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

Welcome back, everyone. In this problem, we want to use linear approximations to estimate the value of 1 divided by 102. We want to choose an appropriate value of A that minimizes the error. A says the estimated value is 0.0088. B 0.0094, C 0.0098, and D 0.0108. Now, how are we going to estimate the value of 1 divided by 102? Well, we need to first find a value that is close to 102. That makes it easy to calculate. So let's let FFX. B equal to 1 divided by X in the same form as 1 divided by 102. And let's let A be equal to 100 because this is close to 102 and 1100 is easy to calculate. OK. Then recall that based on the formula for a linear approximation, L of X is going to be equal to F of A. Plus the derivative of F of A multiplied by X minus A, OK. Now what do we know about FFA since A equals 100? We could use that to solve for FFA to get the derivative of FFA and thus get a formula for L of X or linear approximation. Afterwards we can use 10024 X. Now in that case then. Let's put that in red here. F of A, since A is gonna be 100, is gonna be equal to F of 100. And 100 is equal to 0.01. So let me write it out here. 1 divided by 100 equals 0.01. Next, we know that if FF X is 1 divided by X, then the derivative of FF X is gonna be equal to negative X raise to the power of -2, which equals -1 divided by X2. OK. So in that case then. That means the derivative of FFA since A is 100. is going to be equal to -1 divided by 100 squad, which is gonna be equal to -0.0001. Know that we have values for FFA and the derivative of FFA, it follows then that our linear approximation could be written as F of 100, OK, plus. F of 100 multiplied by X minus 100. Which is gonna be equal to 0.01 minus 0.0001 multiplied by X minus 100. Know that we have a formula 4 or linear approximation that is close to 102, let's substitute, OK. Let's substitute X to be 102. In that case, then that means a linear approximation. I say. Or a linear approximation, which is gonna be approximately equal to F of 102, OK. It's going to be equal to 0.01 minus 0.0001, multiplied by 100 to minus 100. This is gonna be 0.01 minus 0.00.01 multiplied by 2. OK. That's gonna be, well, let me make some space below here. That's gonna be 0.01 minus 0.0002, which equals 0.0098. Thus this tells us. Then that 0.0098 is the estimated value of 1 divided by 102. If we look back on our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.1/203

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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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