From b2ce4623aa41db6db83e75fa79ccbd5370e7c7e9 Mon Sep 17 00:00:00 2001 From: Michael Howard Date: Fri, 9 May 2025 13:05:08 -0500 Subject: [PATCH] Organize and edit --- _toc.yml | 16 ++--- calculus/derivative-rules.md | 20 ++++++- calculus/derivatives.md | 4 ++ calculus/function-identities.md | 79 ++++++++++++++++++++++++ calculus/functions.md | 80 ------------------------- calculus/lhopitals-rule.md | 86 -------------------------- calculus/limits.md | 103 ++++++++++++++++++++++++++++++++ credits.md | 4 +- first-order-odes/definition.md | 4 +- 9 files changed, 216 insertions(+), 180 deletions(-) create mode 100644 calculus/function-identities.md delete mode 100644 calculus/lhopitals-rule.md diff --git a/_toc.yml b/_toc.yml index 841bd5e36..c73fbea87 100644 --- a/_toc.yml +++ b/_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 diff --git a/calculus/derivative-rules.md b/calculus/derivative-rules.md index 318101c6c..6731871cf 100644 --- a/calculus/derivative-rules.md +++ b/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} + ``` 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} diff --git a/calculus/derivatives.md b/calculus/derivatives.md index 80a73c248..21f04a1bd 100644 --- a/calculus/derivatives.md +++ b/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 + @@ -95,9 +96,11 @@ 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} \\ @@ -105,6 +108,7 @@ examples: &= \lim_{h \to 0} \frac{-1}{(x+h)x} \\ &= \frac{-1}{x^2} \end{align} + ``` ## Differentiability diff --git a/calculus/function-identities.md b/calculus/function-identities.md new file mode 100644 index 000000000..0f0aa34f7 --- /dev/null +++ b/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} diff --git a/calculus/functions.md b/calculus/functions.md index 87a6fd4dd..744c94a34 100644 --- a/calculus/functions.md +++ b/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} diff --git a/calculus/lhopitals-rule.md b/calculus/lhopitals-rule.md deleted file mode 100644 index d85a98622..000000000 --- a/calculus/lhopitals-rule.md +++ /dev/null @@ -1,86 +0,0 @@ -# L'Hôpital's Rule - -L'Hôpital's rule is a method that allows you to calculate limits of certain -indeterminate forms using derivatives. - -```{topic} L'Hôpital's Rule - -If $\lim_{x\to a} f(x) = \lim_{x \to a} g(x) = 0$ or $\pm \infty$, then - -\begin{equation} -\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)} -\end{equation} - -This process can be repeated until the above situation no longer applies or -the limit is found not to exist. -``` - -Let's work some examples: - -1. $\displaystyle \lim_{x\to 0} \frac{\sin x}{x}$ - - The limits of the numerator and denominator are both zero, so: - - \begin{equation} - \lim_{x\to 0} \frac{\sin x}{x} = \lim_{x\to 0} \frac{\cos x}{1} = 1 - \end{equation} - -2. $\displaystyle \lim_{x\to 0} \frac{x-\sin x}{x^3}$ - - The limits of the numerator and denominator are both zero, so: - - \begin{align} - \lim_{x\to 0} \frac{x-\sin x}{x^3} - &= \lim_{x\to 0} \frac{1-\cos x}{3x^2} \\ - &= \lim_{x\to 0} \frac{\sin x}{6x} \\ - &= \frac{1}{6} \lim_{x\to 0} \frac{\sin x}{x} = \frac{1}{6} - \end{align} - - where we used L'Hôpital's rule twice, then used the result from the first - example. - -3. $\displaystyle \lim_{x\to \infty} \frac{\ln(x)}{2\sqrt{x}}$ - - The limits of the numerator and denominator are both $\infty$, so: - - \begin{align} - \lim_{x\to \infty} \frac{\ln x}{2\sqrt{x}} - &= \lim_{x\to \infty} \frac{1/x}{1/\sqrt{x}} \\ - &= \lim_{x\to \infty} \frac{1}{\sqrt{x}} = 0 - \end{align} - - where the last limit is obtained by simplification. - -4. $\displaystyle \lim_{x\to \infty} \frac{e^x}{x^2}$ - - The limits of the numerator and denominator are both $\infty$, so: - - \begin{align} - \lim_{x\to \infty} \frac{e^x}{x^2} - &= \lim_{x\to \infty} \frac{e^x}{2x} \\ - &= \lim_{x\to \infty} \frac{e^x}{2} = \infty - \end{align} - - Here, we used L'Hôpital's rule twice, but then ultimately found that the - limit does not exist. - -5. $\displaystyle \lim_{x\to \infty} x\sin(1/x)$ - - This limit is not in a form that's immediately suitable for L'Hôpital's rule, - but it can be made so. - - \begin{align} - \lim_{x\to \infty} x\sin(1/x) - &= \lim_{x\to \infty} \frac{\sin(1/x)}{1/x} \\ - &= \lim_{x\to \infty} \frac{\cos(1/x)(-1/x^2)}{-1/x^2} \\ - &= \lim_{x\to \infty} \cos(1/x) = 1 - \end{align} - - Another option in this case would be to substitute in $y = 1/x$, then note - that $y \to 0$ as $x \to \infty$ so we have - - \begin{equation} - \lim_{y\to 0} \frac{\sin y}{y} = 1 - \end{equation} - - which we already solved! diff --git a/calculus/limits.md b/calculus/limits.md index 5ec42ef46..76639879f 100644 --- a/calculus/limits.md +++ b/calculus/limits.md @@ -501,3 +501,106 @@ A line $x = a$ is a vertical asymptote of $y = f(x)$ if \lim_{x \to a^+}f(x) = \pm\infty \quad {\rm or} \quad \lim_{x \to a^-}f(x) = \pm\infty \end{equation} ``` + +## L'Hôpital's Rule + +```{note} +This technique requires you to be comfortable with taking +[derivatives](./derivatives.md). Come back to it later if you need to study that +first! +``` + +L'Hôpital's rule is a method that allows you to calculate limits of certain +indeterminate forms using derivatives. + +```{topic} L'Hôpital's Rule + +If $\lim_{x\to a} f(x) = \lim_{x \to a} g(x) = 0$ or $\pm \infty$, then + +\begin{equation} +\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)} +\end{equation} + +This process can be repeated until the above situation no longer applies or +the limit is found not to exist. +``` + +Let's work some examples: + +1. $\displaystyle \lim_{x\to 0} \frac{\sin x}{x}$ + + ```{solution} + The limits of the numerator and denominator are both zero, so: + + \begin{equation} + \lim_{x\to 0} \frac{\sin x}{x} = \lim_{x\to 0} \frac{\cos x}{1} = 1 + \end{equation} + ``` + +2. $\displaystyle \lim_{x\to 0} \frac{x-\sin x}{x^3}$ + + ```{solution} + The limits of the numerator and denominator are both zero, so: + + \begin{align} + \lim_{x\to 0} \frac{x-\sin x}{x^3} + &= \lim_{x\to 0} \frac{1-\cos x}{3x^2} \\ + &= \lim_{x\to 0} \frac{\sin x}{6x} \\ + &= \frac{1}{6} \lim_{x\to 0} \frac{\sin x}{x} = \frac{1}{6} + \end{align} + + where we used L'Hôpital's rule twice, then used the result from the first + example. + ``` + +3. $\displaystyle \lim_{x\to \infty} \frac{\ln(x)}{2\sqrt{x}}$ + + ```{solution} + The limits of the numerator and denominator are both $\infty$, so: + + \begin{align} + \lim_{x\to \infty} \frac{\ln x}{2\sqrt{x}} + &= \lim_{x\to \infty} \frac{1/x}{1/\sqrt{x}} \\ + &= \lim_{x\to \infty} \frac{1}{\sqrt{x}} = 0 + \end{align} + + where the last limit is obtained by simplification. + ``` + +4. $\displaystyle \lim_{x\to \infty} \frac{e^x}{x^2}$ + + ```{solution} + The limits of the numerator and denominator are both $\infty$, so: + + \begin{align} + \lim_{x\to \infty} \frac{e^x}{x^2} + &= \lim_{x\to \infty} \frac{e^x}{2x} \\ + &= \lim_{x\to \infty} \frac{e^x}{2} = \infty + \end{align} + + Here, we used L'Hôpital's rule twice, but then ultimately found that the + limit does not exist. + ``` + +5. $\displaystyle \lim_{x\to \infty} x\sin(1/x)$ + + ```{solution} + This limit is not in a form that's immediately suitable for L'Hôpital's rule, + but it can be made so. + + \begin{align} + \lim_{x\to \infty} x\sin(1/x) + &= \lim_{x\to \infty} \frac{\sin(1/x)}{1/x} \\ + &= \lim_{x\to \infty} \frac{\cos(1/x)(-1/x^2)}{-1/x^2} \\ + &= \lim_{x\to \infty} \cos(1/x) = 1 + \end{align} + + Another option in this case would be to substitute in $y = 1/x$, then note + that $y \to 0$ as $x \to \infty$ so we have + + \begin{equation} + \lim_{y\to 0} \frac{\sin y}{y} = 1 + \end{equation} + + which we already solved! + ``` diff --git a/credits.md b/credits.md index bb9db46d4..70ae35e30 100644 --- a/credits.md +++ b/credits.md @@ -71,9 +71,9 @@ This work (c) 2025 is licensed under ## Attributions -Many of these notes were based on my own studies using the excellent textbook: +These notes were based on my own studies using the excellent textbook: > E. Kreyszig. *Advanced Engineering Mathematics*, 9th ed. (John Wiley \& Sons, Inc. Hoboken, NJ, 2006). -Some examples and problem statements were adapted from this text. +Many examples and problem statements were adapted from this text. diff --git a/first-order-odes/definition.md b/first-order-odes/definition.md index c4a6f3a15..82d8d9b02 100644 --- a/first-order-odes/definition.md +++ b/first-order-odes/definition.md @@ -9,14 +9,14 @@ an unknown function of one variable. &x^2 y''' y' + 2e^x y^4 = (x^2 + 2)y^2 \end{align} -The *order* of an ODE is its highest derivative. First-order ODEs in *explicit* +The *order* of an ODE is its highest derivative. First-order ODEs in *implicit* form look like: \begin{equation} F(x, y, y') = 0 \end{equation} -and in *implicit* form look like: +and in *explicit* form look like: \begin{equation} y' = f(x, y)