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

Applications of Taylor series