The ordinary precedence rules apply to mathematical operations: multiplication and division are done ﬁrst, then addition and subtraction. Multiplication and division have the same precedence, as do addition and subtraction; a sequence of mixed multiplications and divisions is evaluated left to right, and likewise for additions and subtractions. Make sure you understand these:

︡3965f67c-e50d-4b0c-86ee-430f433cc1e1︡{"html": "

The ordinary precedence rules apply to mathematical operations: multiplication and division are done\nﬁrst, then addition and subtraction. Multiplication and division have the same precedence, as do addition\nand subtraction; a sequence of mixed multiplications and divisions is evaluated left to right, and likewise\nfor additions and subtractions. Make sure you understand these:\n

"}︡ ︠abbf7aa7-8707-49a7-83d4-8ac0b905d91d︠ 24/3/2; 24/3*2; 24/4+1 ︡b0ba5c67-0320-44b6-a6f4-7a4167922c08︡︡ ︠59c6055e-190f-421b-b468-c3205396f674i︠ %html

You may of course use parentheses to alter the order of operations. Sometimes using extra parentheses is a good idea because it makes it easier to read and understand an expression.

Sage will do arithmetic with rational numbers without converting to decimal form:

︡e9a697ea-620f-4d23-a8b4-5c359dcacba5︡{"html": "

You may of course use parentheses to alter the order of operations. Sometimes using extra parentheses is a\ngood idea because it makes it easier to read and understand an expression.\n

Sage will do arithmetic with rational numbers without converting to decimal form:\n

"}︡ ︠1a18ddb0-9d68-4052-9021-1bf085a3cf10︠ 2/5 + 1/3; 23/45-167/345*2/3 ︡8c8ebd6f-5187-4e46-985a-15eb7c4cd7ce︡︡ ︠506a96be-5692-4df9-82f9-b2f9bfdf8338i︠ %html

Sage knows standard mathematical notation, including two symbols for exponentiation. The underscore character refers to the previous result.

︡13957908-5778-4820-95fb-2311ca537d66︡{"html": "

Sage knows standard mathematical notation, including two symbols for exponentiation. The underscore\ncharacter refers to the previous result.\n

"}︡ ︠cdced045-67d5-48bd-afaa-698b1288c098︠ 2**5; 3^27; factor(_) ︡3ae51307-ea1b-4240-9d3d-2eb2bb074f7f︡︡ ︠e3004f22-8b97-4a42-a040-e125ea3bf640i︠ %html

You can force Sage to approximate values using decimal notation.

︡bb9f6f82-1526-4819-b7da-65e22ce96cb9︡{"html": "

You can force Sage to approximate values using decimal notation.\n

"}︡ ︠50441c45-007b-4d02-9bd7-122f2aa5dedb︠ 12345/98765; n(_) ︡0b4fd256-3366-4362-b09f-74f1e7fc4278︡︡ ︠080984b7-f2ec-4460-8ac4-20dd07aa2bcfi︠ %html

Sage understands $\pi$, $e$, and $i$ (the square root of $ − 1$).

︡6b25f7f3-fa12-4712-ae87-1db47e8a5d57︡{"html": "

Sage understands $\\pi$, $e$, and $i$ (the square root of $ − 1$).

"}︡ ︠d5ed5e12-0b6e-44a2-984b-c3845efd369d︠ 3*pi; n(_); n(e); i^2 ︡e5588269-fd4f-45db-8307-4b1410330a95︡︡ ︠cce9b69c-8e2d-45ba-a2b7-1caa9ebc3490i︠ %html

Sage’s normal output can be a bit hard to read when it is complicated, though it becomes easier with practice, and it can be useful for copying results and pasting them in to new commands. At the top of the worksheet is a checkbox labeled Typeset. Check the box and then evaluate the previous cell; the output will be more familiar. Alternately, you may use the jsmath function to typeset a single result. Uncheck the Typeset box at the top of the page and then execute the next box.

︡8b368c3d-bf92-485b-9a81-ef86d445156d︡{"html": "

Sage’s normal output can be a bit hard to read when it is complicated, though it becomes easier with\npractice, and it can be useful for copying results and pasting them in to new commands. At the top of the\nworksheet is a checkbox labeled Typeset. Check the box and then evaluate the previous cell;\nthe output will be more familiar. Alternately, you may use the jsmath function to typeset\na single result. Uncheck the Typeset box at the top of the page and then execute the next\nbox.\n

"}︡ ︠6e9e5c71-5a52-4ecc-a4de-fc6c50f030ab︠ sqrt(2); jsmath(sqrt(2)) ︡8e9a6e7a-a550-43a6-b676-4d2edf871e55︡︡ ︠8f14f308-952b-4804-8e49-b76869fc95f8i︠ %html

The function $\mathop{ exp}\nolimits (x)$ is an alternate form of ${e}^{x}$.

︡0a06e098-c029-43a0-a9a8-57294373234e︡{"html": "

The function $\\mathop{ exp}\\nolimits (x)$ is\nan alternate form of ${e}^{x}$.\n

"}︡ ︠91715ef5-3ca5-4bfc-9981-33a86e17bad7︠ e^2; exp(2) ︡88787a24-42ed-409f-ae73-9a102851cc96︡︡ ︠4f7ea87f-ed46-47ef-9aac-f99472882f10i︠ %html

1.2 Variables and Expressions

Sage can manipulate expressions involving variables as well as numbers. Variables can be assigned values, either actual numeric values or entire expressions.

︡5351ca12-7251-4818-948c-836015a32860︡{"html": "

1.2 Variables and Expressions

Sage can manipulate expressions involving variables as well as numbers. Variables can be assigned values,\neither actual numeric values or entire expressions.\n

"}︡ ︠75aa3d0c-0473-4956-94a1-7803a9034562︠ f=(x+5)^2; f; expand(f); factor(_) ︡cbdafa2d-eccb-4edb-9be4-94ea4586222f︡︡ ︠dad94203-6525-48b1-a34d-5cca144089e0i︠ %html

For readability you can also put diﬀerent commands on diﬀerent lines; just press return without the shift at the end of the line. Note that only the last result will be displayed in this case, so generally you use this when the other lines produce no output.

