Sage Quick Reference: Calculus William Stein Simplifying and expanding Integrals Sage Version 3.4 Below f must be symbolic (so not a Python function): f(x)dx = integral(f,x) = f.integrate(x) http://wiki.sagemath.org/quickref Simplify: f.simplify_exp(), f.simplify_full(), integral(x*cos(x^2), x) GNU Free Document License, extend for your own use ! f.simplify_log(), f.simplify_radical(), b f(x)dx = integral(f,x,a,b) f.simplify_rational(), f.simplify_trig() a integral(x*cos(x^2), x, 0, sqrt(pi)) ! Expand: f.expand(), f.expand_rational() b Builtin constants and functions f(x)dx numerical_integral(f(x),a,b)[0] a ≈ numerical_integral(x*cos(x^2),0,1)[0] Constants: π = pi e = e i = I = i Equations ! = oo = infinity NaN=NaN log(2) =log2 assume(...): use if integration asks a question ∞ Relations: f = g: f == g, f = g: f != g, φ = golden_ratio γ = euler_gamma f g: f <= g, f & g: f >= g, assume(x>0) 0.915 catalan 2.685 khinchin f <≤ g: f < g, f >≥ g: f > g ≈ ≈ 0.660 twinprime 0.261 merten 1.902 brun Taylor and partial fraction expansion ≈ ≈ ≈ Solve f = g: solve(f == g, x), and Approximate: pi.n(digits=18) = 3.14159265358979324 solve([f == 0, g == 0], x,y) Taylor polynomial, deg n about a: n Builtin functions: sin cos tan sec csc cot sinh taylor(f,x,a,n) c0 + c1(x a) + + cn(x a) solve([x^2+y^2==1, (x-1)^2+y^2==1],x,y) ≈ − · · · − cosh tanh sech csch coth log ln exp ... Solutions: taylor(sqrt(x+1), x, 0, 5) S = solve(x^2+x+1==0, x, solution_dict=True) Partial fraction: Defining symbolic expressions S[0]["x"] S[1]["x"] are the solutions (x^2/(x+1)^3).partial_fraction() Create symbolic variables: Exact roots: (x^3+2*x+1).roots(x) var("t u theta") or var("t,u,theta") Real roots: (x^3+2*x+1).roots(x,ring=RR) Numerical roots and optimization Use * for multiplication and ^ for exponentiation: Complex roots: (x^3+2*x+1).roots(x,ring=CC) Numerical root: f.find_root(a, b, x) 5 2x + √2 = 2*x^5 + sqrt(2) (x^2 - 2).find_root(1,2,x) Typeset: show(2*theta^5 + sqrt(2)) 2θ5 + √2 Maximize: find (m, x ) with f(x ) = m maximal −→ 0 0 Factorization f.find_maximum_on_interval(a, b, x) Symbolic functions Factored form: (x^3-y^3).factor() Minimize: find (m, x0) with f(x0) = m minimal Symbolic function (can integrate, differentiate, etc.): List of (factor, exponent) pairs: f.find_minimum_on_interval(a, b, x) f(a,b,theta) = a + b*theta^2 (x^3-y^3).factor_list() Minimization: minimize(f, start point) Also, a “formal” function of theta: minimize(x^2+x*y^3+(1-z)^2-1, [1,1,1]) f = function(’f’,theta) Limits lim f(x) = limit(f(x), x=a) Multivariable calculus Piecewise symbolic functions: x a → Piecewise([[(0,pi/2),sin(x)],[(pi/2,pi),x^2+1]]) limit(sin(x)/x, x=0) Gradient: f.gradient() or f.gradient(vars) (x^2+y^2).gradient([x,y]) lim f(x) = limit(f(x), x=a, dir=’plus’) x a+ → Hessian: f.hessian() limit(1/x, x=0, dir=’plus’) (x^2+y^2).hessian() lim f(x) = limit(f(x), x=a, dir=’minus’) Jacobian matrix: jacobian(f, vars) x a → − limit(1/x, x=0, dir=’minus’) jacobian(x^2 - 2*x*y, (x,y))

Summing infinite series Python functions Derivatives Defining: d (f(x)) = diff(f(x),x) = f.diff(x) dx ∞ 1 π2 def f(a, b, theta=1): ∂ (f(x, y)) = diff(f(x,y),x) = ∂x n2 6 n=1 c = a + b*theta^2 diff = differentiate = derivative " return c diff(x*y + sin(x^2) + e^(-x), x) Not yet implemented, but you can use Maxima: Inline functions: s = ’sum (1/n^2,n,1,inf), simpsum’ SR(sage.calculus.calculus.maxima(s)) π2/6 f = lambda a, b, theta = 1: a + b*theta^2 −→