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Find the values of $\Delta y$ and $d y$ for the given values of $x$ and $d x$.$$y=\left(x^{2}+2 x\right)^{3}, x=7, \Delta x=0.02$$

$\Delta y=3834.4377$ and $d y=3572.1$

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 8

Differentials and Linear Approximations

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

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soon. This problem? We're looking at the formula. Why? X squared minus two X plus one. We're looking to find d y and we're looking to find Delta Y. Well, the first thing you have to remember is Delta Y is based on the formula, the function of X plus Delta acts minus f of eggs. This will give us Delta Y. So how do we calculate this? Well, we're gonna take our original formula were in a substituted. So right now we've got X plus Delta X squared minus two times X plus Delta X plus one minus and use parentheses. Teoh put in the original formula so that we don't forget to change all the signs. So we need to expand this out and simplify it. To get our final answer, we have X squared plus two x Delta X plus Delta X squared minus chew X minus two Delta X plus one minus X squared plus two x minus one. Okay, the negative one and the positive one will cancel the negative. Two acts in the positive two x will cancel the X squared in the negative. X squared will cancel. That tells us that delta y which is part of our answer is two X dealt axe plus Delta X Squared minus chew Delta X. Now we just have to find d y. Well, I know D Y DX represents the derivative of the original equation. So the derivative the original equation is Chu X minus two to find d y. All I have to do is multiply both sides by D. X so d y equals shoe X minus two d x So here's my d y and here's my delta y.

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