Linearization for Approximation

Question

In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.

f(x) = ∛x, a = 8.5

Accepted Answer

Welcome back everyone. Find the linearization of the function G of X equals square root of X at a point near A equals 16.5, where the function and its derivative are easy to evaluate. We're given 4 answer choices A says x + 6 divided by 8, B X minus 16 divided by 8, C X + 16 divided by 8, and D X + 8 divided by 16. Let's recall the linear approximation formula. L of X is equal to G at point A multiplied by x minus A plus G at point A. So basically this formula tells us that we first will need to identify the derivative of the function at point A. It says near A equals 16.5. Let's analyze the given function G of X equals square root of x. And we know that square root of 16 is equal to 4, so it is easy to evaluate square root of X at 16, which is near 16.5, right? So we're going to choose A equals 16. Now let's identify the derivative G of X. Is the derivative of square root of X. Using the table of basic derivatives, we can tell that this is 1 divided by 2 square root of X, and then we can use this form to evaluate G at 16, which is 1 divided by 2 multiplied by square root of 16, which is 1 divided by 2 multiplied by 4, and that's 1 divided by 8. So we have G at A, and now we need G at A. Which is G at 16, and for this part we need to take the original function square root of X. So we are evaluating square root of 16, which is 4, and now we have the required terms. So the approximation becomes L of X equals G at A or G at 16, which is 1/8. Multiplied by X minus A, so X minus 16, plus G of A, which is 4. And now we can just distribute the terms, right? So, this gives us 1/8 X minus 16 divided by 2, so that's minus 2 + 4. We get X divided by 8 + 2, and we can identify the common denominator, which is 8, right? So we can write the common denominator of 8 and the numerator, we get X plus 2 multiplied by 8, that's 16. So, the final approximation is X + 16 divided by. 8, and this means that the correct answer to this problem corresponds to the answer to C. Thank you for watching.

Linearization for Approximation

In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.

f(x) = ∛x, a = 8.5

Key Concepts

Linearization

Derivative

Choosing a Suitable Point

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