lim
h0
f (a h) h
f (a) lim
x0
f (a x) x
f (a)
.
The left-hand derivative of f at a , is denoted by f ’-(a).
f(a)
lim
h0
f (a h) h
f (a)
lim f (a x) f (a) .
Solution The slope of the tangent line at (4,2) is
f (4) lim f (x) f (4) lim x 2
x4 x 4
x4 x 4
lim
x4
1
x4 (x 4)( x 2) 4
an equation of the tangent line is
Example Let f(x)=│x│.Show that f (x) is not differentiable at 0.
Proof By Definition, we have
f (0) lim
f (0 h)
f (0)
lim
h .
h0
h
h0 h
Since lim h lim h 1and lim h lim h 1,
h0
h
Since f is not defined by the same formula on both sides of 1, we will evaluate this limit by taking one-sided limits.
To the left of 1, f (x) x2. Thus
x2, x 1 f (x)
2x 1, x 1
Solution By definition
f (3) lim f (3 h) f (3)
h0
h
lim (3 h)2 (3)2
h0
h
lim(6 h) h0
6
Now let’s calculate
f (1) lim f (1 h) f (1) .
l Q(x,f(x) f(x)
P(a,f(a)) 0
m lim f (x) f (a) . xa x a
Let h x a Then x a h
So the slope of the second line PQ is
f (a h) f (a)
m pQ
. h
The slope m of the tangent line l is
o f (a)
f (a h) s
Average velocity displacement f (a h) f (a)
time
h
The instantaneous velocity at t=a
v(a) lim f (a h) f (a)
h0
h
Rates of change
y f (x)
c(x) is called the marginal cost R(x) is called the marginal revenue function.
P(x) is called the marginal profit function.
Example Find f (3) and f (1) given that
h0
h
h0
h
1 2x
so ( x ) 1 2x
domain( f ) domain( f )
A function f is differentiable on a closed interval [a, b] if f is differentiable on an open interval (a, b) and both the right-hand derivative f ’+(a) and the left-hand derivative f ’-(b) exist
f(1)
lim
h0
f (1 h) h
f (1)
(1 h)2
lim
h0
h
1
lim (2 h) 2. h0
To the right of 1, f (x) 2x 1. Thus
f(1)
lim
h0
f (1 h) h
f (1)
[2(1 h) 1] 1
lim
h0
h
lim 2 2. h0
And f (1) 2.
xa
xa x a
lim f (x) f (a)
Therefore xa This implies that f is continuous at a.
NOTE 1.The converse of theorem is false 2. f is not continuous at a, then f is not
Chapter2 Derivatives
2.1 The Derivative as a function
The Tangent Problem
Let f be a function and let P(a, f(a)) be a point on the graph of f. To find the slope m of the tangent line l at P(a, f(a)) on the graph of f, we first choose another nearby point Q(x, f(x)) on the graph (see Figure 1) and then compute the slope mPQ of the secant line PQ.
y 2 1 (x 4) or y 1 x 1
4
4
Interpretation of the Derivative as a Rate of Change The derivative f (a) is the instantaneous rate of change
of y f (x)with respect to x when x a.
If x change from x1 to x2,then then change , then the change in x (increment of x )is
x x2 x1
The corresponding change in y is
y f (x2 ) f (x1 )
The average rate of change of y with respect to x
A function f is differentiable on an open interval I if f ’(x) exists for every x in that interval I. Then
x I f (x) lim f (x h) f (x)
h0
h
is a function of x, denoted by
h h h0
h0
h h h0-
h0
h wehave that lim does not exists.
h0 h
Therefore the function f(x) is not differentiable at 0.
Interpretation of the Derivative as the Slope of a Tangent The geometric interpretation of a derivative.
is
y f (x2 ) f (x1 )
x
x2 x1
The instantaneous rate of change of y with respect
to x at x1is
lim y lim f (x2 ) f (x1)
x x2 x1
x2 x1
x2 x1
lim f (x1 x) f (x1)
x0
x
Definition of Derivative
Definition Let y=f(x) be a function defined on an open interval containing a number a.The derivative of f(x) at number a, denoted by f’(a) , is
f (a) lim f (a h) f (a) lim f (a x) f (a)
h0
h
x0
x
if this limit exists.
If we write x=a+h, then h=x-a and h approaches 0 if and only if x approaches a.then
f (a h) f (a)
m lim
.
h0
h
The velocity problem
Suppose an object moves along a straight line according to an equation of motion
s f (t)
f (t) is called the position function of the object
Theorem If a function f is differentiable at a number, then it is continuous at a.
Proof we have f (a) lim f (x) f (a) . xa x a
Hence
lim ( f (x) f (a)) lim[ f (x) f (a) (x a)] 0.