find-the-values-of-delta-y-and-d-y-for-the-given-values-of-x-and-d-x-yleftx22-xright3-x7-delta-x00-3

Question

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

Joseph Lentino · Accepted Answer

soon. This problem? We&#x27;re looking at the formula. Why? X squared minus two X plus one. We&#x27;re looking to find d y and we&#x27;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&#x27;re gonna take our original formula were in a substituted. So right now we&#x27;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&#x27;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&#x27;s my d y and here&#x27;s my delta y.

Joseph Lentino · Answer

Okay, so we&#x27;re starting with why equals X to the third. So the first but we&#x27;re trying to find here is we&#x27;re trying to find d y and Delta y. Okay, so the first thing we have to remember is Delta y is based on the formula f X plus delta X minus f of x. Okay, so in this particular case, we have our formula, which is X to the third. So we have our formula, which we&#x27;re now gonna have X plus Delta X to the third power minus X, the third power. So we know honestly to expand this out to make sure that we have this. All right. Oh, correct. So we&#x27;re gonna have one x to the third. Plus, remember, we were leading coefficient here, which is three x squared Delta X close three x Delta X squared Los Delta X The third, using the binomial expansion minus X to the third, will notice that the X the thirds will cancel right? And so delta why will be equal to three x squared Doubt X plus three x Delta X squared, plus Delta X to the third. The second question is asked us to find D Y Well, we know that D Y d ax is a form of the derivative. So the derivative of my function is three x squared solving for D Why we can see the d y will equal three x squared DX. This is D Y. This is Delta y.

Paul Teng · Answer

Okay, so this problem has four parts A B and C nd andi for East Part. They want you to find the double the second derivative or each function. So starting off with part A, we have a function of Why so Simon X cubed minus five x squared plus x. So if we first take if we take the first derivative, do you Why? Over dx will be happy using combination off the constant multiple rule and, uh, power using the constant multiple rule. We have seven. Leave that alone since the classical efficient. Then using a power rule or execute, we get three x squared. Yeah, negative. Five. We&#x27;ve got one because of the constant multiple rule and derivative of export is two X because of the power role and finally derivative of access Just one. So, um, can simplify this to get 21 x word minus 10 x plus one. So if we were to see the second derivative secondary to D squared y over G X squared it again had to use the power rule Bring down the to or 21 x where you have 21 bring down a to get to X or through development of X squared and then for negatives. Te next derivative of Magnetics just negative. Tencent&#x27;s the review of Accidents one and finally plus one at the end. We&#x27;ll is just a constant, so that ends up becoming zero. So our answer for part may because 42 x minus 10 That is your final answer. Now your final wolf. Your answer for party now Part B ask you to differentiate 12 X squared minus two X plus three. So this is us. This is slightly easier since we&#x27;re dealing with quadratic instead of a cubic. So if we were to 80 y dx off this, I would have to bring down that to you. The powerful and you were do that two times 12 is 24. We get up 24 x minus the derivative off native. Two exits just needed to. And finally plus three the constant. So that just goes to zero. So this is our first derivative. And if you were take our second derivative D squared. Why? Over DX squared Oh, 24 x becomes 24. Sense of derivative of X is just one and 2400 and 24 a negative two. That&#x27;s a constant, so that becomes zero. So our second rooted for Part B is just 24 and we were going to start. See, we have quotient X plus one over X. But we can simplify this by separating our numerator so we can have X over X plus one over X. We could rewrite this as one because X over X is equal to one. Plus, uh, one over X can be written as can be rewritten as X to the negative one. So now that we have this, we can go ahead and take the first derivative D Y over D X, the derivative of one that&#x27;s just a constant so that become zero and the derivative of extra native one to bring out a native one and multiply by X could a negative to, since native one minus one is just named too. So this is our first derivative, our second derivative eastward. Why over DX square, we have to again use our power role, so bringing down leaving that negative alone&#x27;s and that is a constant and the constant multiple. They remove all this that we&#x27;ve told us to leave that constant womb. So then we have to bring down our negative two and multiplied by accident. Negative three. And this just goes to two x two, native three. And as your answer for part C o D or D asked for you to, uh, on the second derivative of a product of two expressions five x squared. I&#x27;ve explored my eyes three times. Seven x q plus X. So the first thing we want to do is to expand these two expressions by multiplying these two expressions. If we were to do that, foiling will get 35 exit 1/5 minus 60 Next Cube minus react. So this is our is a nice polynomial function, which is power functions that we were to take the first derivative. Do you? Why over TX we will have 35. That&#x27;s a constant multiple. We&#x27;ll leave that alone times Bring down the five, the derivative of X. If it this I&#x27;ve exited fourth because of the power will next. Negative 16. We&#x27;ve been alone because of the constant multiple rule and the derivative of executed is three expert because of ah, powerful and finally the derivative of negative three exit. Just make it a three. So this is our first derivative and our second derivative D squared y over DX. Word equals soon. Uh, well, I should probably simplify this first 35 35 times five is 175. So 175 times x ray before actually 1/4 minus 16 times three is 40 days and 48 x squared, minus three. All right, so now take second derivative. We have to leave 175 alone again because constant multiple So then the derivative of four x x to afford is for excuse because of the power role. Then next we have negative 48 1948. And leave that alone because it&#x27;s a constant call efficient in front of the X squared variable and the derivative of X squared is two x power rule, and negative three is a constant as a negative three so derivative that become zero. So simplifying this we get 175 times four at 707 100 x cubes minus 40 times to which is 96 x And this is your answer for party

Taylor Shimono · Answer

So for this problem, we can see that we&#x27;re going to need to separate our equation out so that we have the white terms on one side and the ex terms on the other. I&#x27;m so we could just go ahead and cross multiply here. I&#x27;m so with you that we get why Cube, do you? Why is equal to seven x squared DX? And now that we have our separation of variables, we can go ahead and integrate each side of our equation. I&#x27;m so the interval of White Cube isn&#x27;t the power rule is just 1/4. Why? To the fourth power. Um, and this is going to be equal to the integral of a seven x squared, which again, according to the power rule in seven over, three times x cubed. And then we can&#x27;t forget our plus seat. So we only need to write one. Plus, he here because we&#x27;re going to have a constant on the left hand side and the constant on the right hand side. But we could just combine them into one constancy. Um and then lastly, we need to find our constant, um, by plugging in a wife. Three is equal to two. I&#x27;m so we know that we&#x27;re going to plug in to for wife is equal to 7/3 at times three cubed plus c So let me simplify 1/4 times two to the fourth Power is just going before and this is equal Thio seven times nine which is 63 plus e. So when we saw what we find, that C is equal to 59. So our equation for why in terms of X is going to be a 1/4 y to the fourth power is equal to 7/3 ex Camped plus C, which is 59 and then lastly I&#x27;m just going Thio, get rid of this 1/4 and y to the fourth power I get Why is equal to the fourth route of four times our original equation which was 7/3 x cubed plus 59

Ashley High · Answer

forgiven this problem. Then we know that why equals three. So we plug it in so I can subtract three from both sides. So I need factors of positive for that will give me negative for so I can use negative three and negative one. So X minus three equals zero. So I mean Exc Unequal three and I have X minus one equals zero. So X equals one so x can equal three or X equal one. Those are my two answers.

Problem

Find the values of $\Delta y$ and $d y$ for the g…

Problem 18 Easy Difficulty

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

Answer

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

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Catherine R.

Kayleah T.

Caleb E.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 24

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Kayleah T.

Caleb E.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

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

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Kayleah T.

Caleb E.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Catherine R.

Kayleah T.

Caleb E.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus