math (1029) - Calc: tangents & differentiation

talked about two things in maths today, the first was the tangent of a curve.

each point on a curve can only have one tangent. a tangent is a linear function that touches a curve at a point without crosing it.

a tangent can also be described as the limit of secants along the curve (coz secants all have 2 points, coming into one point that eventually only has 1 point - the tangent)

most curves can have tangents anywhere (except on discontinuous limits) an exception would be absolute value graphs, on the very corner of the graph, tangents cannot exist (because its possible to have heaps of tangents (in theory) but thats not possible.. that wold not be a tangent) this is also true on piecemeal graphs that are split between a curved half and a straight half. the tangent does not exist on the straight side on the corner.

the formula for differentiation by first principles goes as follows:

f(x+h) - f(x)
f`(x) = lim ___________
h -> 0 h

this is what we call the derivative of a function. we were working on this during the late parts of the lesson, and i was falling asleep.. so have to go over it again.

but basically the question they give you, you just sub it into the formula.