html5 colgroud,mth229.github.io/derivatives.tex at 51251e8ce00b61b18510f9d87165ea86b4dac9a2 · mth229...

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\section{Questions to be handed in for project 6:}\newline

To get started, we load \texttt{Plots} so that we can make plots and the \texttt{Roots} package for later usage.\begin{verbatim}

using Plots

gadfly()

using Roots

\end{verbatim}

\rule{\textwidth}{1pt}

\subsubsection{Quick background}\newline

Read about this material here: \href{http://mth229.github.io/derivatives.html}{Approximate derivatives in julia}.\newline

For the impatient, the slope of the tangent line to the graph of $f(x)$ at the point $(c,f(c))$ is given by the following limit:

$$

\lim_{h \rightarrow 0} \frac{f(c + h) - f(c)}{h}.

$$

\newline

The notation for this – when the limit exists – is $f'(c)$, the intuition is that this is the limit of the slope of a sequence of secant lines connecting the points $(c, f(c))$ and $(c+h, f(c+h))$. In general the derivative of a function $f(x)$ is the function $f'(x)$, which returns the slope of the tangent line for each $x$ where it is defined.\subsubsection{Approximate derivatives}\newline

Approximating the slope of the tangent line can be done \href{XXX}{several ways}. The \textit{forward difference quotient} takes a small value of $h$ and uses the values $(f(x+h) - f(x))/h$ as an approximation.\newline

For example to estimate the derivative of $x^x$ at $c=1$ with \texttt{h=1e-6} we could have\begin{verbatim}

f(x) = x^x

c, h = 1, 1e-6

(f(c+h) - f(c))/h

\end{verbatim}

\begin{verbatim}

1.000001000006634\end{verbatim}

\newline

In \texttt{julia} we can write a function that does this, allowing us to pass in any function:\begin{verbatim}

forward(f, c; h=1e-6) = (f(c+h) - f(c))/h

\end{verbatim}

\begin{verbatim}

forward (generic function with 1 method)\end{verbatim}

\newline

We can define an \textit{operator} – something which takes a function and returns a function modifying the above slightly:\begin{verbatim}

Df(f; h=1e-6) = x -> forward(f,x,h=h)

\end{verbatim}

\begin{verbatim}

Df (generic function with 1 method)\end{verbatim}

\subsubsection{Automatic derivatives}\newline

In the \texttt{Roots} package, an operator \texttt{D} (using Euler's notation) is given which uses a numeric approach to compute the derivative. This is more accurate, but conceptually a bit more difficult to understand and does not work for all functions. It is also used like an operator, e.g., \texttt{D(f)} is a function derived from the function \texttt{f}:\begin{verbatim}

using Roots

fp(x) = D(sin)(x)# define a function fp or use D(sin) directly

fp(pi) # finds cos(pi). Also D(sin)(pi)

\end{verbatim}

\begin{verbatim}

-1.0\end{verbatim}

\newline

The usual notation for a deriviative can be defined, though it isn't the default. If we use this command (ignoring the warning):\begin{verbatim}

Base.ctranspose(f::Function) = D(f)

\end{verbatim}

\begin{verbatim}

ctranspose (generic function with 53 methods)\end{verbatim}

\newline

then, we can differentiate using the "usual" notation:\begin{verbatim}

f(x) = sin(x)

f'(pi) # same as D(f)(pi)

f''(pi) # same as D(f,2)(pi)

\end{verbatim}

\begin{verbatim}

-1.2246467991473532e-16\end{verbatim}

\subsubsection{Symbolic derivatives}\newline

The \texttt{D} function gives accurate numeric values for first, second, and even higher-order derivatives. It does not however, return the expression one would get were these computed by hand. The \texttt{diff} function from \texttt{SymPy} will find symbolic derivatives, similar to what is achieved when differentiating "by hand."\newline

The \texttt{diff} function can be called with a function:\begin{verbatim}

using SymPy

f(x) = exp(x) * sin(x)

diff(f)

\end{verbatim}

$$e^{x} \sin{\left (x \right )} + e^{x} \cos{\left (x \right )}$$\newline

It can also be called with a symbolic expression:\begin{verbatim}

x, a = symbols("x, a")

\end{verbatim}

\begin{verbatim}

(x,a)\end{verbatim}

\begin{verbatim}

@vars x a

\end{verbatim}

\begin{verbatim}

diff(x * sin(x))

