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An alternate definition for approximation is as -

If a and a+h belongs to the domain of a differentiable function ’ f ’ , then the approximate value of f (a +h) will be f (a+h) » f (a) + hf ‘ (a), f ‘(a) ¹ 0. By using this, approximate problems can be solved easily.

Example 50

Find correct to 4 decimal places, the approximate value of

Solution : Let f (x) =

Take a = 8 and h = 0.05 then

Now the approximation formula is


Example 51

Find the approximate value of tan -1 (0.99)

Solution : Let f (x) = tan-1 (x)

\ f ‘ (x) =

Take a = 1 and h = 0.01 then

f (a) = f (1) = tan -1 (1) = radians.

By the approximation formula

f ( a+h ) » f (a) + h f ‘ (a)

Example 52

Find the approximate value of sin 310 assuming that 10 = 0.0175 radians and cos 300 = 0.8660.

Solution : Let f (x) = sin (x) then f ‘ (x) = cos x

Take a = 300 = p/6 Radians. h = 10 = 0.0175 radians.

Then f (a) = f (300) = sin 300 = 0.5

and f ‘ (a) = f ‘ (30) = cos 300 = 0.8660

using the approximation formula i.e.

f (a + h) » f (a) + h f ‘ (a) we get

\ sin 310 = f (300 ) + ( 0.0175 ) f ‘ (300)

\ sin 310 = 0.5000 + ( 0.0175 ) (0.8660)

\ sin 310 = 0.5152

Errors

Let y = f (x) be a differential function of x and Dx an error in x, then

The approximate error in y is Dy = f ‘ (x) Dx

i.e. Dy = f ‘ (x) dx

Approximate relative error in y is

Approximate percentage error in y is

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Index

5.1 Tangent And Normal Lines
5.2 Angle Between Two Curves
5.3 Interpretation Of The Sign Of The Derivative
5.4 Locality Increasing Or Decreasing Functions 5.5 Critical Points
5.6 Turning Points
5.7 Extreme Value Theorem
5.8 The Mean-value Theorem
5.9 First Derivative Test For Local Extrema
5.10 Second Derivative Test For Local Extrema
5.11 Stationary Points
5.12 Concavity And Points Of Inflection 5.13 Rate Measure (distance, Velocity And Acceleration)
5.14 Related Rates
5.15 Differentials : Errors And Approximation

Chapter 6





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