Local Linearity. The idea is that as you zoom in on a graph at a specific point, the graph should eventually look linear. This linear portion approximates the tangent line to the curve at the specific point.
A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.
Derivatives will fail to exist at:
The derivative is defined at the end points of a function on a closed interval.
The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.
nDeriv(1/x,x,0) returns 1000000 on a TI-84, but -∞on a TI-89
nDeriv(abs(x),x,0) returns 0 on a TI-84, but +/-1 on a TI-89
difference quotients (the same way you’ve been doing it).
Forward difference quotient:
Backward difference quotient:
Symmetric difference quotient:
If a and b are any two points in an interval on which f is differentiable, then takes on every value between and .
Between a and b, must take on every value between and .
Intermediate Value Theorem for Derivatives