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| Deletions are marked like this. | Additions are marked like this. |
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| raise ValueError, "f must have a sign change in the interval (%s,%s)"%(a,b) | raise ValueError("f must have a sign change in the interval (%s,%s)"%(a,b)) |
| Line 155: | Line 155: |
| pretty_print(html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')) | pretty_print(html(r'$\int_{a}^{b}{f(x) dx} {\approx} \sum_i{f(x_i) \Delta x}$')) |
| Line 446: | Line 446: |
| == Coordinate Transformations (FIXME in Jupyter) == | == Coordinate Transformations == |
| Line 466: | Line 466: |
| u_percent=slider(0,1,0.05,label="<font color='red'>u</font>", default=.7), v_percent=slider(0,1,0.05,label="<font color='blue'>v</font>", default=.7), |
u_percent=slider(0,1,0.05,label="u", default=.7), v_percent=slider(0,1,0.05,label="v", default=.7), |
| Line 497: | Line 497: |
| jacobian=abs(T.diff().det()).simplify_full() | jacobian(u,v)=abs(T.diff().det()).simplify_full() |
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| pretty_print(table([[uvplot,xyplot]])) | show(graphics_array([uvplot,xyplot])) |
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| pretty_print(html('$f(x)\;=\;%s$'%latex(f))) pretty_print(html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))) |
pretty_print(html(r'$f(x)\;=\;%s$'%latex(f))) pretty_print(html(r'$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))) |
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| pretty_print(html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>')) pretty_print(html('Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|')) pretty_print(html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.')) pretty_print(html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.')) pretty_print(html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.')) pretty_print(html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.')) |
pretty_print(html(r'<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>')) pretty_print(html(r'Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|')) pretty_print(html(r'and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.')) pretty_print(html(r'From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.')) pretty_print(html(r'It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.')) pretty_print(html(r'As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.')) |
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| pretty_print(html("<h1>The midpoint rule for a function of two variables</h1>")) | pretty_print(html(r"<h1>The midpoint rule for a function of two variables</h1>")) |
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| pretty_print(html("$$\int_{"+str(R16(y_start))+"}^{"+str(R16(y_end))+"} "+ "\int_{"+str(R16(x_start))+"}^{"+str(R16(x_end))+"} "+func+"\ dx \ dy$$")) pretty_print(html('<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+'</p>')) |
pretty_print(html(r"$\int_{"+str(R16(y_start))+r"}^{"+str(R16(y_end))+r"} "+ r"\int_{"+str(R16(x_start))+r"}^{"+str(R16(x_end))+r"} "+latex(SR(func))+r"\ dx \ dy$")) pretty_print(html(r'<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+r'</p>')) |
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| from numpy import linspace | from numpy import linspace, asanyarray, diff |
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| y_val = map(scaled_ff,x_val) | y_val = [*map(scaled_ff,x_val)] |
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| pretty_print(html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n, | pretty_print(html(r"$$\sum_{i=1}^{i=%s}w_i\left(%s\right)= %s\approx %s =\int_{-1}^{1}%s \,dx$$"%(n, |
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| pretty_print(html('\n<center><font color="red">$\lim_{(x,?)\\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+' and <font color="red">$\lim_{(?,y)\\rightarrow(x_0,y_0)} f(x,y) =%s$</font></center>'%str(limit_y))) | pretty_print(html(r'<center><font color="red">$\lim_{(x,?)\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+r' and <font color="red">$\lim_{(?,y)\rightarrow(x_0,y_0)} f(x,y) =%s$</font></center>'%str(limit_y))) |
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| pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty $.')) | pretty_print(html(r'Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\rightarrow \infty $.')) |
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| pretty_print(table([[surface],['hi']])) | show(surface) |
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| == Lateral Surface Area (FIXME in Jupyter) == | == Lateral Surface Area == |
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Note that this works in Sage cell, but causes a zip file error in Jupyter |
|
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| == Parametric surface example (FIXME in Jupyter) == | == Parametric surface example == |
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Note that this works in Sage cell, but causes a zip file error in Jupyter. |
|
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| u(t) = u v(t) = v w(t) = w |
Sage Interactions - Calculus
goto interact main page
Contents
-
Sage Interactions - Calculus
- Root Finding Using Bisection
- Newton's Method
- A contour map and 3d plot of two inverse distance functions
- A simple tangent line grapher
- Numerical integrals with the midpoint rule
- Numerical integrals with various rules
- Some polar parametric curves
- Function tool
- Newton-Raphson Root Finding
- Coordinate Transformations
- Taylor Series
- Illustration of the precise definition of a limit
- A graphical illustration of sin(x)/x -> 1 as x-> 0
- Quadric Surface Plotter
- The midpoint rule for numerically integrating a function of two variables
- Gaussian (Legendre) quadrature
- Vector Calculus, 2-D Motion
- Vector Calculus, 3-D Motion
- Multivariate Limits by Definition
- Directional Derivatives
- 3D graph with points and curves
- Approximating function in two variables by differential
- Taylor approximations in two variables
- Volumes over non-rectangular domains
- Lateral Surface Area
- Parametric surface example
- Line Integrals in 3D Vector Field
Root Finding Using Bisection
by William Stein
Newton's Method
Note that there is a more complicated Newton's method below.
