|
YOU are better than YOU think. Show yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
refinement.. |
-/[]\-
||
_ / \
|||||||||||||||||||||||||||| .
|
| |
Parameters in Limits
If k is a real number which is constant or does not depend on x, then
limx® a k f(x) = k limx® a f(x)
For example,
limx® 3 k x2 = k limx® 3 x2 = k 32 =
k9 = 9k
when k = 1, 10, 55 or any number you choose.
If a number denoted by a letter k or q or p appears in a limit involving
another (dummy) variable, we call the number and the letter that denotes
it, a parameter.
To learn more about describing and talking about numbers and
quantities. read the long essay [What
is a Variable] and read
Volume 2, Three Skills For Algebra (chapters
8 & 9 ).
A parameter a or x will occur in the forthcoming limit-based introduction of
derivatives.
- lim x® A a x2 +bx+c = a A2 +bA+c
Here a, b, c and A are parameters. The right hand side depends on the
parameters a, b and c, and A. When a =3, b = 4, c =10 and A = 2. The
foregoing equation or template becomes
lim x® 2 3 x2 +4x+10 = 3*22
+4*2+10 = 12 + 8 + 10 = 30
- Let f(x) = x2. Then with the parameter a
= 4 (or another value)
f(a+h) - f(a) = (a+h)2-a2 = a2
+ 2h + h2 = 2ha + h2 =h(2a + h)
Therefore
lim
h® 0 |
f(a+h)-f(a)
h |
= |
lim
h® 0 |
h(2a + h)
h |
= |
lim
h® 0 |
2a + h
|
= 2a
|
Re-read the foregoing with a =3, 8, 99, r (another parameter) or x. What changes?
-
Let f(x) = x2. Now rewrite the foregoing with an
x instead of a.
f(x+h) - f(x) = (x+h)2-x2 = x2
+ 2h + h2 = 2hx + h2 =h(2x + h)
Therefore
lim
h® 0 |
f(x+h)-f(x)
h |
= |
lim
h® 0 |
h(2x + h)
h |
= |
lim
h® 0 |
2x + h
|
= 2x
|
In examples 2 above, the limit process
applied to the a and h dependent expression
eliminates the h dependence and results in an a-dependent
expression 2a. Likewise in example 3, with x in place of the number a in example
2, the evaluation of the limit
results in an expression 2x which depends only on x and not on the eliminated
or dummy variable h. In the limit evaluations above, the variables x and a
are parameters, and values of the limits are parameter dependent.
In the discussion of derivatives for curves or functions y = f(x) in
the next or one of the next lessons, the evaluation of
gives an x-dependent result,
| g(x) = |
lim
h® 0 |
f(x+h)-f(x)
h |
The formula for g(x) also written as f '(x) can be obtained from the
formula for f(x) via a limit-based calculation or via calculation rules,
obtained from the properties of limits, which go directly from the formula for
f(x) to the formula for g(x) without an explicit evaluation of limits.
| |
www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
Calculus Videos
0. First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn: Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
3. Derivative Motivation
3. Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x) (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3. Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
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)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5 Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere
To Learn More, visit Volumes 2 and 3.
Advanced Topics
Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity
To Learn More: Visit Real Analysis.
[ Back ] [ Up ] [ Next ]
|