Video: Estimating Derivatives of a Given Function

Given that 𝑦 = 𝑓(π‘₯) is a function for four known values, where 𝑓(2) = 3, 𝑓(6) = 3.75, 𝑓(7) = 4, and 𝑓(11) = 4.25, estimate 𝑓′(7).

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Video Transcript

Given that 𝑦 is equal to 𝑓 of π‘₯ is a function for four known values, where 𝑓 of two is equal to three, 𝑓 of six is equal to 3.75, 𝑓 of seven is equal to four, and 𝑓 of 11 is equal to 4.25, estimate 𝑓 prime of seven.

In this question, we’ve been asked to estimate the derivative of 𝑓 at seven and we’ve been given 𝑓 of π‘₯ values near seven. Therefore, we can use the numerical method in order to estimate this derivative. We have that 𝑓 prime of π‘Ž is roughly equal to 𝑓 of π‘Ž minus 𝑓 of 𝑏 over π‘Ž minus 𝑏 plus 𝑓 of 𝑐 minus 𝑓 of π‘Ž over 𝑐 minus π‘Ž all over two, where 𝑏 is less than π‘Ž, which is less than 𝑐. And we want to choose the closest possible values to π‘Ž for 𝑏 and 𝑐. In our case, since we’re trying to find 𝑓 prime of seven, π‘Ž is equal to seven. And the closest π‘₯-values on either side of seven, for which we’ve been given their 𝑓 values, is six and 11. So we can let six be equal to 𝑏 and 11 be equal to 𝑐.

Next, we can simply substitute these values into our formula. We have that 𝑓 prime of seven is roughly equal to 𝑓 of seven minus 𝑓 of six over seven minus six plus 𝑓 of 11 minus 𝑓 of seven over 11 minus seven all over two. Now, we know the values of 𝑓 of six, 𝑓 of seven, and 𝑓 of 11 since they’ve been given to us in the question. So we can substitute these values in, which leaves us with this. And next, we can simplify the fractions in the numerator to give us 0.25 over one plus 0.25 over four all over two.

Now, we can write 0.25 over one as 0.25. And we can write 0.25 over four as one-fourth multiplied by 0.25. Next, we can rewrite the 0.25s as one-fourth. And next, we can multiply through and then add the two fractions in the numerator and then finally divide five over 16 by two to give us that 𝑓 prime of seven is roughly equal to five over 32. Our solution can also be written in decimal form as 𝑓 prime of seven is approximately equal to 0.15625.

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