Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Leibnitz Notation
We previously found that if f  x   x then f   x  
1
2 x
.
This may also be written:
if f  x   y  x then
dy
1

dx 2 x
using Leibnitz notation. Thus,
dy df
d


f  x
dx dx dx
The expressions dy and dx are commonly called differentials.
The expressions f   a  and
dy
dx
derivative of y  f  x  at x  a.
both mean to evaluate the
xa
We will next consider rules for differentiation that allow us to find
derivatives without the use of limits.
This may also be written as:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
From the Sum and Constant Multiplier Rules, we may
state:
 f  g    f    g   
f   g
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Finding Horizontal Tangents
Find the points where the graph of the function y = t3 – 12t + 4
has horizontal tangents.
How is the graph of f (x) = x3 – 12x related to the graph of its
derivative, f '(x) = 3x2 – 12?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
In Figure 5, given the graph of f (x), is (A) or (B) the graph of f '(x)?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
It is shown in the text that the derivative of bx with respect to x is
proportional to bx, or
The following table gives values for the proportionality factor, mb,
for some select bases.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
From the table on the previous slide, it would appear that for some
base valued between 2.5 and 3, the proportionality constant mb must
equal one.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Show that the equation of the tangent
line to y = 3ex – 5x2 at x = 2 is
y = 2.17(x – 2) +2.17.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 8 illustrates how the
secant lines at a jump
discontinuity fail to converge
on the tangent line at one side
or the other of the discontinuity.
This leads to Theorem 3.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example
Show that the function y  3 x is continous at x  0, but not
differentiable.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework




Homework Assignment #11
Read Section 3.3
Page 139, Exercises: 1 – 73 (EOO), 71
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company