by William Stein
A contour map and 3d plot of two inverse distance functions
by William Stein
A simple tangent line grapher
by Marshall Hampton
Numerical integrals with the midpoint rule
by Marshall Hampton
Numerical integrals with various rules
by Nick Alexander (based on the work of Marshall Hampton)
Some polar parametric curves
by Marshall Hampton. This is not very general, but could be modified to show other families of polar curves.
Function tool
Enter symbolic functions f, g, and a, a range, then click the appropriate button to compute and plot some combination of f, g, and a along with f and g. This is inspired by the Matlab funtool GUI.
Newton-Raphson Root Finding
by Neal Holtz
This allows user to display the Newton-Raphson procedure one step at a time. It uses the heuristic that, if any of the values of the controls change, then the procedure should be re-started, else it should be continued.
Coordinate Transformations
by Jason Grout
Taylor Series
by Harald Schilly
Illustration of the precise definition of a limit
by John Perry
I'll break tradition and put the image first. Apologies if this is Not A Good Thing.
A graphical illustration of sin(x)/x -> 1 as x-> 0
by Wai Yan Pong
Quadric Surface Plotter
by Marshall Hampton. This is pretty simple, so I encourage people to spruce it up. In particular, it isn't set up to show all possible types of quadrics.
The midpoint rule for numerically integrating a function of two variables
by Marshall Hampton
Gaussian (Legendre) quadrature
by Jason Grout
The output shows the points evaluated using Gaussian quadrature (using a weight of 1, so using Legendre polynomials). The vertical bars are shaded to represent the relative weights of the points (darker = more weight). The error in the trapezoid, Simpson, and quadrature methods is both printed out and compared through a bar graph. The "Real" error is the error returned from scipy on the definite integral.
Vector Calculus, 2-D Motion
By Rob Beezer
A fast_float() version is available in a worksheet
Vector Calculus, 3-D Motion
by Rob Beezer
Available as a worksheet
Multivariate Limits by Definition
by John Travis
http://sagenb.mc.edu/home/pub/97/
Directional Derivatives
This interact displays graphically a tangent line to a function, illustrating a directional derivative (the slope of the tangent line).
3D graph with points and curves
By Robert Marik
This sagelet is handy when showing local, constrained and absolute maxima and minima in two variables. Available as a worksheet
Approximating function in two variables by differential
by Robert Marik
Taylor approximations in two variables
by John Palmieri
This displays the nth order Taylor approximation, for n from 1 to 10, of the function sin(x2 + y2) cos(y) exp(-(x2+y2)/2).
Volumes over non-rectangular domains
by John Travis
Lateral Surface Area
by John Travis
http://sagenb.mc.edu/home/pub/89/
Note that this works in Sage cell, but causes a zip file error in Jupyter
Parametric surface example
by Marshall Hampton
Note that this works in Sage cell, but causes a zip file error in Jupyter.
Line Integrals in 3D Vector Field
by John Travis
