Differentials to estimate value

Question

Use differentials to estimate the value of 4sqrt16.2. Compare the answer to the exact value of 4sqrt16.2.

 

Round your answers to six decimal places, if required.

 

estimate =

 

 ____________

 

 

 exact value =

 

 ____________

Answer 1

Let y = 4 √x. Then, dy/dx = 2/√x

While this is not stated, as 16 has a square root, this is the point from which the approximation was intended (why AI systems have such trouble with math).

Then, our approximation for 4√16.2 is f(16) + f'(16)(16.2 - 16)

Then, our approximation will be 4√16 + 2/√16 (.2) = 16 + .1 = 16.1

4√16.2 is, rounded to 6 digits, 16.099689.

The approximation is within .001 of the exact value.