Approximation by Differentials solved questions

CHAPTER 18
Approximation 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.)

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