Differences between revisions 108 and 111 (spanning 3 versions)
 ⇤ ← Revision 108 as of 2020-06-02 14:03:09 → Size: 62939 Editor: kcrisman Comment: ← Revision 111 as of 2020-06-02 14:13:05 → ⇥ Size: 62966 Editor: kcrisman Comment: Deletions are marked like this. Additions are marked like this. Line 527: Line 527: 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))) Line 571: Line 571: pretty_print(html('

A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$

'))    pretty_print(html('Below is the unit circle, so the length of the red line is |sin(x)|'))    pretty_print(html('and the length of the blue line 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'

A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$

'))    pretty_print(html(r'Below is the unit circle, so the length of the red line is |sin(x)|'))    pretty_print(html(r'and the length of the blue line 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 $\lim_{x -> 0} \sin(x)/x =1$.')) Line 628: Line 628: pretty_print(html("

The midpoint rule for a function of two variables

")) pretty_print(html(r"

The midpoint rule for a function of two variables

")) Line 643: Line 643: 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('

Numerical approximation: ' + str(num_approx)+'

')) 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'

Numerical approximation: ' + str(num_approx)+r'

'))

# Sage Interactions - Calculus

## 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 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 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  ## 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 (FIXME in Jupyter)

by John Travis ## Parametric surface example (FIXME in Jupyter)

by Marshall Hampton ## Line Integrals in 3D Vector Field

by John Travis interact/calculus (last edited 2020-08-11 14:10:09 by kcrisman)