︡3a93db51-53f6-45f0-8c39-410b58820f4a︡{"html": "

For readability you can also put diﬀerent commands on diﬀerent lines; just press return without the shift\nat the end of the line. Note that only the last result will be displayed in this case, so generally you use this\nwhen the other lines produce no output.\n

"}︡ ︠a308c0d6-5c52-4f45-874f-36f08ab56b69︠ f=(x+5)^2 expand(f) factor(_) ︡a5f05f30-d13c-4046-ab80-809870d0fd0f︡︡ ︠fb6779dd-588f-45b8-bf07-4e847b04b967︠ f=(x+5)^2 expand(f); factor(_) ︡5a5f8022-0e8b-47b3-884f-31c700c6908c︡︡ ︠2c349d21-bbb7-46ec-8def-a336c68dd44di︠ %html

We normally think of $f = {(x + 5)}^{2}$ as a function, but it is not oﬃcially a function.

︡16199dc7-7dcb-44bd-b798-f58361067556︡{"html": "

We normally think of $f = {(x + 5)}^{2}$\nas a function, but it is not oﬃcially a function.\n

"}︡ ︠9275d408-4f3d-4d87-b26e-61385394e02a︠ f(5) ︡fc76d449-7260-4469-afb7-8da77d662ef0︡︡ ︠d79e0d7d-529e-40c6-83b0-df1b7a4e2277i︠ %html

So that gives the result you expect, but only after a warning. Here’s how to make $f$ an oﬃcial function:

︡0ab4abfd-cf52-4eb0-8a51-855fbb1a020e︡{"html": "

So that gives the result you expect, but only after a warning. Here’s how to make\n$f$ an\noﬃcial function:\n

"}︡ ︠0b90e55f-7cc3-4660-898b-dba7f119f984︠ f(x) = (x+5)^2; f(5) ︡b4c04d05-f8c7-4b37-8af3-56be85b2f354︡︡ ︠e02b5317-0d06-4b0c-9fd0-fbbbdb7d3863i︠ %html

You should get in the habit of deﬁning functions this way. You can see the value of $f$ by evaluating it:

︡39ab520b-61ae-4833-8dca-d8f4354566c4︡{"html": "

You should get in the habit of deﬁning functions this way. You can see the value of\n$f$ by\nevaluating it:\n

"}︡ ︠8fce23d2-7168-4df1-9642-e2bad24d1ed3︠ f ︡2c161199-1ee0-4522-b2fb-f3959a22e5e1︡︡ ︠1bc1c609-67ac-4f09-97c2-b87d114f1fb1i︠ %html

This notation indicates that Sage thinks of $f$ as a function, not merely an expression. Compare to this:

︡913b334c-5178-4cd6-97a7-4c92f55c5bbb︡{"html": "

This notation indicates that Sage thinks of $f$\nas a function, not merely an expression. Compare to this:\n

"}︡ ︠888399ef-d101-44e6-91e7-eaf4554ac80d︠ g=(x+5)^2; g ︡6bdcd876-569b-4a75-ab89-83d2590d1e2e︡︡ ︠53586e98-91a6-4e2d-81e6-d3a5f429e8e9i︠ %html

If you have a mere expression that you would like to be a function, you have a couple of options. You can redeﬁne it, but the expression you have might be the result of many calculations, not a simple deﬁnition. To turn the previous $g$ into a function you can do this:

︡81769305-bff4-42dc-a259-b0ce797bb4f0︡{"html": "

If you have a mere expression that you would like to be a function, you have a couple of options. You can\nredeﬁne it, but the expression you have might be the result of many calculations, not a simple deﬁnition. To turn\nthe previous $g$\ninto a function you can do this:\n

"}︡ ︠4708a771-0d03-403f-965f-2d2d205def59︠ h(x) = g; h ︡2601b278-fda0-4b76-9267-af00382a64c6︡︡ ︠b3cf3891-db37-4eb1-a8ec-aa7234c657a1i︠ %html

There’s no need to use a diﬀerent name if you don’t need the original one:

︡b4ccbe5f-0588-4c23-8981-85a8c0e56dbb︡{"html": "

There’s no need to use a diﬀerent name if you don’t need the original one:\n

"}︡ ︠d32ef0b5-7da4-44ac-acd2-21bd8124705c︠ g(x) = g; g ︡1d55a40f-0c90-46f5-9836-7a42ba146eb8︡︡ ︠993b5434-6f7b-4ddc-ae06-9cedccc5d352i︠ %html

Here’s something a little more realistic:

︡79b9bc02-e0b2-48d1-a227-e929fd2c1a8d︡{"html": "

Here’s something a little more realistic:\n

"}︡ ︠792e89c3-77f1-413a-9eee-73c7b505e344︠ expand((x+4)^2 * (2*x+1)); g(x) = _; g; g(3) ︡5a6a002b-3f3d-493d-8e84-18c36db48761︡︡ ︠07ba583d-8790-4578-95fa-483db7df1af7i︠ %html

You may need to apply a function to a list of input values; here is something a bit easier than doing each one separately:

︡aeefa98b-ecee-4466-8dc6-67f9699fd603︡{"html": "

You may need to apply a function to a list of input values; here is something a bit easier than doing\neach one separately:\n

"}︡ ︠8210abf4-d38b-48f7-bec1-edcbffc64225︠ L = [1,2,3,10]; map(g,L) ︡991ca2d8-0195-41d3-8767-effb8072bf71︡︡ ︠542a557f-c33c-4ab3-995d-c6e60dd3c7aai︠ %html

In ordinary mathematical notation you are used to leaving out some multiplication symbols; Sage does not allow this:

︡645a2b35-d902-4b4a-a274-0e37e6237eea︡{"html": "

In ordinary mathematical notation you are used to leaving out some multiplication symbols; Sage does\nnot allow this:\n

"}︡ ︠0d144d3a-324e-4801-8831-9e75aee21a48︠ 2x ︡0d939aa6-91cb-46cb-8d6d-0b79c80b40d5︡︡ ︠807331d0-fec0-461e-a967-b42a780e6c2bi︠ %html

Type an asterisk between the 2 and the $x$ and evaluate that cell again.

