Approximate Error Differentials
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available. Most of the classes have practice problems with solutions available on the practice problems pages. Also most classes have using differentials to approximate error assignment problems for instructors to assign for homework (answers/solutions to the assignment
Use Differentials To Approximate The Maximum Error
problems are not given or available on the site). Algebra [Notes] [Practice Problems] [Assignment Problems] Calculus I differentials to approximate change [Notes] [Practice Problems] [Assignment Problems] Calculus II [Notes] [Practice Problems] [Assignment Problems] Calculus III [Notes] [Practice Problems] [Assignment Problems] Differential Equations [Notes] Extras Here are some extras topics differentials to approximate square root that I have on the site that do not really rise to the level of full class notes. Algebra/Trig Review Common Math Errors Complex Number Primer How To Study Math Close the Menu Current Location : Calculus I (Notes) / Applications of Derivatives / Differentials Calculus I [Notes] [Practice Problems] [Assignment Problems] Review [Notes] [Practice Problems] [Assignment Problems]
Differentials To Approximate Volume
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Use Differentials To Approximate The Quantity
Integers Adding Integers Adding Integers on a Number Line Subtracting Integers Subtracting Integers on a Number Line Multiplying Integers Dividing Integers Exponents and Integers Factors and Multiples > http://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx Divisibility of Integers Even Numbers (Integers) Odd Numbers (Integers) Divisibility Rules What are Factors and Multiples Integer Factorization What is a Prime Number Composite Numbers How do you do Prime Factorization Greatest Common Divisor (GCD) Least Common Multiple (LCM) Fractions > What is Fraction Proper Fractions Improper Fractions Mixed Numbers/Fractions Mixed Numbers on a Number Line Equivalent Fractions http://www.emathhelp.net/notes/calculus-1/differentials/using-differentials-to-estimate-errors/ Reducing Fractions Adding Fractions with Like Denominators Subtracting Fractions with Like Denominators Adding Fractions with Unlike Denominators Subtracting Fractions with Unlike Denominators Converting Mixed Numbers to Improper Fractions Converting Improper Fractions to Mixed Numbers Adding Fractions with Whole Numbers Subtracting Fractions with Whole Numbers Adding Mixed Numbers Subtracting Mixed Numbers Comparing Fractions Multiplying Fractions Multiplying Mixed Numbers Dividing Fractions by Whole Number Dividing Fractions Dividing Mixed Numbers Reciprocals Negative Exponents Rational Numbers Decimals > What is Decimal Decimals Place Value Rounding Decimals Decimal Number Line Comparing Decimals Powers of 10 Scientific Notation Decimal Fractions Converting Decimals To Fractions Converting Fractions to Decimals Adding Decimals Subtracting Decimals Multiplying Decimals Dividing Decimals by Whole Numbers Dividing Whole Numbers by Decimals Dividing Decimals Repeating (Recurring) Decimals Absolute Value Percents > What is Percent Converting Decimals to Percents Converting Percents to Decimals Irrational Numbers > Squares and Square Roots Perfect Square Cubes and Cube Roots Perfect Cube Nth Root What is Irrational Number Irrational Numbers on a Number Line Real Numbers Powers and Exponents > Frac
so that each side is 0.5 inches plus or minus 0.01 inches. Then its volume is http://ltcconline.net/greenl/courses/105/Applications/difern.htm V = x3 So that V ' = 3x2= 3(0.5)2 https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm = 0.75 Dy @ (0.75)(0.01) = .0075 cu inches. So that the volume of the die is approximately in the range (0.5)3 +- 0.0075 = 0.125 0.0075 or between 0.1175 and 0.1325 cubic inches We can use differentials to approximate We let differentials to f(x) = x1/2 Since f(1 + Dx) - f(1) @ f '(1) Dx We have f(1 + Dx) @ f '(1) Dx + f(1) f(1) = 1, f '(1) = 1/2, Dx = .01 we have f(1 + Dx) @ 1/2 (.01) + 1 = 1.005 (The true value is 1.00499) Exercise: A spherical bowl differentials to approximate is full of jellybeans. You count that there are 25 1 beans that line up from the center to the edge. Give an approximate error of the number of jelly beans in the jar for this estimate. Relative Error and Percent Error DefinitionThe relative error is defined as Error Relative Error = Totalwhile the percent error is defined by Error Percent Error = x 100% Total Example The level of sound in decibels is equal to V = 5/r3 Where r is the distance from the source to the ear. If alistener stands 10 feet 0.5 feet for optimal listening, how much variation will there be in the sound? What is the relative and percent error? Solution 15 V' = -15r-4 = = -0.0015 10,000 DV @ (-0.0015)(.5) = -0.00075 V @ 0.005 0.00075 We have a percent error of 0.00075 Percent Error @ = 15% 0.005 Back to Math 105 Home Page e-mail Questions and Suggestions
with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations in algebraic context. At this mathematical level our presentation can be briefer. We can dispense with the tedious explanations and elaborations of previous chapters. 6.2 THE CHAIN RULE AND DETERMINATE ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: [6-1] ∂R ∂R ∂R dR = —— dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation to DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations. They are also called determinate error equations, because they are strictly valid for determinate errors (not indeterminate errors). [We'll get to indeterminate errors soon.] The coefficients in Eq. 6.3 of the fractional errors are of the form [(x/R)(∂R/dx)]. These play the very important role of "weig