What Derivatives Tell Us.

0:00 What Derivatives Tell Us. Increasing and decreasing. Critical Points. Concave up and concave down. Inflection Points.

10:08 Example in detail.

Sketch the Curve Example.

Use the Guidelines of Calculus to sketch the curve of f(x)=((2x^2)/(x^2-1)).

A. Domain.

B. Intercepts.

C. Symmetry.

D. Asymptotes.

E. Intervals for Increase and Decrease.

F. Local Maximum and Minimum Values.

G. Concavity and Points of Inflection.

H. Sketch the Graph.

Maxima and Minima.

0:00 Maxima and Minima.

0:33 Absolute Maxima and Minima.

2:08 Examples within Graphs.

12:21 Extreme Value Theorem.

13:02 Local Maximum and Local Minimum.

13:59 Local Extreme Value Theorem.

14:44 Critical Points.

16:57 Example: Find the critical points of the following function f(x) = 3x^2- 4x +2.

18:00 Example: Find the critical points of the following function f(x) = (1 / x ) +ln (x).

18:59 Example: Find the critical points of the following function f(x) = x ^2 (SQRT ( x+5)).

20:35 Example: Determine the location and value of the Absolute Extreme Values of f ( x ) = x^2 -10 on [ -2, 3].

22:57 Example: Determine the location and value of the Absolute Extreme Values of f ( x ) = x^4 -4 x^3 +4 X^2 on [ -1, 3].

Curve Sketching.

Inc. & Dec. Concave Up. Concave Down, Abs Max, and Abs Min. The Extreme Value Th.

Graphing Functions using Derivatives.

0:00   Graphing Functions using Derivatives. X-Intercepts.

First Derivative Test. Decreasing. Increasing. Critical Points. Local Min and Local Max. Second Derivative Test. Concave Up. Concave Down. Inflection Points.

0:25 Example: f(x) = (x-3)(x+3)^2.

6:29 Example: f(x) = x^3-6x^2+9x.

9:32 Example: f(x) =x^4-6x^2.

Continuity I

Sketch the graph of a function that is continuous except for the stated discontinuity.

Discontinuous, but continuous from the right at 2.