Approximate Relative Error Differentials
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Whole Number Place Value of Whole Numbers Rounding Whole Numbers Whole Numbers on a Number Line Comparing Whole Numbers Adding Whole Numbers Subtracting Whole Numbers Multiplying Whole Numbers using differentials to approximate error Multiplication Table Dividing Whole Numbers (Long Division) Division with Remainder Integers differentials to approximate change > Negative Numbers What is Integer Number Rounding Integers Number Line with Integers Ordering and Comparing Integers Adding
Differentials To Approximate Square Root
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 > Divisibility of Integers Even
Differentials To Approximate Volume
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 Reducing Fractions Adding Fractions with Like differentials to approximate cube root 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 > Fractional (Rational) Exponents Adding Exponents Subtracting Exponents Multiplying Exponents Dividing Exponents Prop
Solutions 1. Approximations If a quantity x (eg, side of a square) is obtained by measurement and a quantity y (eg, area of the square) is calculated as a function of x, say y = f(x), then any error involved in the measurement of x produces an error in the calculated value of y
Use Differentials To Approximate The Value Of The Expression
as well. Recall from Section 4.3 Part 2 that the Section 8.3 Part 1, we have: use differentials to approximate the quantity That is, the error in x is dx and the corresponding approximate error in y is dy = f '(x) dx. Fig. 1.1 approximate relative error matlab Fig. 1.2 – 1st and 2nd axes: if 1,000 = xa – 1 then xa = 1,001, – 1st and 3rd axes: if 1,000 = xa + 1 then xa = 999, therefore xa is somewhere in [999, 1,001]. http://www.emathhelp.net/notes/calculus-1/differentials/using-differentials-to-estimate-errors/ Example 1.1 Solution Let s be the side and A the area of the square. Then A = s2. The error of the side is ds = 1 m. The approximate error of the calculated area is: dA = 2s ds = 2(1,000)(1) = 2,000 m2. EOS Note that we calculate dA from the equation A = s2, since the values of s and ds are given. To find the differential of A we must have an equation relating A to s. So even if the measured value of the side is given http://www.phengkimving.com/calc_of_one_real_var/08_app_of_the_der_part_2/08_04_approx_of_err_in_measrmnt.htm we still define the variable s that takes on as a value the measured value. In general, when the measured value say V of a quantity and the error say E in the measurement are given, we define a variable say x for the quantity, so that x = V and dx = E, which will be used later on in the solution. When using the quantity, first use the variable x, not the value V, then use the value V when a value is to be obtained. Go To Problems & Solutions Return To Top Of Page 2. Types Of Errors A measurement of distance d1 yields d1 = 100 m with an error of 1 m. A measurement of distance d2 yields d2 = 1,000 m with an error of 1 m. Both measurements have the same absolute error of 1 m. However, intuitively we feel that measurement of d2 has a smaller error because it's 10 times larger and yet has the same absolute error. Clearly the effect of 1 m out of 1,000 m is smaller than that of 1 m out of 100 m. This leads us to consider an error relative to the size of the quantity being expressed. This relative error is accomplished by representing the absolute error as a fraction of the quantity being expressed. For example, the relative error for d1 is 1 m / 100 m = 1/100 = 0.01 and that for d2 is 1 m / 1,000 m = 1/1,000 = 0.001. As desired the relative error for d2 is smaller than that for d1. The percentage error is the absolute error as a percentage of the quantity being expressed. For example
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