|
Starter Guide (Views)
Real Player Videos
4. Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
Starter & Warm Up Lessons
1. Usual Review/Starter Lessons
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material
| |
Using the Second Derivative
to classify critical points as
maximums or minimums.
Two Site
Reviews
- Magellan, the McKinley Internet Directory, 1996:
Mathphobics, this site may ease your fears of the subject, perhaps
even help you enjoy it. The tone of the little lessons and
"appetizers" on math and logic is unintimidating, sometimes
funny and very clear. There are a number of different angles offered,
and you do not need to follow any linear lesson plan. Just pick and
peck. The site also offers some reflections on teaching, so that
teachers can not only use the site as part of their lesson, but also
learn from it. (Magellan is no longer online)
- The
World-Wide Web Virtual Library Education by Country - Canada 1,
2005. Why Slopes: Appetizers and Lessons for Math and Reason. This
online classroom offers appetizers and lessons for math from
arithmetic to calculus or why slopes; for deductive reason (logic) and
critical thinking; and for learning in general. Included here are
opinions on the communication of skills and mathematics instruction.
The logic appetizers are math free. Each appetizer is different. If
one is not to your liking try another. Most are from three books on
understanding and explaining math and reason.
may encourage a visit to site entrance www.whyslopes.com. |
Let y = f(x) be a function. If the second derivative f''(x) is non-zero at a
critical point x = a (that is a point where the first derivative is zero - f'(a)
= 0) then the sign of the second derivative may be used to classify the point x
= a as a local maximum or a local minimum.
Second Derivative Test:
Suppose y = f(x) and f'(a) = 0 and the second derivative f"(x) is
continuous about an interval centered at x = a. Then
- x = a is a locally a minimum if f"(a) > 0,
- x = a is a locally a maximum if f"(a) < 0; and
Here if f"(a) = 0, the sign of f"(x) gives no result.
Memory Aid for what the sign of f"(a) implies: For f(x) = x2
has f"(x) = 2 > 0 and the critical point x = 0 is a minimum.
Remark: When there are many maxima, the first and
second derivative based tests alone can not say which the greatest or absolute
maximum. For that function values are required. When there are many minima,
the first and second derivative based tests alone can not say which the
greatest or absolute minimum. For that function values are required.
Example 1: Classification of Critical Points as Max or Min from
sign of second derivative.
"Proof" of the 2nd Derivative Test:
- If f'(a) = 0 and f"(a) > 0 then linear approximation of y'
=f '(x) by f''(a)(x-a) implies near x = a that y' = f '(x) is negative for x
< a and positive for x < a, and hence x = a is local minimum due to
the first derivative test.
- If f'(a) = 0 and f"(a) < 0 then linear approximation of y'
=f '(x) by f''(a)(x-a) implies near x = a that y' = f '(x) is positive for x
< a and negative for x < a, and hence x = a is local maximum because
of the first derivative test.
See your textbook (or the linear approximation theorem in site pages) for a
fuller explanation of the second derivative test.
| |
|
www.whyslopes.com
site
search
Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Skill
& Concept
Review or Development
1. Decimal
Arith - Video Based ]
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations -
making alg easier
4. Formulas
forwards & Backwards - unifying theme for Algebra
5. Proportionality,
Back- & For-wards - theme at work.
6. Logic
- Math Free, good for precision in work & studies
7. Euclidean-Geometry
(leanly)
8. Slopes
and Lines
9. Why
Study Slopes - a context
10. Quadratics
11 Polynomials
12 Factored
Polys - a context
13 Functions
- For-& Back -wards
14 Number
Theory, Richly
15. Exponents,
Radicals & logs.
16 Calculus
- Examples & Advice
17. Real
Analysis
18
Electric
Circuits Etc (So So)
19 Maps,
Similarity & Trig, (alt view)
20 Complex
numbers
21 Logic
with Symbols+truth tables
22 Consistent
Story Telling
23. Even
More Logic
|
|