We saw above how to expand and factor polynomials. If a polynomial has integer or rational coeﬃcients, the “factor” command tries to factor it into polynomials with integer coeﬃcients, perhaps times a rational number:

︡7f488d80-cf7d-4481-86ca-334ed0f60f29︡{"html": "

Type an asterisk between the 2 and the $x$\nand evaluate that cell again.\n

We saw above how to expand and factor polynomials. If a polynomial has integer or rational\ncoeﬃcients, the “factor” command tries to factor it into polynomials with integer coeﬃcients, perhaps\ntimes a rational number:\n

"}︡ ︠2cde27b7-a87d-47b9-84c1-b589cd426aa1︠ expand((x+2)*(x-1/3)); factor(_) ︡575b6d85-d092-4808-b3de-946a3e81b8f5︡︡ ︠e2a41375-6644-4f2e-88fd-a7d626b4b073i︠ %html

If Sage can’t do this, it leaves the polynomial alone, even if it has a “nice” factorization:

︡296d37b1-3199-4e3b-b6b7-c780c2370bed︡{"html": "

If Sage can’t do this, it leaves the polynomial alone, even if it has a “nice” factorization:\n

"}︡ ︠927ee268-8804-478e-a86c-6d583cf44659︠ factor(x^2-2); factor(x^2-x-1) ︡01d23295-5b0d-4c0f-b88a-ea82342f7938︡︡ ︠1c6208cc-18a6-41f3-853e-f28ae357aaeci︠ %html

1.3 Substitution

We use “variables” in a variety of ways, to stand for “true variables”, for functions, or for unspeciﬁed constants. We might say, for example, $f(x) = A{(x + 5)}^{2}$, using all of these. If we try this, there is already a problem:

︡a5c544b3-0d0d-4ce4-9d0b-6f9a2aadc8d0︡{"html": "

1.3 Substitution

We use “variables” in a variety of ways, to stand for “true variables”, for functions, or for unspeciﬁed constants. We might\nsay, for example, $f(x) = A{(x + 5)}^{2}$,\n \n \nusing all of these. If we try this, there is already a problem:\n

"}︡ ︠c30c1190-f185-4858-9c80-f31e1a70c999︠ f(x)=A*(x+5)^2 ︡35db4a16-5321-44e0-8f38-49ae67f5fb7f︡︡ ︠5a99654a-9a23-4afb-b678-5ed2be7eead9i︠ %html

Although Sage understands that $x$ is a variable, we must tell it explicitly about others:

︡01783f38-3cc6-479b-9365-defb0cb4c36f︡{"html": "

Although Sage understands that $x$\nis a variable, we must tell it explicitly about others:\n

"}︡ ︠822e6ca0-b3c5-44b8-8948-c73f6abf90fe︠ A=var("A"); f(x)=A*(x+5)^2 ︡39544ef3-a186-4a20-b05f-916b4159b924︡︡ ︠8840e03e-2b04-49ec-b23d-efcefceea394i︠ %html

We can manipulate this as you would expect:

︡da2db64c-bd2f-4ecf-9367-31852364b6a2︡{"html": "

We can manipulate this as you would expect:\n

"}︡ ︠ceb4b4d6-607f-4dc5-afd4-9143dd389bb7︠ expand(f); factor(_) ︡0811bb47-d078-438a-9d25-e379a10bb7fb︡︡ ︠12803ba0-e02c-4d12-8be6-4439a50d8e49i︠ %html

Notice that strangely the result of expand is a function but the result of factor is not!

We may then want to assign the constant $A$ a value. We can do this temporarily like this:

︡e941aa09-56ce-41d1-a08f-b425c58d750d︡{"html": "

Notice that strangely the result of expand is a function but the result of factor is not!\n

We may then want to assign the constant $A$\na value. We can do this temporarily like this:\n

"}︡ ︠0d84e9b5-bc68-4fda-b6cb-eafa8e094b10︠ f(A=2); f(3,A=2); f; A ︡58d5b612-0688-4a83-b705-6ec009432970︡︡ ︠e874a912-5a0d-4f67-8e75-ac46aa97b649i︠ %html

So we see that the deﬁnition of $f$ has not changed, it still contains the symbol $A$. This happens even if we give $A$ a permanent value:

︡e338dbbe-7de1-4e40-a6c4-804576edcad9︡{"html": "

So we see that the deﬁnition of $f$ has not\nchanged, it still contains the symbol $A$.\nThis happens even if we give $A$\na permanent value:\n

"}︡ ︠79898e5a-6172-4ee6-be90-e0afc1c844fb︠ A=2; f; A ︡49c1cdb4-4638-4cc7-bf89-d8e34fd29da2︡︡ ︠4d468da2-6e34-4b6f-8e9c-eae56bbfaa8di︠ %html

This is probably a bad idea, since it is at least confusing. If you would like a new function with $A$ replaced by 2 you can do something like this:

︡4278b2f5-827e-4b1d-94de-496fb1df9587︡{"html": "

This is probably a bad idea, since it is at least confusing. If you would like a new function with\n$A$\nreplaced by 2 you can do something like this:\n

"}︡ ︠1fe5ac19-95c9-412b-a991-3e8792e2d77b︠ g(x) = f(A=2); g; g(5) ︡15306153-b4d2-4650-a149-82bd04ccd81b︡︡ ︠918ebf11-2c63-4187-96b9-c93a3f665823i︠ %html

Let’s restore $A$ to its status as a variable with no value:

︡af98d637-2281-461f-a27b-798b74f6a742︡{"html": "

Let’s restore $A$\nto its status as a variable with no value:\n

"}︡ ︠1436cc13-ea15-497e-bd14-a36d4d306d40︠ A; A=var("A"); A ︡fe705672-2069-494c-a81c-943a5b51d3d3︡︡ ︠92d56e84-1a2e-41e7-859e-7a2c9038eeb9i︠ %html

You can subsitute expressions for variables, not just values.

