Differences between revisions 81 and 83 (spanning 2 versions)
 ⇤ ← Revision 81 as of 2016-10-11 01:41:11 → Size: 62694 Editor: kcrisman Comment: ← Revision 83 as of 2016-10-11 01:49:50 → ⇥ Size: 63360 Editor: kcrisman Comment: Deletions are marked like this. Additions are marked like this. Line 118: Line 118: html('

Tangent line grapher

') pretty_print(html('

Tangent line grapher

')) Line 158: Line 158: html('

Numerical integrals with the midpoint rule

')    html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$') pretty_print(html('

Numerical integrals with the midpoint rule

'))    pretty_print(html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')) Line 249: Line 249: html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$') pretty_print(html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$')) Line 336: Line 336: html('
$f = %s$
'%latex(f))    html('
$g = %s$
'%latex(g))    html('
$h = %s = %s$
'%(lbl, latex(h))) pretty_print(html('
$f = %s$
'%latex(f)))    pretty_print(html('
$g = %s$
'%latex(g)))    pretty_print(html('
$h = %s = %s$
'%(lbl, latex(h)))) Line 517: Line 517: html("$T(u,v)=%s$"%(latex(T(u,v))))    html("Jacobian: $%s$"%latex(jacobian(u,v)))    html("A very small region in $xy$ plane is approximately %0.4g times the size of the corresponding region in the $uv$ plane"%jacobian(u_val,v_val).n()) pretty_print(html("$T(u,v)=%s$"%(latex(T(u,v)))))    pretty_print(html("Jacobian: $%s$"%latex(jacobian(u,v))))    pretty_print(html("A very small region in $xy$ plane is approximately %0.4g times the size of the corresponding region in the $uv$ plane"%jacobian(u_val,v_val).n())) Line 539: Line 539: html('$f(x)\;=\;%s$'%latex(f))    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1)) 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))) Line 553: Line 553: html("

Limits: ε-δ

")html("This allows you to estimate which values of δ guarantee that f is within ε units of a limit.")html("
• Modify the value of f to choose a function.
• ")html("
• Modify the value of a to change the x-value where the limit is being estimated.
• ")html("
• Modify the value of L to change your guess of the limit.
• ")html("
• Modify the values of δ and ε to modify the rectangle.
")html("If the blue curve passes through the pink boxes, your values for δ and/or ε are probably wrong.") pretty_print(html("

Limits: ε-δ

"))pretty_print(html("This allows you to estimate which values of δ guarantee that f is within ε units of a limit."))pretty_print(html("
• Modify the value of f to choose a function.
• "))pretty_print(html("
• Modify the value of a to change the x-value where the limit is being estimated.
• "))pretty_print(html("
• Modify the value of L to change your guess of the limit.
• "))pretty_print(html("
• Modify the values of δ and ε to modify the rectangle.
"))pretty_print(html("If the blue curve passes through the pink boxes, your values for δ and/or ε are probably wrong.")) Line 583: Line 583: html('

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

')    html('Below is the unit circle, so the length of the red line is |sin(x)|')    html('and the length of the blue line is |tan(x)| where x is the length of the arc.')     html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.')    html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.')    html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.') 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 $\lim_{x -> 0} \sin(x)/x =1$.')) Line 640: Line 640: html("

The midpoint rule for a function of two variables

") pretty_print(html("

The midpoint rule for a function of two variables

")) Line 655: Line 655: html("$$\int_{"+str(R16(y_start))+"}^{"+str(R16(y_end))+"} "+ "\int_{"+str(R16(x_start))+"}^{"+str(R16(x_end))+"} "+func+"\ dx \ dy$$")    html('

Numerical approximation: ' + str(num_approx)+'

') 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)+'

