Differences between revisions 82 and 83
 ⇤ ← Revision 82 as of 2016-10-11 01:42:43 → Size: 62750 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 228: Line 228: pretty_print(html(r''' html(r''' Line 238: Line 238: ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer))) ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)) 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)