Differential Calculus

Besides the examples on this page, please see the discussion in ["BasicCalculus"].

Functions

Piecewise fcns, polynomials, exponential, logs, trig and hyperboic trig functions.

Limits

SAGE can compute \lim_{x\rightarrow 0}\frac{\sin(x)}{x}:

sage: limit(sin(x)/x,x=0)
1

Laws and properties

Continuity

Differentiation

SAGE can differentiate x^2\log(x+a) and \tan^{-1}(x)=\arctan(x):

sage: diff(x^2 * log(x+a), x)
2*x*log(x + a) + x^2/(x + a)
sage: derivative(atan(x), x)
1/(x^2 + 1)

Laws

SAGE can verify the product rule

sage: function('f, g')
(f, g)
sage: diff(f(t)*g(t),t)
f(t)*diff(g(t), t, 1) + g(t)*diff(f(t), t, 1)

the quotient rule

sage: diff(f(t)/g(t), t)
diff(f(t), t, 1)/g(t) - (f(t)*diff(g(t), t, 1)/g(t)^2)

and linearity:

sage: diff(f(t) + g(t), t)
diff(g(t), t, 1) + diff(f(t), t, 1)
sage: diff(c*f(t), t)
c*diff(f(t), t, 1)

Rates of change, velocity

Derivatives of polys, exps, trigs, log

Chain rule

Implicit differentiation

Higher derivatives

Applications

Related rates

Maximum and minimum values

You can find critical points of a piecewise defined function:

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f1 = x^0
sage: f2 = 1-x
sage: f3 = 2*x
sage: f4 = 10*x-x^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: f.critical_points()
[5.0]

Optimization problems

Indeterminate Forms, L'Hopital's rule

Newton’s Method

Sequences and series

(Some schools teach this topic as part of integral calculus.)

Sequences

Series

Tests for Convergence

• o The Comparison Test o Absolute and Conditional Convergence o The Ratio Test o The Root Test

Power series

• o Shift the Origin o Convergence of Power Series

Taylor series

Applications of Taylor series