Download ln x

Ch 3.2 Differentiability
Graphical, Numerical, Algebraic by
Finney Demana, Waits, Kennedy
Estimate the Derivative
Using the function y = ln x, make a table of values for x that
would be useful to estimate the derivative of y = ln x at x = 1
and then estimate the derivative.
x
ln x
.97
.98
.99
1
1.01 1.02 1.03
Estimate the Derivative
Using the function y = ln x, make a table of values for x that
would be useful to estimate the derivative of y = ln x at x = 1
and then estimate the derivative.
x
.97
.98
.99
1
1.01
1.02
1.03
ln x
-.0305
-.0202
-.0101
0
.00995
.0198
.02956
.00995 - (-.0101) .02005
y(1) =

= 1.0025
1.01 - .99
.02
 1
Local Linearity
Activity
1986 AB 4
Let f be the function defined as follows:

 x - 1 + 2, for x < 1
f(x) =  2

ax + bx, for x  1, where a and b are constants
1.
If a = 2 and b = 3, is f continuous for all x? Justify your answer.
2. Describe all values of a and b for which f is a continuous function.
3. For what values of a and b is f both continuous and differentiable?
4. Using these values of a and b and the definition of the derivative, find f '(x) at
x = 1.