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This WiZiQ
Calculus Tutorial Link gives the date and time of
a whyslopes. com interactive session in which students may see
written and spoken answers to start of calculus questions on
say on functions, limits, continuity, and derivatives. First
session is free.
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YOU are better than YOU think. Show
yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
For online automated help in senior
high school maths & calculus, visit quickmath.com
For Automatic Calculus and Algebra Help with derivatives,
integrals, graphs, linear equations, matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different
range of services, some free, some not, all based on webmathematica.
Good luck.
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Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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Limits Evaluation by immediate or delayed substitution
Immediate substitution of x = a into an expression f(x)
is possible when and only when a the expression or function f(x) is
continuous at x = a.
Limits, Algebraic Evaluation
Here are some limits we are going to evaluate algebraically using the
algebraic described properties of limits in the previous lesson.
A = lim x® 2
3x+4
B = lim x®
6 5 x2-
8x
C = lim x® -2
4x3-3x+ 1
D = lim y ®
-2 4y3-3y+ 1
and
Solutions
For the calculation
A = lim x® 2
3x+4
as x ® 2 we observe or assume
3x ® 6 and so observe or assume 3x+4 ®
10. We can also write more briefly
A = lim x® 2
3x+4 = 6+ 4 = 10 (result)
Second
B = lim x® 6
5 x2- 8x
= 5(6)2 - 8(6)
= 5(30) - 48
= 150 - 48 = 102 (result)
| The limit evaluation process x®
6 applied to 5 x2- 8x, an expression
dependent on x, results in a number -25 which does not
depend on x. This limit evaluation process eliminates the x
dependence. When we apply a limit process to an formula or
function f(x) which eliminates the x-dependence, we call the
letter or placeholder x, a dummy variable. |
Third
C = lim x® -2 4x3-3x+
1
= 4 (-2)3 - 3(-2) + 1
= 4(-8) +6 + 1 = -32+ 7
= -25 (result)
Fourth we can evaluate
D = lim y ® -2
4y3-3y+ 1 = -25
directly by same reasoning we did for C. Simply replace the x by a y.
| More on Dummy Variables: The letters x and
y in the expressions for C and D have the same roles. In the
expressions x® -2 and y ®
-2 they both represent the ideas of a number approaching the value
-2. But the results for C and D do not depend on our choice of
letters in the limit expressions for them.
In the evaluation of a limit
the value L of the limit does not depend on x, limit
evaluation eliminates that dependence, but the value of L may
depend on a. |
Fifth, for the evaluation of the limit
we observe the attempt to evaluate inside expression
by the immediate substitution of 5 for x (x =5) in the yields 0/0, a
fraction with a zero in the denominator, a fraction which has no numerical
definition or value. It is undefined. But observe the inside expression
|
f(x) = |
x-5
x2-5x |
= |
x-5
(x-5)x |
= |
1
x |
® |
1
5 |
when x ® 5 |
The foregoing suggests the values of the inside expression
f(x) =
x-5
x2-5x
approach 1/5 or 0.2 as x approaches 5 even thought f(0) is not defined.
Note: The nuance, subtlely or technicality here is that in
the evaluation of a limit
the value of the inside expression f(x) at the limiting value x = a of
x as x approaches a is not of interest. It does not have to be defined.
Limit evaluation here is independent of what happens at x = a. Limit
evaluation is based gives the limiting value of f(x) when x is restricted
to smaller and smaller a-deleted intervals centred at x = a, that is
intervals in which the value a has been removed. |
The foregoing discussion needs to be understood, but when we evaluate the
limit
we write less. In particular we write
|
E = |
lim
x® 5 |
x-5
x2-5x | |
= |
lim
x® 5 |
x-5
(x-5)x | |
= |
lim
x® 5 |
1
x | |
= |
1
5 |
(exact
answer) |
The avoidance of 0/0 by replacing the initial expression by another
represents a delay substitution.
Please leave your results as a fraction. Here
the answer can be expressed exactly as a decimal but in correcting your
results and any work leading to your results it easier to recognize a fraction
(reduced to lowest forms) than it is to recognize a decimal. So do exact
arithmetic with fractions and radicals (square roots etc instead of using your
calculator. This instructionto do exact arithmetic and avoid decimals,
or postpone their use until all possible exact arithmetic is done with whole
numbers and fractions provides a standard to meet in your mastery of calculus.
I have done the limit evaluation process over several lines. You could do it
in one lines.
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www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
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.
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