How do we find our approximation for 2.95?

2 Answers
Sep 24, 2016

See explanation

Explanation:

2.9=29/10 is a rational number of the form integer/integer..

And so,29#/1010)^5 = 20511149/100000 is rational.

The exact value in decimals is 205,11149.

Successive rounded approximations are

3-significant digits ( sd:): 205

4-sd: 205.1

5-sd: 205.11

6-sd: 205.111

6-sd: 205.1115.

You can try instead the binomial expansion for

2.95=(30.1)5=35(1130)5

Sep 24, 2016

We can also use the approximation of y=x5 around x=3.

The derivative of y is dydx=5x4, so the slope of the tangent line around x=3 is 5(34)=405.

The point it intersects is (3,35)=(3,243).

Thus the tangent line is y243=405(x3)y=405x972.

Thus, an approximation for 2.95 would be:

y=405(2.9)972=202.5.

This compares to the actual value of 2.95=205.11149.