︡5d865f43-5f84-4f98-b0e2-045f01581f46︡{"html": "

You can subsitute expressions for variables, not just values.\n

"}︡ ︠6cfd2286-cb92-4602-bb13-7fc4d2d505b2︠ g(x) = f(A=x^3-1); g; h(x) = f(2*x+3,A=x^3-1); h ︡946405a1-7da2-49f5-844d-ccd4c51ed33d︡︡ ︠ff2772e8-c5c7-492f-971c-5c34c539b0bci︠ %html

Make sure you understand how $h$ is deﬁned. This substitution method works quite generally on expressions that may not have names:

︡037179c1-9643-4513-aafe-ea3337df8889︡{"html": "

Make sure you understand how $h$\nis deﬁned. This substitution method works quite generally on expressions that may not have\nnames:\n

"}︡ ︠5b77a4fb-62d9-4b26-b80f-36963225b737︠ (A^2+2)(A=3*x+1) ︡3168afd8-20df-49fa-be52-df9919fb6d30︡︡ ︠d64680e1-2b20-4489-a835-d1a095325022i︠ %html

1.4 Solving Equations

Sage will solve a limited class of equations exactly and a broader class approximately. The solve command will give exact answers if possible, including complex answers.

︡e258950c-0b68-43cb-b0b6-f5a596d02281︡{"html": "

1.4 Solving Equations

Sage will solve a limited class of equations exactly and a broader class approximately. The solve command\nwill give exact answers if possible, including complex answers.\n

"}︡ ︠9505817c-4986-4302-a241-5f26fb480a3a︠ solve(x^2-x-10==10,x); ︡c5888616-8363-4644-98fe-0c6bf759b7d4︡︡ ︠665e4f44-a730-4467-b4b6-81899b4b7e17i︠ %html

If you specify just a function, Sage assumes you want to set it equal to zero and solve.

︡ed5bd0ff-b0f2-4e30-86d0-f0a8186db13d︡{"html": "

If you specify just a function, Sage assumes you want to set it equal to zero and solve.\n

"}︡ ︠8e6f929e-da90-41c1-b630-bee45089f402︠ solutions=solve(g(x),x); solutions ︡693857ee-3f93-45be-97ce-19ca8178c31b︡︡ ︠b84ab650-cb4f-4191-ba96-3cbe2ba4c3dci︠ %html

Even when Sage can solve the equation exactly, you may want to see numerical approximations. You can use the n function on each solution, or with a bit of programming do them all at once. First we generate the solutions in a diﬀerent form:

︡2012ce39-5a18-4186-be93-d0a50dfc0d4f︡{"html": "

Even when Sage can solve the equation exactly, you may want to see numerical approximations. You can\nuse the n function on each solution, or with a bit of programming do them all at once. First we generate\nthe solutions in a diﬀerent form:\n

"}︡ ︠bebdddb2-b574-4af2-b101-e42d38ff84bb︠ solutions=solve(g(x),x,solution_dict=True); solutions; ︡bce311bc-2a3c-4a0a-9758-533edb6bc8b7︡︡ ︠58bf95e6-e020-4787-997b-a059b2e15309i︠ %html

Then we can extract just the numerical values:

︡8b3aa374-d445-413f-a934-317fbe4a33fb︡{"html": "

Then we can extract just the numerical values:\n

"}︡ ︠33d78e35-e141-4a06-84d5-14f2755a4014︠ newsolutions = [ s[x] for s in solutions]; newsolutions ︡ee5c15bd-5834-402e-a52a-04be20993924︡︡ ︠b61d0b9e-73ac-41dd-a990-0dbfcd57b156i︠ %html

Then use the map function to apply n to each number on the list:

︡3fcad7ef-5fda-4931-bd02-e5c1d0db116c︡{"html": "

Then use the map function to apply n to each number on the list:\n

"}︡ ︠90da66c8-5b30-41d7-8eee-655291f2e162︠ map(n,newsolutions) ︡b5438415-158b-44b5-8aec-2a80802e8d60︡︡ ︠facb4706-8ac9-4146-8bf2-8b902e18c153i︠ %html

Or we could combine the last two steps:

︡efd930a1-60ab-4a7f-8037-c77ea77eb09d︡{"html": "

Or we could combine the last two steps:\n

"}︡ ︠9a0a8dc6-8d15-4e61-9fa2-a204e7911be2︠ [ n(s[x]) for s in solutions] ︡74e0c1a1-2816-430c-a2bb-e30c13a47fdf︡︡ ︠95a25560-260d-4f5b-9e4e-81e1379c1690i︠ %html

If Sage can’t solve an equation it will simply reprint the equation. You can approximate a solution, but you have to limit the search to a range, which you might discover by graphing the function or by substituting some values into the function. For example, using $f(x) = {x}^{5} + 4{x}^{3} − 3{x}^{2} + 10$, you might see that $f(0) = 10$ and $f(−2) = −66$, so there must be a root between $0$ and $ − 2$.

