align}
-\sin \theta &= \frac{y}{r} & \csc \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\
-\cos \theta &= \frac{x}{r} & \sec \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\
-\tan \theta &= \frac{y}{x} = \frac{\sin \theta}{\cos \theta} &
-\cot \theta &= \frac{x}{y} = \frac{\cos \theta}{\sin \theta}
-\end{align}
-
-\begin{align}
-\sin^2 \theta + \cos^2 \theta = 1 \\
-1 + \tan^2 \theta = \sec^2 \theta \\
-1 + \cot^2 \theta = \csc^2 \theta
-\end{align}
-
-\begin{align}
-\cos(A+B) = \cos A \cos B - \sin A \sin B \\
-\sin(A+B) = \sin A \cos B - \cos A \sin B
-\end{align}
-
-\begin{align}
-\cos 2 \theta = \cos^2 \theta - \sin^2 \theta \\
-\sin 2 \theta = 2 \sin \theta \cos \theta
-\end{align}
-
-\begin{align}
-\cos^2 \theta = \frac{1 + \cos 2 \theta}{2} \\
-\sin^2 \theta = \frac{1 - \cos 2 \theta}{2}
-\end{align}
-
-\begin{align}
-\sin(\theta + 2 \pi) = \sin \theta \\
-\cos(\theta + 2 \pi) = \cos \theta
-\end{align}
-
-\begin{align}
-\sin(- \theta) = - \sin \theta \\
-\cos(- \theta) = cos \theta
-\end{align}
-
-### Exponential functions
-
-\begin{align}
-a^x a^y &= a^{x+y} \\
-\frac{a^x}{a^y} &= a^{x-y} \\
-(a^x)^y = (a^y)^x &= a^{xy} \\
-a^x b^x &= (ab)^x \\
-\frac{a^x}{b^x} &= \left (\frac{a}{b} \right)^x
-\end{align}
-
-### Logarithmic functions
-
-Definition:
-
-\begin{align}
-y &= \log_{a}x \\
-x &= a^y
-\end{align}
-
-Natural Log:
-
-\begin{equation}
-\ln x = \log_{e} x
-\end{equation}
-
-Common log:
-
-\begin{equation}
-\log x = \log_{10} x
-\end{equation}
-
-\begin{align}
-\ln(bx) &= \ln(b) + \ln(x) \\
-\ln\left(\frac{b}{x}\right) &= \ln(b) - \ln(x) \\
-\ln(x^r) &= r \ln(x) \\
-\log_{a}x &= \frac{\ln(x)}{\ln(a)} \\
-\end{align}