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Jumps and Limited Error Control
In some cases unlimited error control is not possible at the point x = a. It fails in the following case:
There is an e > 0 such that for every d > 0, there is some x satisfying the conditionThis means as the input x to the function y = f(x) becomes a better approximation to the number a, there is no guarantee the difference |f(x)-f(a)| will be smaller than the error control target e. This concept is illustrated by functions whose graphs have a few jumps in them. The height of the largest jump near a point x = a indicates how small the target tolerance e or ½ ·10-n can be in the discussion of error control.
|x-a| < d and |f(x)-f(a)| > e.
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Example of a function with jumps and with places where the function is not defined. |
Unlimited error control is possible in the following circumstances:
These circumstances appear when f(x) is continuous at x = a.
For each target tolerance e > 0, there is a tolerance d > 0 such that the condition
|x-a| < d and |f(x)-f(a)| £ e.
Computations on machines with finite accuracy precision arithmetic, restrict the number n of decimals places that can be accurately computed. Every computing machine which calculates to finitely many binary or decimal places, suffers from such a limit. Small discontinuities in calculations appear, except in those case where exact arithmetic can be done. For example, on a computing machine which computes to at most n0 decimal places, the existence of a rule of the form
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On a computing machine which computes to at most n0 decimal places, the error control of a single addition and multiplication are guaranteed to only n0 binary (or decimal) places. Digits beyond the n0 place are uncertain. If several such calculations are done, with numbers in one calculation being used in the next, errors accumulate and accuracy is lost. The calculations in question may have to be reorganized to improve accuracy.
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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
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