Page 10 - Math SL HB Sem 2

b) Rate ofchange interpretation ofthe derivative
If y: (x), then P is the function whose value at x is the instantaneous rate of
change of y with rcspect to x at the point x.

TAI{GENTS
u=fq

tlngent

,otnlof
co,tiod @,!hD

noamal
x=a

Consider a curve y =/i'.r,).
If A is the point with x-coordinate a, then the slope of the tangent at this point is
f(a).
The equation of the tangent is

Y- f(o) = y'1o1 .f equating slopes).
x-a
or y -f(a) :f (a)(t -a)

NORMALS

A normal to a curve is a line which is perpendicular to the tangent at the point of
contact.

Thus, the slope of a normal at r = a is -J-
f (a)

Exzunple 4 :

: f(a+h)-f(a)
Use the first principles formula 7'(a) tim to find:
h
h-+o

a) the slope ofthe tangent to /(:)= x'? +2x at x=5
!
b) the instantaneous rate of change of f G) = at "r = -3
x