Use linear approximations to estimate the following quantities...

Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

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Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with linear approximations. This problem says use linear approximations to estimate the value of 1 divided by the cube root of 727. Choose an appropriate value of A that minimizes the error. We're given four possible choices as our answers. For choice A, we have 2,189 divided by 19,683. For choice B, we have 2,312, divided by 19,683. For choice C, we have 3,287 divided by 15,383. And for choice D, we have 3,319 divided by 15,383. Now we're asked to estimate the value of 1 divided by the cube root of 727 using linear approximations. So recall for a linear approximation, we have L of X is equal to F of A. Plus F of A. Multiplied by the quantity of X minus A. Now, for this problem here, we're going to have F of X. Be equal to 1 divided by the cube root of X. And we'll also need to take the first derivative of this function. So we'll calculate F of X, and that's going to be equal to the derivative with respect to X of the quantity of X raised to the minus 1/3 power. So here we've just rewritten 1 divided by the cube root of X as X raised to the minus 1/3 power. So taking this derivative, this is going to be equal to -1 divided by 3, multiplied by x raised to the -4 divided by 3. Which we can rewrite as being equal to. -1 divided by the quantity of 3 multiplied by X raised to the 4 divided by 3. So now we just need to choose a value for A. We're told to choose an appropriate value for A that minimizes the error. And we're going to choose A to be equal to. 729. The reason why we chose 729 is it's very close to 727, but it's also a perfect cube. So thou allow us to easily calculate f of A and F of A. So for F of A. This is going to be equal to F of 729. Which is going to be equal to. 1 divided by the cube root of 729. Which is equal to 1 divided by 9. Now we also need to calculate F of A. So we'll have F of A. is equal to F of 729. Which is equal to -1 divided by the quantity of 3 multiplied by the quantity of 729 in quantity raised to the 4 divided by 3. Simplifying this expression, this is going to be equal to -1 divided by 3 multiplied by 9 raised to the 4th power. Which is going to be equal to -1 divided by the quantity of 3, multiplied by 6,561. Which is equal to -1 divided by 19,000 683. Now, the final quantity that we need to identify before we can use our linear approximation, is our value for X, and X is going to be equal to 727, since we're looking for one divided by the cube root of 727. So, at this point, we can substitute in these values into our linear approximation expression. And we will have L of 727. is equal to F of A, which is 1 divided by 9. Plus F A, which is -1 divided by 19,683. And that is multiplied by the quantity of X minus A, which is going to be 727 minus 729. So now we just need to simplify this expression. So this is equal to 1 divided by 9 minus. 1 divided by 19,683, and that is multiplied by minus 2. Which is going to be equal to 1 divided by 9, plus 2 divided by 19,683. So now we want to combine these two fractions with a common denominator of 19,683. Doing so, I will get 2,187. Plus 2, and that quantity is divided by 19,683. And so our final answer is going to be equal to 2,189 divided by 19,683. This here is the answer that we are looking for. And now, if we look at our answer choices and compare our calculated answer to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.1/³√510

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Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
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