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.

ln 1.05

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 the natural log of 0.98. Choose an appropriate value of A that minimizes the error. We're given four possible choices as our answers. For choice A, we have minus 0.08. For choice B, we have 0.06. For choice C, we have -0.02, and for choice D, we have 0.04. Now, we're asked to estimate the value of the natural log of 0.98 using linear approximations. So, we call for linear approximations. 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 our case here, our F of X is going to be equal to the natural log of X. And when we take a derivative with respect to X of this function, we will get F of X, which is equal to 1 divided by X. So recall that the derivative with respect to X of the natural log of X is equal to 1 divided by X. So now we need to choose an appropriate value for A that minimizes the error. So we need to choose a value for A that is close to 0.98, and we're going to choose A equal to 1, since 1 is very close to 0.98. There is another reason for choosing A equal to 1, and that's because F of A is going to be equal to F of 1. Which is equal to the natural log of 1, which is equal to 0. So this means that um our F of A here is easy to calculate, since it's going to be the natural log of 1, which is value that we know, which is equal to 0. Now we also need to find F A, which is equal to F1. Which is equal to one divided by one, which is equal to one. And then finally, we're going to set X equal to 0.98 in our expression for the linear approximation, since we're trying to find the natural log of 0.98. So, substituting all these values into our expression for the linear approximation, you will get L of 0.98. Which is equal to F of A, which is 0, plus F of A, which is 1. Multiplied by the quantity of X minus A, which is 0.98 minus 1. Evaluating this expression, this is going to be equal to minus 0.02. So this here is the answer that we were looking for, and when we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice C. 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.ln 1.05

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Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
ln 1.05