2016-04-29.sagews

Math 480 - Homework 4: Due 6pm on April 29, 2016

There are several problems and all have equal weights.

Solve each problem using Sage unless otherwise indicated. In particular, if there is some calculus question, which you could easily do by hand or in your head, you should still show exactly how to solve it using Sage if possible. Of course, do think with your brain!

For any part that asks for a symbolic result, use show(...), so the output is easier for us to read.

Problem 1 -- defining and evaluating a function:

1. Define in Sage $f(x) = \sinh(x^2+\sqrt{x-1}) + e^{\pi x} + \arcsin(x) + \frac{1}{x^3-x-e}$.

2. Compute $f(1/2)$ symbolically (exactly).

3. Compute $f(1/2)$ numerically (so a decimal expansion).

4. Plot $f(x)$ from $-1$ to $1$.

f(x) = sinh(x^2 + (x-1)) + e^(pi*x) + asin(x) + 1/(x^2 - x- e)
f(0.5)
N(f(0.5))
plot(f, (x, -1, 1))

1/6*pi - 1.00000000000000/(1.00000000000000*e + 0.250000000000000) + e^(0.500000000000000*pi) - 0.252612316808168 4.74456860615189

Problem 2 -- finding zeros numerically:

Let $f(x) = \displaystyle x^2 + \sin(x)$

1. Draw a plot of $f$ on the interval $[-2,2]$.

2. Differentiate $f$

3. Integrate $f$

4. Find all the zeros of $f(x)$ numerically.

f(x) = x^2 + sin(x)
plot(f, -2,2)
f.find_root(-2,-1/2)
f.find_root(-1/2,1/2)
# (use brain) no other zeros because $x^2$ is big.

-0.8767262153950622 5.78218635220173e-23

Problem 3 -- The Cauchy Distribution

Let $\displaystyle\space f(x;\space x_0,\space\gamma) = \frac{1}{\pi\gamma}\left[\frac{\gamma^2}{(x - x_0)^2 + \gamma^2}\right]$

1. Plot and find the area under the curve of $f([-2, 2])$ for the following values of $x_0$ and $\gamma$ on the interval $x = [-4,4]$

• $x_0 = 0,\hspace{3mm} \gamma = 1$

• $x_0 = 2,\hspace{3mm} \gamma = 2$

• $x_0 = 0,\hspace{3mm} \gamma = 0.5$

1. Integrate $f$ from $-\infty$ to $x$ using the dummy variable $t$ as in $f(t;\space x_0,\space\gamma)$.

2. Plot the resulting function from 3 for $x_0 = 0,\hspace{3mm} \gamma = 0.5$.

# 1a.
gamma = 1
x_0 = 0
f(x) = 1/(pi*gamma) * (gamma^2/((x - x_0)^2 + gamma^2))
p = plot(f, -2, 2, fill=True) + plot(f, -4, 4)
show(p)
show(integral(f, x, -2, 2))

2*arctan(2)/pi

# 1b.
gamma = 2
x_0 = 2
f(x) = 1/(pi*gamma) * (gamma^2/((x - x_0)^2 + gamma^2))
p = plot(f, -2, 2, fill=True) + plot(f, -4, 4)
show(p)
show(integral(f, x, -2, 2))

$\displaystyle \frac{\arctan\left(2\right)}{\pi}$

# 1c.
gamma =0.5
x_0 = 0
f(x) = 1/(pi*gamma) * (gamma^2/((x - x_0)^2 + gamma^2))
p = plot(f, -2, 2, fill=True) + plot(f, -4, 4)
show(p)
show(integral(f, x, -2, 2))

$\displaystyle \frac{2.0 \, \arctan\left(4\right)}{\pi}$

var('gamma', 'x_0')
f(t) = 1/(pi*gamma) * (gamma^2/((t - x_0)^2 + gamma^2))
show(integral(f(t), t, -oo, x))

(gamma, x_0)

$\displaystyle \frac{\gamma {\left(\frac{\pi}{\gamma} - \frac{2 \, \arctan\left(-\frac{x - x_{0}}{\gamma}\right)}{\gamma}\right)}}{2 \, \pi}$

# 2. & 3.
f(t) = 1/(pi*gamma) * (gamma^2/((t - x_0)^2 + gamma^2))
F(x) = integral(f(t), t, -oo, x)
show(F)
print ""
plot(F,-4, 4)

$\displaystyle x \ {\mapsto}\ \frac{0.5 \, {\left(\pi + 2 \, \arctan\left(2 \, x\right)\right)}}{\pi}$

Problem 4 -- a function with no elementary antiderivative:

Let $f(x) = \sin(x^2) + \exp(1/x)$

1. Draw a plot of $f$ on the interval $[1/2, 4]$.

2. Differentiate $f$

3. Integrate $f$

Problem 5: Limits

1. Compute $\lim_{x\to 0} \sin(x)/x$

2. Use Sage to verify that strange and amazing fact $\lim_{x\to 0} (\cos x)^{1/x^2} = \frac{1}{\sqrt{e}}$.

# Solution

limit(sin(x)/x, x=0)
limit(cos(x)^(1/x^2), x=0)

1 e^(-1/2)

Problem 6: Taylor Series

Let $f(x) = \sin(x^2)$

1. Find the 3rd degree taylor series, $p_3(x)$, of $f$ where $x_0 = 2\pi$

2. Plot the 10th degree taylor series $p_{10}(x)$ where $x_0 = 2\pi$ alongside $f$

• Plot on the interval $x=[\pi, 3\pi

# 1
f(x) = sin(x^2)
p_3(x) = taylor(f(x), x, 2*pi, 3)
show(p_3)

$\displaystyle x \ {\mapsto}\ \frac{4}{3} \, {\left(2 \, \pi - x\right)}^{3} {\left(8 \, \pi^{3} \cos\left(4 \, \pi^{2}\right) + 3 \, \pi \sin\left(4 \, \pi^{2}\right)\right)} - {\left(2 \, \pi - x\right)}^{2} {\left(8 \, \pi^{2} \sin\left(4 \, \pi^{2}\right) - \cos\left(4 \, \pi^{2}\right)\right)} - 4 \, \pi {\left(2 \, \pi - x\right)} \cos\left(4 \, \pi^{2}\right) + \sin\left(4 \, \pi^{2}\right)$

# 2
f(x) = sin(x^2)
p_10(x) = taylor(f(x), x, 2*pi, 10)

P1 = plot(p_10(x), pi, 3*pi, ymax=2, ymin=-2)
P2 = plot(f(x), pi, 3*pi, linestyle='--')

show(P1 + P2)

Problem 7 - Gradient Vector Field:

1. Compute the gradient of $f(x,y) = 3\sin(x) - 2\cos(2y) - x - y$.

2. Plot the 2-dimensiona vector field defined by the gradient of $f$ in the rectangle $-2 \leq x,y \leq 2$.

f(x,y) = 3*sin(x) - 2*cos(2*y) - x- y
plot_vector_field(f.gradient(), (x,-2,2), (y,-2,2))

Problem 8 - Symbolic Sums:

1. Compute $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$.

2. Compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

3. Compute $\sum_{n=1}^{\infty} \frac{1}{n^3}$ both symbolically (in terms of the Riemann Zeta function) and numerically.

4. Compute $\sum_{n=1}^{\infty} \frac{1}{n^4}$.

5. Compute $\sum_{n=1}^k \sin(n)$.

# Solution

%var n
show(sum((-1)^n/n, n, 1, oo))

$\displaystyle -\log\left(2\right)$

%var n
show(sum(1/n^2, n, 1, oo))

$\displaystyle \frac{1}{6} \, \pi^{2}$

%var n
show(sum(1/n^3, n, 1, oo))
N(sum(1/n^3, n, 1, oo))

$\displaystyle \zeta(3)$

1.20205690315959

%var n
show(sum(1/n^4, n, 1, oo))

$\displaystyle \frac{1}{90} \, \pi^{4}$

Problem 9 -- Unit Conversion:

Use Sage's units functionality (written by a UW undergrad -- David Ackerman!)

1. Convert 68 degrees Fahrenheit to Celcius. Hint: use 68*units.temperature.fahrenheit to define fahrenheit.

2. Convert 15 milliseconds to hours.

3. Convert 2016 degrees kelvins to degrees Fahrenheit.

4. Convert 9.8 meters per second squared to feet per second squared.

# Solution:

a = 68*units.temperature.fahrenheit
a.convert(units.temperature.celsius)

20*celsius

Problem 10 - 3d Plotting:

1. Draw a 3d plot of a torus.

2. Draw a single 3d plot the has the five regular polytopes in it: tetrahedron, cube, octahedron, dodecahedron, icosahedron. All five must be visible.

3. Draw a 3d plot of the "Mexican hat function" (see, e.g., https://en.wikipedia.org/wiki/Mexican_hat_wavelet). [Hint: you have to make a choice of parameter $\sigma$ so that it looks like Mexican hat.]

# Solution:

u, v = var('u,v')
f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v))
parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), texture="red", mesh=2)

3D rendering not yet implemented

tetrahedron() + cube().translate((1,0,0)) + octahedron().translate((2,0,0)) + dodecahedron().translate((3,0,0)) + icosahedron().translate((4,0,0))

3D rendering not yet implemented

sigma = -1/2
f(x,y) = 1/(pi*sigma^4) * (1-((x^2+y^2)/(2*sigma^2)))*exp(-(x^2+y^2)/(2*sigma^2))
plot3d(f, (x,-2,2), (y,-2,2))

3D rendering not yet implemented