︡36367dc9-4c74-46f9-86f6-7daeda21ecde︡{"html": "

If Sage can’t solve an equation it will simply reprint the equation. You can approximate\na solution, but you have to limit the search to a range, which you might discover by\ngraphing the function or by substituting some values into the function. For example, using\n$f(x) = {x}^{5} + 4{x}^{3} − 3{x}^{2} + 10$, you might\nsee that $f(0) = 10$ and\n$f(−2) = −66$, so there must be\na root between $0$\nand $ − 2$.\n

"}︡ ︠e579b5f8-a4de-433e-8e1d-507d3560281e︠ solve(x^5+4*x^3-3*x^2+10,x); find_root(x^5+4*x^3-3*x^2+10,-2,0) ︡d5cd4d0b-761c-4ed0-afec-6b674eeb43b1︡︡ ︠2780cab4-56d9-4561-9861-1b64e62729ed︠ find_root(tan(x)==x+1,0,pi/2); ︡b12eff89-283e-4c27-b903-52c07b260418︡︡ ︠65f1e484-d9cc-46a3-96ad-033690f9eb89i︠ %html

Sage will also solve systems of equations; note the square brackets forming a list of equations.

︡b3504f0e-4c9a-4028-a9ee-f15655d3be14︡{"html": "

Sage will also solve systems of equations; note the square brackets forming a list of equations.\n

"}︡ ︠442bc022-ab4b-4ed1-a668-0127b007abb8︠ y=var("y"); solve([3*x+5*y==6,4*x-6*y==-8],x,y) ︡4b446249-80aa-4347-914c-0dd87a7d5f64︡︡ ︠ea06f80b-5bc4-4b8a-be49-95d5b6dbb435i︠ %html

A bit more involved:

︡0c913aea-9524-4d0f-a898-1f5a7538ef32︡{"html": "

A bit more involved:\n

"}︡ ︠4ac97619-b1c7-4628-bf85-d86e66f8ec78︠ h1=3*x-6*y-5; h2=-4*x*y+7; solve([h1,h2],x,y); jsmath(_) ︡54bf6fb1-bf33-4da9-82ce-7e45e6c9e0c9︡︡ ︠028424e0-2f7e-4e84-8bec-edbf2eeb2767i︠ %html

2 Plotting

2.1 Functions of one variable

Sage can do two and three dimensional graphing. There are many ways to adjust the plots that Sage produces; we’ll cover just the basics here.

It’s easy to plot ordinary one-variable functions.

︡0aa579a7-053e-4d73-8d48-55fdfb435e46︡{"html": "

2 Plotting

2.1 Functions of one variable

Sage can do two and three dimensional graphing. There are many ways to adjust the plots that Sage\nproduces; we’ll cover just the basics here.\n

It’s easy to plot ordinary one-variable functions.\n

"}︡ ︠5c0b8ca9-e217-40af-98ae-92f4a8184079︠ plot(g,-5,5); ︡aef56263-b531-401d-ba5e-c7a6b50a9ee3︡︡ ︠f2e1b77e-09b0-4b88-a011-305c38101e9ei︠ %html

The vertical scale makes it hard to see what’s going on in that graph; you can limit the vertical scale for a better look, using “ymin” and “ymax”.

︡7d6378b9-3e09-4022-8883-a6804b3eec7f︡{"html": "

The vertical scale makes it hard to see what’s going on in that graph; you can limit the vertical scale for a\nbetter look, using “ymin” and “ymax”.\n

"}︡ ︠44438f86-df80-434b-819a-b18cdc4450d6︠ plot(g,-5,5,ymax=10); ︡35ff8bce-3acd-40cc-a87f-4793307a9ac7︡︡ ︠29dfb936-f70d-4de8-b2dd-db2f03d76785i︠ %html

Some functions have vertical asymptotes, but Sage will just connect the dots:

︡517c2758-7905-4d98-9ab7-5e9e341fd4bd︡{"html": "

Some functions have vertical asymptotes, but Sage will just connect the dots:\n

"}︡ ︠ece7b26c-a7a0-4001-a25f-914fc1e6fb36︠ plot(1/(x-1),-5,5,ymin=-10,ymax=10); ︡85a95e6b-80c7-4fb6-8cfd-3814eb5ae84c︡︡ ︠4e27e188-5004-4c9b-97a1-b8f1c7492c1di︠ %html

You can get rid of the spurious bit in the middle:

︡3636fe53-f890-46ab-8908-8fe2a4d30ea4︡{"html": "

You can get rid of the spurious bit in the middle:\n

"}︡ ︠8295fbca-6b7d-4f13-bb99-804740434a4e︠ plot(1/(x-1),-5,5,ymin=-10,ymax=10,exclude=[1]); ︡dfff6e21-96b4-41a5-88f7-985dc1ac1c88︡︡ ︠f920c29c-31c0-4a97-aea4-bcc81ca55108i︠ %html

You can also put a list of excluded values in the brackets; try plotting $f(x) = 1∕({x}^{2} − 1)$ between $ − 5$ and $5$.

If you right-click on a plot you can download it as a png image.

You can put multiple plots on the same axes:

︡a98ee8bd-5a9b-4523-a653-099fa82753a5︡{"html": "

You can also put a list of excluded values in the brackets; try plotting\n$f(x) = 1∕({x}^{2} − 1)$ between\n$ − 5$ and\n$5$.\n \n \n

If you right-click on a plot you can download it as a png image.\n

You can put multiple plots on the same axes:\n

"}︡ ︠9e8773bb-9a23-4c72-9c59-a0ab7d10fdcd︠ plot([g,1/(x-1)],-5,5,ymin=-110,ymax=20,exclude=[1]); ︡52c89729-fc85-42a1-b99a-975534a05781︡︡ ︠62bc90b5-5791-4ba6-ae1f-6d3f16b389f2i︠ %html

It’s often better to do the plots separately but display them together:

︡5fe565f3-bf07-454f-a1fd-a993da289e42︡{"html": "

It’s often better to do the plots separately but display them together:\n

"}︡ ︠9b68f3e4-4ec4-4186-976d-1fbeb4e5e330︠ p1=plot(g,-5,5,ymin=-110,ymax=20,color='red') p2=plot(1/(x-1),-5,5,ymin=-110,ymax=20,exclude=[1]) p1+p2 ︡386cd9a6-d768-4003-9cef-115f2c7735d7︡︡ ︠40c65986-2f6c-4176-8742-b2b4bd6eccd5i︠ %html

Sage normally uses diﬀerent scales on the two axes; you can force it to use the same scales to see the “true shape”.

