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
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