21-111, Applications of the derivative, Ch2

I am still doing Calculus 1, 21-111 from Carnegie Mellon UC. Today I was going through Chapter 2. Everything seemed to be clear and obvious.

  • Describing Graphs of Functions
  • The First- and Second-Derivative Rules
  • The First- and Second-Derivative Tests and Curve Sketching
  • Curve Sketching (Conclusion)
  • Optimization Problems
  • Further Optimization Problems
  • Applications of Derivatives to Business and Economics

And as always, a little bit of notes/ theory from the topics learned.

First-Derivative Rule

  • If f′(a) > 0, then f(x) is increasing at x = a. If f′(a) < 0, then f(x) is decreasing at x = a.
  • If f′(a) = 0, the function might be increasing or decreasing or have a relative extreme point at x = a.

Second-Derivative Rule

  • If f′′(a) > 0, then f(x) is concave up at x = a. If f′′(a) < 0, then f(x) is concave down at x = a.

What’s Concave Up or Down?

concave_graph

And combined results

concave_derivative_

The First-Derivative Test (for local extreme points) Suppose that f′(a) = 0.

  • If f′ changes from positive to negative at x = a, then f has a local maximum at a.
  • If f′ changes from negative to positive at x = a, then f has a local minimum at a.
  • If f′ does not change sign at a, then f has no local extremum at a.

The Second-Derivative Test (for local extreme points)

  • If f′(a) = 0 and f′′(a) < 0, then f has a local maximum at a.
  • If f′(a) = 0 and f′′(a) > 0, then f has a local minimum at a.

I wont cover the topic on how it can be used in business, but I really suggest you to go through the chapter yourself to get the understanding.

A great chance to practice your skills is to complete interactive exercises on Khan.

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