︡d0a5105c-dc51-41ac-80fa-a61b2617a706︡{"html": "

Sage normally uses diﬀerent scales on the two axes; you can force it to use the same scales to see the “true\nshape”.\n

"}︡ ︠ea41044e-796c-489e-a833-7798b0bd2758︠ (p1+p2).show(aspect_ratio=1) ︡fd7c91c5-d28c-438a-a30d-761be4adde25︡︡ ︠905ca20e-fec9-4c20-93dd-b7b4da8819c6i︠ %html

Here’s a better example: the line doesn’t look like it has slope 1 in the ﬁrst graph.

︡6ce80343-eca5-46d7-8794-cc5f5ece8bac︡{"html": "

Here’s a better example: the line doesn’t look like it has slope 1 in the ﬁrst graph.\n

"}︡ ︠cdebcb15-3adb-4d66-9ea1-3d342cad3b4d︠ p1=plot(x+1,-5,5); p1; p1.show(aspect_ratio=1) ︡478e1541-e060-4165-9476-f974a2cfff81︡︡ ︠fe941673-6d77-4f3c-b3c4-a952bd5b1769i︠ %html

Sage will do parametric plots in two or three dimensions.

︡a35cb926-74e4-497e-b2ac-60bf755e8ec8︡{"html": "

Sage will do parametric plots in two or three dimensions.\n

"}︡ ︠c53a86d6-99ee-4514-a596-7eae0466e44b︠ parametric_plot3d([cos(x),sin(x),cos(6*x)],(x,0,2*pi)) ︡c0b53e51-71ec-4e08-ab02-edd35322a1fc︡︡ ︠feef11d3-67d0-4cca-a623-70d721b6391fi︠ %html

Plots in polar coordinates are really special examples of parametric plots.

︡cafe4de3-725f-4d78-abf1-7b39b79dcdd9︡{"html": "

Plots in polar coordinates are really special examples of parametric plots.\n

"}︡ ︠69c39b12-f05c-468e-93ba-19226e427d68︠ polar_plot(1+cos(x),0,2*pi) ︡77e8f07b-9ca9-49bf-9847-d0785ff540a0︡︡ ︠801bdb38-dad2-4000-a7a7-4985cf0f07cc︠ parametric_plot([cos(x)*(1+cos(x)),sin(x)*(1+cos(x))],(x,0,2*pi)) ︡86e61474-7a4b-468f-8ae1-b3ffc50a115e︡︡ ︠994f0fe1-a56a-478b-9966-4feae140ee91i︠ %html

You can also do parametric surface plots:

︡5fb30604-fd4b-49bb-9925-4c222175fa27︡{"html": "

You can also do parametric surface plots:\n

"}︡ ︠d5c45c8c-2af6-4bf5-a482-cb778cc48644︠ (u,v)=var("u v") parametric_plot3d([v*cos(2*u),v*sin(2*u),u],(u,0,2*pi),(v,0,1)) ︡b1a6495e-8e55-483e-ac34-e0dca917f036︡︡ ︠cc13a36c-15d1-4b64-aebd-6433226a4ef7i︠ %html

You may notice that the edges of this surface appear more polygonal than round; you can change the number of points in either direction on the plot. The $u$ direction is the one we want to increase; the $v$ direction is along radial lines, so a very small value is ﬁne.

︡f965bdd9-6678-4bbe-9c0e-3605840bc1d1︡{"html": "

You may notice that the edges of this surface appear more polygonal than\nround; you can change the number of points in either direction on the plot. The\n$u$ direction is the one we\nwant to increase; the $v$\ndirection is along radial lines, so a very small value is ﬁne.\n

"}︡ ︠ad0ac423-d84c-4745-8468-3977fa559fe6︠ (u,v)=var("u v") parametric_plot3d([v*cos(2*u),v*sin(2*u),u],(u,0,2*pi),(v,0,1),plot_points=[200,5]) ︡c38f5454-f664-4c01-bfcf-6d9ea88828bd︡︡ ︠549afba8-b5fd-4f3f-afe6-e8b5afb13560i︠ %html

Here’s a fancier plot: a sphere inside a cylinder, and the cylinder is partially transparent.

︡995d98f6-e79b-48b9-8965-4baf2b0fb60c︡{"html": "

Here’s a fancier plot: a sphere inside a cylinder, and the cylinder is partially transparent.\n

"}︡ ︠b3482639-3f41-4746-882b-66f9af113186︠ (u,v)=var("u v") s=parametric_plot3d([sin(v)*cos(u),sin(v)*sin(u),cos(v)],(u,0,2*pi),(v,0,pi),color='red',aspect_ratio=1) c=parametric_plot3d([v,cos(u),sin(u)],(u,0,2*pi),(v,-1,2),opacity=0.6) s+c ︡6286c74a-2c80-4618-bf0d-56f93cb847cb︡︡ ︠f3de39f7-6ab6-4052-bb45-d8560d25d820i︠ %html

Note that Sage has a bit of trouble where the cylinder and the sphere meet. You can ﬁx this by cheating a little: make the cylinder just slightly bigger.

