Padmanabhan Seshaiyer

1
Introductory Calculus with Business Applications
Math 108 (Section 003)
Padmanabhan Seshaiyer Mathematical
Sciences George Mason University Email
pseshaiy_at_gmu.edu Lecture 7 (October 20, 2009)
2
Intermediate Value Theorem
Padmanabhan Seshaiyer George Mason University

• Show that   has a
  root somewhere in the interval 1, 2.

3
2.1 The Derivative
Padmanabhan Seshaiyer George Mason University

• If air resistance is neglected, an object dropped
  from a great height will fall s(t)16t2 feet in t
  seconds.
• What is the objects velocity after t2 seconds?
• How is the velocity found in part (a) related to
  the graph of s(t)?

4
Definition of the derivative
Padmanabhan Seshaiyer George Mason University

• Find the derivative of the function f(x) 16x2.

5
Applications of derivatives
Padmanabhan Seshaiyer George Mason University

• Find the tangent line to the function f(x)x3 at
  the point x 1.
• A manufacturer determines that when x thousand
  units of a particular commodity are produced, the
  profit generated will be
• P(x) 400x2 6800x 12000
• dollars. At what rate is profit changing with
  respect to the level of production x when 9000
  units are produced?

6
Significance of the sign of f (x)
Padmanabhan Seshaiyer George Mason University

• If the function f is differentiable at xc, then
• f is increasing at x c if f (c) gt 0
• f is decreasing at x c if f (c) lt 0

7
Derivative Notation
Padmanabhan Seshaiyer George Mason University

• Find the rate at which is changing
  with respect to x when x1.

8
Differentiability and Continuity
Padmanabhan Seshaiyer George Mason University

• Discuss the differentiability of the function
  yx at x0.

9
Announcements
Padmanabhan Seshaiyer George Mason University

• Suggested HW Problems
• Section 2.1 1-40, 43-54 (odd problems)
• EXAM II
• Chapter 1