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Approximation by Differentials solved questions
Approximation by Differentials solved questions
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CHAPTER 18Approximation by Differentials
18.1 State the approximation principle for a differentiable function/(*).
Let x be a number in the domain of /, let A* be a small change in the value of x, and let Ay =
f(x + *x)-f(x) be the corresponding change in the value of the function. Then the approximation principle
asserts that Ay = f ' ( x ) • AJC, that is, Ay is very close to /'(*)' Ax for small values of AJC.
In Problems 18.2 to 18.8, estimate the value of the given quantity.
18.2
Let let x = 49, and let Ax = 2. Then A: + Ax = 51,
Note that The approximation principle tells us that Ay =
/'W-A*, (Checking a table of square roots shows that this is actually
correct to two decimal places.)
18.3
Let f ( x ) = Vx, A; = 81, AA: =-3. Then ;c + Ax = 78,
So, by the approximation principle, Hence,
(Comparison with a square root table shows that this is correct to two decimal places.)
18.4
Let /(jc)=v% AT = 125, Ax = -2. Then x + A* = 123,
So, by the approximation principle,
5-0.03 = 4.97. (This is actually correct to two decimal places.)
18.5 (8.35)2'3.
Let f ( x ) = x 2 ' 3 , x = 8 , A A : = 0 . 3 Then
5 . x + A J C = 8 . 3 5 , A y = ( 8 . 3 5 ) 2 ' 3 - 8 2 ' 3 = ( 8 . 3 5 ) 2 ' 3 Also,
- 4.
So, by the approximation principle, (8.35)2'3 - 4 ~ \ • (0.35), (8.35)2'3 = 4 + 0.35/3 =
4 + 0.117 = 4.117. (The actual answer is 4.116 to three decimal places.)
18.6 (33)-"5.
Let f(x) = x~ll\ A: = 32, A* = l. Then
Also, So, by the approximation
principle, (This is correct to three decimal
places.)
18.7
Let Then Also,
So, by the approximation principle,
(This is correct to three decimal places.)
138
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