︡36fda9f9-e2f3-487e-9482-1303acb050f1︡{"html": "

Note that Sage has a bit of trouble where the cylinder and the sphere meet. You can ﬁx this by cheating a\nlittle: make the cylinder just slightly bigger.\n

"}︡ ︠68b6e2b5-1124-4890-a07a-229beeab1c4b︠ (u,v)=var("u v") s=parametric_plot3d([sin(v)*cos(u),sin(v)*sin(u),cos(v)],(u,0,2*pi),(v,0,pi),color='red',aspect_ratio=1) c=parametric_plot3d([v,1.02*cos(u),1.02*sin(u)],(u,0,2*pi),(v,-1,2),opacity=0.6) s+c ︡28e37cb8-9a60-4c2c-ae76-4606663b5e98︡︡ ︠f9e83479-4c4d-4912-923e-1fc955513b1ai︠ %html

You can plot quite complicated surfaces using multiple parametric plots together. The intersection of three cylinders of the same size at right angles to one another forms a solid. Note that we have plotted each color as four surfaces when in fact one would do; try plotting just the red surface with a single plot, using the range $(u, 0, 2π)$ and see what happens; can you explain what has gone wrong? You might try adding the original cylinders to this plot, colored appropriately, and partially transparent. First we set up some useful functions.

︡7a57b676-1cca-4021-9f11-c346dd24085c︡{"html": "

You can plot quite complicated surfaces using multiple parametric plots together. The intersection of three\ncylinders of the same size at right angles to one another forms a solid. Note that we have plotted each color as\nfour surfaces when in fact one would do; try plotting just the red surface with a single plot, using the\nrange $(u, 0, 2π)$\nand see what happens; can you explain what has gone wrong? You might try adding the original\ncylinders to this plot, colored appropriately, and partially transparent. First we set up some useful\nfunctions.\n

"}︡ ︠3d1070dd-5ad0-4193-968c-43c1ec89faad︠ (x,y,u,t,a,b)=var("x y u t a b") minimum(a,b)=((a+b)-abs(b-a))/2 x=cos(u); y=sin(u) v=t*minimum(sqrt(1-x^2),sqrt(1-y^2)) ︡c2435962-8ab2-44da-aa69-3a1fa5db6a7a︡︡ ︠6a3d4dc3-db9b-462d-b45e-d1adce41df60i︠ %html

Now notice that each of the colors is essentially the same, with the coordinates plotted against diﬀerent axes for the diﬀerent colors.

︡3f0b1489-c954-4967-8b7e-d8ac32058c88︡{"html": "

Now notice that each of the colors is essentially the same, with the coordinates plotted against diﬀerent\naxes for the diﬀerent colors.\n

3 Calculus

3.1 Limits

Sage can compute many typical limits as encountered in calculus. Here’s an easy one, for a continuous function.

︡2f1cb95f-d02e-46f0-add5-613dd6ccdc61︡{"html": "

3 Calculus

3.1 Limits

Sage can compute many typical limits as encountered in calculus. Here’s an easy one, for a continuous\nfunction.\n

"}︡ ︠e810cd89-e8d0-46f4-9ea4-fa9758d0fe3c︠ f(x)=x^2+4*x+30 limit(f,x=3); f(3) ︡0156bf70-f94e-4e60-8d9c-16157415cd2b︡︡ ︠0cd9638f-94f8-4f72-901e-0b5c3bcd4cffi︠ %html

And everyone’s favorite non-trivial limit:

︡c9ec9471-308a-420c-8e7e-8e4c6a54cabd︡{"html": "

And everyone’s favorite non-trivial limit:\n

"}︡ ︠89ed6810-a878-427b-b478-cbc7488c37c7︠ g(x)=sin(x)/x limit(g,x=0); limit(g,x=infinity) ︡2d6713ed-bae8-41d5-b571-6b66e57d7de1︡︡ ︠f69fa48c-5c96-482d-9d3c-62934a83d571i︠ %html

You can specify one-sided limits in two ways.

︡8bf6cc1a-e4d7-403f-98b3-136713b9621f︡{"html": "

You can specify one-sided limits in two ways.\n

"}︡ ︠dd4ff007-beab-456d-a169-a3d44c2124e4︠ g(x)=x/abs(x) limit(g,x=0); limit(g,x=0,dir='-'); limit(g,x=0,dir='right') ︡32330b4b-cce0-4420-90c0-3bc5bca5e1eb︡︡ ︠e68b486f-66eb-4ab2-a7b4-2c8e1b4ef2a4i︠ %html

3.2 Diﬀerentiation

Here’s a typical easy calculus problem: ﬁnd a derivative, ﬁnd critical values, ﬁnd the $y$-coordinates for each.

︡8531a522-7af0-4cca-b385-4bd2c87e4883︡{"html": "

3.2 Diﬀerentiation

Here’s a typical easy calculus problem: ﬁnd a derivative, ﬁnd critical values, ﬁnd the\n$y$-coordinates\nfor each.\n

"}︡ ︠38e6e657-b469-4d97-8a1f-ccdc50c5e135︠ f(x)=x^3+4*x^2+3*x+4 fp=diff(f) critvals=solve(fp,x,solution_dict=True) f; fp; critvals; jsmath([expand(f(s[x])) for s in critvals]) ︡8ece0736-f535-41a0-bfe6-954f7d9750f5︡︡ ︠7dd3e91e-a824-4bfe-b39d-1036a89b0257i︠ %html

3.3 Implicit Diﬀerentiation

Sage has no built-in implicit diﬀerentiation command, but we can do it using diﬀerentiation, in essentially the way we do it by hand. We tell Sage that $y$ is a function of $x$, then proceed essentially as usual. Here is the standard example computing $y' = −x∕y$ for $y$ deﬁned by ${x}^{2} + {y}^{2} = 1$. Note that we convert this to ${x}^{2} + {y}^{2} − 1 = 0$, which guarantees that the derivative of the right side is zero. (In this case the derivative of the right side, 1, was already zero, but in general it might be more complicated.)