\end{verbatim}

$$x \cos{\left (x \right )} + \sin{\left (x \right )}$$\newline

If there is another symbol, then a second argument is passed to specify which one is being having its derivative taken:\begin{verbatim}

diff(x*sin(a*x), x)

\end{verbatim}

$$a x \cos{\left (a x \right )} + \sin{\left (a x \right )}$$\subsubsection{Questions}\begin{itemize}\item Calculate the slope of the secant line of $f(x) = 3x^2 + 5$ between $(3,f(3))$ and $(5, f())$.\end{itemize}

\\begin{answer}

type: numeric

reminder: slope of secant line

answer: [23.999, 24.001]

answer_text: [23.999, 24.001]

\\end{answer}

\begin{itemize}\item Verify that the derivative of $f(x) = \sin(x)$ at $\pi/3$ is $1/2$ by finding the following limit using a table:\end{itemize}

$$

\lim_{h \rightarrow 0} \frac{f(\pi/3 + h) - f(\pi/3)}{h}

$$

\newline

(Use \texttt{[hs ys]} to look at your generated data, as was done in the limits project.)

\\begin{answer}

type: longtext

reminder: Verify derivative using a table

answer_text: \verb+[hs map(h->(sin(pi/3+ h)-sin(pi/3))/h, hs)]+

rows: 3

cols: 60

\\end{answer}

\begin{itemize}\item Let $f(x) = 1/x$ and $c=4$. Find the approximate derivative (forward) when \texttt{h=1e-6}.\end{itemize}

\\begin{answer}

type: numeric

reminder: approx forward derivative \verb+(1/x)'(4)+

answer: [-0.06259998438234705, -0.06239998438234705]

answer_text: [-0.063, -0.062]

\\end{answer}

\begin{itemize}\item Let $f(x) = x^x$ and $c=4$. Find the approximate derivative (forward) when \texttt{h=1e-8}.\end{itemize}

\\begin{answer}

type: numeric

reminder: approx forward derivative \verb+(x^x)'(4)+

answer: [610.8913613847911, 610.8913633847911]

answer_text: [610.891, 610.891]

\\end{answer}

\begin{itemize}\item Let $f(x) = (x + 2)/(1 + x^3)$. Plot both $f$ and its approximate derivative on the interval $[0,5]$. Identify the zero of the derivative. What is its value? What is the value of $f(x)$ at this point?\end{itemize}\newline

What commands produce the plot?

\\begin{answer}

type: longtext

reminder: what commands produce a plot of f=(x+2)/(1+x^3) and its derivative over [0,5]?

rows: 3

cols: 60

\\end{answer}

\newline

What is the zero of the derivative on this interval?

\\begin{answer}

type: numeric

reminder: zero of derivative

answer: [0.18436715263814152, 0.5843671526381415]

answer_text: [0.184, 0.584]

\\end{answer}

\newline

What is the value of $f$ at this point:

\\begin{answer}

type: numeric

reminder: value of f(x0)

answer: [2.0562447684254272, 2.4562447684254276]

answer_text: [2.056, 2.456]

\\end{answer}

\begin{itemize}\item Let $f(x) = (x^3 + 5)(x^3 + x + 1)$. The derivative of this function has one real zero. Find it. (You can use \texttt{fzero} with the derivative function after plotting to identify a bracketing interval.)\end{itemize}

\\begin{answer}

type: numeric

reminder: zero of f'(x)

answer: [-1.3642244478879977, -1.3640244478879977]

answer_text: [-1.364, -1.364]

\\end{answer}

\begin{itemize}\item Let $f(x) = \sin(x)$. Following the example on p124 of the Rogawski book we look at a table of values of the forward difference equation at $x=\pi/6$ for various values of $h$. The true derivative is $\cos(\pi/6) = \sqrt{3}/2$.\end{itemize}\newline

Make the following table. \begin{verbatim}

f(x) = sin(x)

c = pi/6

hs = [(1/10)^i for i in 1:12]

ys = [forward(f, c, h=h) for h in hs] - sqrt(3)/2

[hs ys]

\end{verbatim}

\begin{verbatim}

12x2 Array{Any,2}:

0.1 -0.0264218

0.01 -0.00251441

0.001 -0.000250144

0.0001 -2.50014e-5

1.0e-5 -2.50002e-6

1.0e-6 -2.49917e-7

1.0e-7 -2.51525e-8

1.0e-8 -2.39297e-9

1.0e-9 1.42604e-8

1.0e-10 1.80794e-7

1.0e-11 -1.48454e-6

1.0e-12 4.06657e-6\end{verbatim}

\newline

What size \texttt{h} has the closest approximation?

