4130.sagews

var('x,y,z')

\newcommand{\Bold}[1]{\mathbf{#1}}\left(x, y, z\right)

var('t')

\newcommand{\Bold}[1]{\mathbf{#1}}t

v(t) = [t*e**-t, 2*arctan(t),2*e**t]
v

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(t e^{\left(-t\right)},\,2 \, \arctan\left(t\right),\,2 \, e^{t}\right)

t = v.derivative(t)
t

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(-t e^{\left(-t\right)} + e^{\left(-t\right)},\,\frac{2}{t^{2} + 1},\,2 \, e^{t}\right)

t(0)/abs(t(0))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{1}{3},\,\frac{2}{3},\,\frac{2}{3}\right)

v(t) = [cos(t),3*t,2*sin(2*t)]
v

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(\cos\left(t\right),\,3 \, t,\,2 \, \sin\left(2 \, t\right)\right)

t = v.derivative(t)
t

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(-\sin\left(t\right),\,3,\,4 \, \cos\left(2 \, t\right)\right)

t(0)/abs(t(0))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(0,\,\frac{3}{5},\,\frac{4}{5}\right)

v(t) = [t,t**2,t**3]
v

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(t,\,t^{2},\,t^{3}\right)

t1 = v.derivative(t)
t1

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(1,\,2 \, t,\,3 \, t^{2}\right)

t2 = v.derivative(t,2)
t2

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(0,\,2,\,6 \, t\right)

t1.cross_product(t2)

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(6 \, t^{2},\,-6 \, t,\,2\right)

t1(1)/abs(t1(1))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{1}{14} \, \sqrt{14},\,\frac{1}{7} \, \sqrt{14},\,\frac{3}{14} \, \sqrt{14}\right)

v(t) = [1+2*sqrt(t),t**3-t,t**3+t]
v

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(2 \, \sqrt{t} + 1,\,t^{3} - t,\,t^{3} + t\right)

t1 = v.derivative(t)
t1, t1(1)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t \ {\mapsto}\ \left(\frac{1}{\sqrt{t}},\,3 \, t^{2} - 1,\,3 \, t^{2} + 1\right), \left(1,\,2,\,4\right)\right)

v(1)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(3,\,0,\,2\right)

t1(1).dot_product(vector([x,y,z])), v(1)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(x + 2 \, y + 4 \, z, \left(3,\,0,\,2\right)\right)

v(t) = [e**-t, e**-t*sin(t), e**-t]
v, v(0)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t \ {\mapsto}\ \left(e^{\left(-t\right)},\,e^{\left(-t\right)} \sin\left(t\right),\,e^{\left(-t\right)}\right), \left(1,\,0,\,1\right)\right)

t1 = v.derivative(t)
t1

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \left(-e^{\left(-t\right)},\,-e^{\left(-t\right)} \sin\left(t\right) + e^{\left(-t\right)} \cos\left(t\right),\,-e^{\left(-t\right)}\right)

t1(0)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-1,\,1,\,-1\right)

v(0) +t*t1(0)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-t + 1,\,t,\,-t + 1\right)

v(t) = [t,e**-t, 2*t-t**2]
v, v(0)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t \ {\mapsto}\ \left(t,\,e^{\left(-t\right)},\,-t^{2} + 2 \, t\right), \left(0,\,1,\,0\right)\right)

v(0) + t*v.derivative(t)(0)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t,\,-t + 1,\,2 \, t\right)

v(t) = [t*cos(t), t, t*sin(t)]
v,v(pi)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t \ {\mapsto}\ \left(t \cos\left(t\right),\,t,\,t \sin\left(t\right)\right), \left(-\pi,\,\pi,\,0\right)\right)

v(pi)+t*v.derivative(t)(pi)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-\pi - t,\,\pi + t,\,-\pi t\right)

print 'problem 31'

problem 31

def par(ex, t_s, t_e): 
    return parametric_plot3d(ex,(t,t_s,t_e),thickness=10)

f1(t) = [t,t**2,t**3]
f2(t) = [sin(t),sin(2*t),t]
f1,f2

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t \ {\mapsto}\ \left(t,\,t^{2},\,t^{3}\right), t \ {\mapsto}\ \left(\sin\left(t\right),\,\sin\left(2 \, t\right),\,t\right)\right)

t_s, t_e = -.25, 1
graph = par(f1,t_s,t_e)
graph += par(f2,t_s,t_e)
graph += point([0,0,0], size=20,color='red')
show(graph)

t1 = f1.derivative(t)
t2 = f2.derivative(t)
t1,t2

\newcommand{\Bold}[1]{\mathbf{#1}}\left(t \ {\mapsto}\ \left(1,\,2 \, t,\,3 \, t^{2}\right), t \ {\mapsto}\ \left(\cos\left(t\right),\,2 \, \cos\left(2 \, t\right),\,1\right)\right)

dp = t1(0).dot_product(t2(0))
mag = abs(t1(0)) * abs(t2(0))
angle = acos(dp/mag)
angle = round(angle,3)
print 'angle between tangent vectors at origin: %s radians' % angle
print 'or %s degrees' % round(angle*180/pi,3)

angle between tangent vectors at origin: 1.15 radians or 65.89 degrees

print 'problem 33'

problem 33

var('i,j,k')

\newcommand{\Bold}[1]{\mathbf{#1}}\left(i, j, k\right)

f(t) = 16*t**3*i - 9*t**2*j + 25*t**4*k
f

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ 25 \, k t^{4} + 16 \, i t^{3} - 9 \, j t^{2}

int = integral(f,t)
int

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ 5 \, k t^{5} + 4 \, i t^{4} - 3 \, j t^{3}

int(1)-int(0)

\newcommand{\Bold}[1]{\mathbf{#1}}4 \, i - 3 \, j + 5 \, k

print 'problem 35'

problem 35

f(t) = 3*sin(t)**2*cos(t)*i + 3*sin(t)*cos(t)**2*j+2*sin(t)*cos(t)*k
f

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ 3 \, i \sin\left(t\right)^{2} \cos\left(t\right) + 3 \, j \sin\left(t\right) \cos\left(t\right)^{2} + 2 \, k \sin\left(t\right) \cos\left(t\right)

int = integral(f,t)
int

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ i \sin\left(t\right)^{3} - j \cos\left(t\right)^{3} - k \cos\left(t\right)^{2}

int(pi/2)-int(0)

\newcommand{\Bold}[1]{\mathbf{#1}}i + j + k

print 'problem 37'

problem 37

f(t) = e**t*i + 2*t*j + ln(t)*k
f

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ i e^{t} + 2 \, j t + k \log\left(t\right)

int = integral(f,t)
int

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ j t^{2} + {\left(t \log\left(t\right) - t\right)} k + i e^{t}

int.differentiate(t)

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ i e^{t} + 2 \, j t + k \log\left(t\right)

print 'problem 39'

problem 39

f(t) = 2*t*i + 3*t**2*j + sqrt(t)*k
f

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ 3 \, j t^{2} + 2 \, i t + k \sqrt{t}

int = f.integral(t)
int

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ j t^{3} + i t^{2} + \frac{2}{3} \, k t^{\left(\frac{3}{2}\right)}

int(1)

\newcommand{\Bold}[1]{\mathbf{#1}}i + j + \frac{2}{3} \, k

int -= 2/3*k
int

\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ j t^{3} + i t^{2} + \frac{2}{3} \, k t^{\left(\frac{3}{2}\right)} - \frac{2}{3} \, k

int(1)

\newcommand{\Bold}[1]{\mathbf{#1}}i + j