︡bcc8a0a5-667b-4804-a17a-8a52f373176b︡{"html": "

3.3 Implicit Diﬀerentiation

Sage has no built-in implicit diﬀerentiation command, but we can do it using\ndiﬀerentiation, in essentially the way we do it by hand. We tell Sage that\n$y$ is a\nfunction of $x$,\nthen proceed essentially as usual. Here is the standard example computing\n$y' = −x∕y$ for\n$y$ deﬁned by\n${x}^{2} + {y}^{2} = 1$. Note that we\nconvert this to ${x}^{2} + {y}^{2} − 1 = 0$,\nwhich guarantees that the derivative of the right side is zero. (In this case the derivative of the right side,\n1, was already zero, but in general it might be more complicated.)\n

"}︡ ︠798f215a-bf30-4370-815f-4409b583a607︠ y=function('y',x) temp=diff(x^2+y^2-1) solve (temp,diff(y)) ︡034155f8-fd02-4186-b5cd-e32b3acbb32d︡︡ ︠6c945cd7-42ef-49ce-a058-57c7c3fced2ei︠ %html

3.4 Integration

The “integral” command computes an antiderivative. Here, we take the derivative to check the antiderivative.

︡00f6ffba-dd06-49a7-98fc-9899cb8872fc︡{"html": "

3.4 Integration

The “integral” command computes an antiderivative. Here, we take the derivative to check the\nantiderivative.\n

"}︡ ︠c9879e34-6875-43f2-a92f-39825c20ef83︠ f(x)=(2*x-3)*sqrt(x-3) g(x)=integral(f,x) g; diff(g); factor(_) ︡f5ef584b-aefd-4a49-bbf9-ef29a96378c8︡︡ ︠b8a494f8-e29d-47e8-93df-d34c9131708fi︠ %html

If Sage can’t handle an integral, it does nothing.

︡3726d9bc-ea66-4156-a208-c75536a90632︡{"html": "

If Sage can’t handle an integral, it does nothing.\n

"}︡ ︠5f2bdba0-1cbc-49bc-8343-69a6b1a742f6︠ integral(tan(x^2),x) ︡3b6f5484-fe3a-485f-b98d-d01a6ac09d25︡︡ ︠13d619c3-e780-4585-92be-bef25c9a2408i︠ %html

We could of course do deﬁnite integration by substituting and subtracting, but the integral command will also do it.

︡eef37010-5bac-4b68-bf22-18afdd29aa17︡{"html": "

We could of course do deﬁnite integration by substituting and subtracting, but the integral command\nwill also do it.\n

"}︡ ︠f25bd5f8-362e-4ee3-9b1e-8ccccd7d0573︠ integral(f(x),x,5,10); g(10)-g(5) ︡ed9250ab-2111-4f5d-af03-0560c7d6e591︡︡ ︠9098bd9c-d270-40b3-9899-cd02f841d1c0i︠ %html

Sage can do some improper integrals.

︡fe7bc736-8261-48cc-917c-c800673af9f8︡{"html": "

Sage can do some improper integrals.\n

"}︡ ︠1ce5c6b7-0ba0-455d-b8cd-23adf7ec9d58︠ f(x)=1/sqrt(x); integral(f(x),x,0,1); integral(f(x),x,1,infinity) ︡8fab4f38-7b2e-4302-b9a5-6b74423a153a︡︡ ︠3f890c63-7486-46d4-bdfb-c970483904d3︠ f(x)=1/x^2; integral(f(x),x,1,infinity) ︡8a2464fd-293f-4ca9-ad39-df6696b18fdf︡︡ ︠0f103547-f502-4aca-b8d7-e8967d544214i︠ %html

The function ${e}^{−{x}^{2}}$ doesn’t have a “nice” antiderivative, though it can be expressed in terms of the “erf” function. For most values of $x$, erf(x) can only be approximated, but an improper integral turns out to be quite nice. (The function ${e}^{−{x}^{2}}$ is a Gaussian function, a “bell curve”.)

︡4464566c-963b-491b-bfec-085a02cd8c82︡{"html": "

The function ${e}^{−{x}^{2}}$\ndoesn’t have a “nice” antiderivative, though it can be expressed in terms of the “erf” function. For most values\nof $x$,\nerf(x) can only be approximated, but an improper integral turns out to be quite nice. (The function\n${e}^{−{x}^{2}}$ is a\nGaussian function, a “bell curve”.)\n

"}︡ ︠7eb6e498-acc2-47c3-8ae9-25b1b5dfcae3︠ f(x)=exp(-x^2); integral(f,x); integral(f(x),x,0,1); integral(f(x),x,-infinity,infinity) ︡d1a1bc82-11e8-4c83-80a6-a5c5eff5d504︡︡ ︠a786f513-abcc-41eb-8e1e-4535290bd206i︠ %html

Sometimes the result is mysterious.

︡eeb0831f-9bd3-4f9d-a648-37dfac783c0d︡{"html": "

Sometimes the result is mysterious.\n

"}︡ ︠3ab99342-dc34-47cb-a98e-4b4cc9fb117f︠ f(x)=sin(x); integral(0,infinity) ︡b71353e0-322c-4e5c-abd0-3a62c8e588c9︡︡ ︠6558e613-108f-4b07-8ed5-b26691347c45i︠ %html

Sage can’t ﬁnd antiderivatives of some functions. In such cases you can approximate the value of a deﬁnite integral using the $n$ function.

︡4487c861-6abc-40ad-acc2-fa834e545594︡{"html": "

Sage can’t ﬁnd antiderivatives of some functions. In such cases you can approximate the value of a deﬁnite integral\nusing the $n$\nfunction.\n

"}︡ ︠864846ba-a5fb-4bc4-b340-cb89aa8852c5︠ integral(tan(x^2),x,0,1); n(_) ︡9da7ab47-1864-420e-a930-b1ee7dc37163︡︡ ︠d091afc9-d272-4d70-a40d-bba86652941bi︠ %html