\\begin{answer}

type: radio

reminder: smallest error

values: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12

labels: 1e-1 | 1e-2 | 1e-3 | 1e-4 | 1e-5 | 1e-6 | 1e-7 | 1e-8 | 1e-9 | 1e-10 | 1e-11 | 1e-12

answer: 8

\\end{answer}

\begin{itemize}\item The \texttt{D} operator is easy to use. Here is how we can plot both the sine function and its derivative\end{itemize}\begin{verbatim}

using Plots, Roots# to load plot and D

f(x) = sin(x)

plot([f, D(f)], 0, 2pi) # or plot(f, 0, 2pi); plot!(D(f), 0, 2pi)

\end{verbatim}

\begin{html}

\end{html}

\newline

(If you defined the \texttt{'} notation to work, then it is even easier: \texttt{plot([f,f'], 0, 2pi)}.)\newline

Make a plot of $f(x) = \log(x+1) - x + x^2/2$ and its derivative over the interval $[-3/4, 4]$. The commands are:

\\begin{answer}

type: longtext

reminder: Commands to plot f and its derivative

answer_text: \verb+plot([f,D(f)], -3/4, 4)+

rows: 3

cols: 60

\\end{answer}

\newline

Is the derivative always increasing?

\\begin{answer}

type: radio

reminder: is derivative always increasing?

values: 1 | 2

labels: true | false

answer: 2

\\end{answer}

\begin{itemize}\item The function $f(x) = x^x$ has a derivative for $x > 0$. Use \texttt{fzero} to find a zero of its derivative. What is the value of the zero?\end{itemize}

\\begin{answer}

type: numeric

reminder: zero of derivative of \( x^x \)

answer: [0.36777944117144235, 0.3679794411714423]

answer_text: [0.368, 0.368]

\\end{answer}

\begin{itemize}\item Using the \texttt{diff} function from the \texttt{SymPy} package, identify the proper derivative of $x^x$:\end{itemize}

\\begin{answer}

type: radio

reminder:

values: 1 | 2 | 3 | 4

labels: $x^x$ | $x^(x+1) / (x+1)$ | $x^x*(log(x) + 1)$ | $x*x^(x-1)$

answer: 3

\\end{answer}

\begin{itemize}\item Using the \texttt{diff} function, find the derivative of the inverse tangent, $\tan^{-1}(x)$ (\texttt{atan}). What is the function?\end{itemize}

\\begin{answer}

type: radio

reminder:

values: 2 | 3 | 1

labels: $(-1)\tan^{-2}(x) \cdot (\tan^2(x) + 1)$ | $1/(1+x^2)$ | $(-1)\tan^{-2}(x)$

answer: 2

\\end{answer}

\subsection{Some applications}\begin{itemize}\item Suppose the height of a ball falls according to the formula $h(t) = 300 - 16t^2$. Find the rate of change of height at the instant the ball hits the ground.\end{itemize}

\\begin{answer}

type: numeric

reminder: value of derivative when h is 0

answer: [-138.5641646055102, -138.5639646055102]

answer_text: [-138.564, -138.564]

\\end{answer}

\begin{itemize}\item A formula for blood alcohol level in the body based on time is based on the number of drinks and the time \href{http://en.wikipedia.org/wiki/Blood_alcohol_content}{wikipedia}.\end{itemize}\newline

Suppose a model for the number of drinks consumed per hour is \begin{verbatim}

n(t) = t <= 3 ? 2 * sqrt(3) * sqrt(t) : 6.0

\end{verbatim}

\begin{verbatim}

n (generic function with 1 method)\end{verbatim}

\newline

Then the BAL for a 175 pound male is given by\begin{verbatim}

bal(t) = (0.806 * 1.2 * n(t)) / (0.58 * 175 / 2.2) - 0.017*t

\end{verbatim}

\begin{verbatim}

bal (generic function with 1 method)\end{verbatim}

\newline

From the plot below, describe when the peak blood alcohol level occurs and is the person ever in danger of being above 0.10?\begin{verbatim}

plot(bal, .5,7)

\end{verbatim}

\begin{html}

\end{html}

\\begin{answer}

type: longtext

reminder: Describe when peak BAL occurs, is it ever above 0.10?