')) Line 727: Line 727: html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,        latex(f), approximation, integral, latex(scaled_func))) pretty_print(html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,        latex(f), approximation, integral, latex(scaled_func)))) Line 1021: Line 1021: html('Enter $(x_0 ,y_0 )$ above and see what happens as $R \\rightarrow 0$.')    html('The surface has a limit as $(x,y) \\rightarrow$ ('+str(x0)+','+str(y0)+') if the green region collapses to a point.') pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as $R \\rightarrow 0$.'))    pretty_print(html('The surface has a limit as $(x,y) \\rightarrow$ ('+str(x0)+','+str(y0)+') if the green region collapses to a point.')) Line 1042: Line 1042: html('The red curves represent a couple of trajectories on the surface. If they do not meet, then')        html('there is also no limit. (If computer hangs up, likely the computer can not do these limits.)')        html('\n
$\lim_{(x,?)\\rightarrow(x_0,y_0)} f(x,y) =%s$'%str(limit_x)+' and $\lim_{(?,y)\\rightarrow(x_0,y_0)} f(x,y) =%s$
'%str(limit_y)) pretty_print(html('The red curves represent a couple of trajectories on the surface. If they do not meet, then'))        pretty_print(html('there is also no limit. (If computer hangs up, likely the computer can not do these limits.)'))        pretty_print(html('\n
$\lim_{(x,?)\\rightarrow(x_0,y_0)} f(x,y) =%s$'%str(limit_x)+' and $\lim_{(?,y)\\rightarrow(x_0,y_0)} f(x,y) =%s$
'%str(limit_y))) Line 1081: Line 1081: html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty$.')    html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.') pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty$.'))    pretty_print(html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.')) Line 1172: Line 1172: html(r'Function $f(x,y)=%s$ '%latex(f(x,y))) pretty_print(html(r'Function $f(x,y)=%s$ '%latex(f(x,y)))) Line 1186: Line 1186: html(r'
$\quad f(%s,%s)\quad $$\quad %s \quad f(%s,%s)\quad$$\quad %s$
\              '%(latex(x0),latex(y0),z0.n())) pretty_print(html(r'\              '%(latex(x0),latex(y0),z0.n()))) Line 1220: Line 1220: html('Points x0 and y0 are values where the exact value of the function \ pretty_print(html('Points x0 and y0 are values where the exact value of the function \ Line 1222: Line 1222: and approximation by differential at shifted point are compared.') and approximation by differential at shifted point are compared.')) Line 1240: Line 1240: html(r'Function $f(x,y)=%s \approx %s$ '%(latex(f(x,y)),latex(tangent(x,y))))  html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori)))  html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay))))  html(r'Value of the function in shifted point is $%s$'%f(x0+deltax,y0+deltay))  html(r'Value on the tangent plane in shifted point is $%s$'%latex(approx_value))  html(r'Error is $%s$'%latex(abs_error)) pretty_print(html(r'Function $f(x,y)=%s \approx %s$ '%(latex(f(x,y)),latex(tangent(x,y)))))  pretty_print(html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori))))  pretty_print(html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay)))))  pretty_print(html(r'Value of the function in shifted point is $%s$'%f(x0+deltax,y0+deltay)))  pretty_print(html(r'Value on the tangent plane in shifted point is $%s$'%latex(approx_value)))  pretty_print(html(r'Error is $%s$'%latex(abs_error))) Line 1282: Line 1282: html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$') pretty_print(html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$')) Line 1440: Line 1440: html(r'Lateral Surface $\approx$ %s'%str(line_integral_approx)) pretty_print(html(r'Lateral Surface $\approx$ %s'%str(line_integral_approx))) Line 1524: Line 1524: html(r'

$\int_{C} \left \langle M,N,P \right \rangle dr$ = $%s$

'%latex(line_integral)) pretty_print(html(r'

$\int_{C} \left \langle M,N,P \right \rangle dr$ = $%s$

'%latex(line_integral)))

# Sage Interactions - Calculus

by William Stein

## Newton's Method

Note that there is a more complicated Newton's method below.

by William Stein

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.

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 (FIXME)

by Rob Beezer

Available as a worksheet

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

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).

by John Travis

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)