Question

Consider the following. f(x) = x4 − x 9 compare the values of δy and dy if x changes from 1 to 1.07. (round your answers to four decimal places.) δy = 0.2408 dy = -0..9 what if x changes from 1 to 1.02? (round your answers to four decimal places.) δy = 0.0624 dy = 0.2448 incorrect: your answer is incorrect. does the approximation δy ≈ dy become better as δx gets smaller? yes/No

Answer 1

The values of δy and dy are calculated when x changes from 1 to 1.07 and 1 to 1.02. The approximation δy ≈ dy becomes better as δx gets smaller.

Step-by-step explanation:

The question asks us to compare the values of δy and dy when x changes from 1 to 1.07 and from 1 to 1.02. Let's calculate each value:

For x changing from 1 to 1.07:

δy = (f(1.07) - f(1)) = (1.07^4 - 1) - (1.07^9 - 1) ≈ 0.2408

dy = 0.0117(1.07^3) - 0.0117(1.07^8) ≈ -0.9

For x changing from 1 to 1.02:

δy = (f(1.02) - f(1)) = (1.02^4 - 1) - (1.02^9 - 1) ≈ 0.0624

dy = 0.0117(1.02^3) - 0.0117(1.02^8) ≈ 0.2448

So, the values of δy and dy change as x changes. Now, let's answer whether the approximation δy ≈ dy becomes better as δx gets smaller:

Yes, the approximation δy ≈ dy becomes better as δx gets smaller because it becomes closer to the true value of δy.