answer_text: No, it isn't

rows: 3

cols: 60

\\end{answer}

\begin{itemize}\item Plot the derivative of \texttt{bal} over the time $[0.5, 7]$. Is this function ever positive?\end{itemize}

\\begin{answer}

type: radio

reminder: Is bal' ever positive

values: 2 | 3 | 1

labels: No, it never is | Yes, initially | Yes, after 3

answer: 2

\\end{answer}

\subsubsection{Tangent lines}\newline

The tangent line to the graph of $f(x)$ at $x=c$ is given by $y = f(c) + f'(c)(x-c)$. It is fairly easy to plot both the function and its tangent line – we just need a function to compute the tangent line.\newline

Here we write an operator to return such a function. The operator needs to know both the function name and the value $c$ to find the tangent line at $(c, f(c))$ (notice the \texttt{x->} bit indicating the following returns a function):\begin{verbatim}

tangent(f, c) = x -> f(c) + f'(c)*(x-c) # returns a function

\end{verbatim}

\begin{verbatim}

tangent (generic function with 1 method)\end{verbatim}

\newline

Here we see how to use it:\begin{verbatim}

f(x) = x^2# replace me

plot([f, tangent(f, 1)], 0, 2) # or plot(f, 0, 2); plot!(tangent(f, 1), 0, 2)

\end{verbatim}

\begin{html}

\end{html}

\begin{itemize}\item For the function $f(x) = 1/(x^2 + 1)$ (The witch of Agnesi), graph $f$ over the interval $[-3,3]$ and the tangent line to $f$ at $x=1$.\end{itemize}

\\begin{answer}

type: longtext

reminder: Commands to plot witch of Agnesi over [-3,3] with tangent line at x=1

rows: 3

cols: 60

\\end{answer}

\begin{itemize}\item Let $f(x) = x^3 -2x - 5$. Find the intersection of the tangent line at $x=3$ with the $x$-axis.\end{itemize}

\\begin{answer}

type: numeric

reminder: intersection of tangent line with x axis

answer: [2.3598999999999997, 2.3601]

answer_text: [2.36, 2.36]

\\end{answer}

\begin{itemize}\item Let $f(x)$ be given by the expression below. \end{itemize}\begin{verbatim}

f(x; a=1) = a * log((a + sqrt(a^2 - x^2))/x ) - sqrt(a^2 - x^2)

\end{verbatim}

\begin{verbatim}

f (generic function with 1 method)\end{verbatim}

\newline

The value of \texttt{a} is a parameter, the default value of $a=1$ is fine.\newline

For $x=0.25$ and $x=0.75$ the tangent lines can be drawn with\begin{verbatim}

plot([f, tangent(f, 0.25), tangent(f, 0.75)], 0.01, 0.8)

\end{verbatim}

\begin{html}

\end{html}

\newline

Verify that the length of the tangent line between $(c, f(c))$ and the $y$ axis is the same for $c=0.25$ and $c=0.75$. (For any $c$, the distance formula can be used to find the distance between the point $(c, f(c))$ and $(0, y_0)$ where, $y_0$ is where the tangent line at $c$ crosses the $y$ axis.)

\\begin{answer}

type: longtext

reminder: Verify lengths of two lines are same

answer_text: Write a function to compute length squared: \verb#c^2 + (f(c)-(f(c)+D(f)(c)*(0-c)))^2#

rows: 3

cols: 60

\\end{answer}

\subsubsection{Higher-order derivatives}\newline

Higher-order derivates can be approximated as well. For example, one can use \texttt{D(f,2)} \textit{or}, if defined, \texttt{f''} to approximate the second derivative.\begin{itemize}\item Find the second derivative of $f(x) = \sqrt{x \cdot exp(x)}$ at $c=3$.\end{itemize}

\\begin{answer}

type: numeric

reminder:

answer: [3.0177551230478667, 3.0197551230478665]

answer_text: [3.018, 3.02]

\\end{answer}

\begin{itemize}\item Find the zeros in $[0, 10]$ of the second derivative of the function $f(x) = \sin(2x) + 3\sin(4x)$ using \texttt{fzeros}.\end{itemize}

\\begin{answer}

type: radio

reminder:

values: 3 | 1 | 2

labels: 13 numbers: 0.420534, 1.20943, ..., 9.00424, 9.84531 | 13 numbers: 0.0, 0.806238, ..., 8.61854, 9.42478 | 13 numbers: 0.0, 0.869122, ..., 8.55566, 9.42478

answer: 2

\\end{answer}

\end{document}