Question Video: Find the Taylor Representation of a Function Mathematics

Video Transcript

The following table shows the value of function 𝑓 and some of its derivatives at 𝑥 equals negative two. Write the first five terms of the Taylor series of 𝑓.

So, we have a table here with some values. And at this point, it’s worth me pointing out that this notation means the second, third, and fourth derivatives of 𝑓. We’ve been asked to write the first five terms of the Taylor series of 𝑓, so let’s remind ourselves of the Taylor series expansion. This allows us to express a function as a sum of its derivatives about a point.

The Taylor series of a function 𝑓 about the point 𝑎 is the sum from 𝑛 equal zero to infinity of the 𝑛th derivative of 𝑓 at the point 𝑎 over 𝑛 factorial multiplied by 𝑥 minus 𝑎 to the power 𝑛. Recall that 𝑛 factorial is the product of all the integers less than or equal to 𝑛 but greater than or equal to one. We can equivalently express this sum as 𝑓 of 𝑎 add 𝑓 dash of 𝑎 over one factorial multiplied by 𝑥 minus 𝑎 add the second derivative of 𝑓 at 𝑎 over two factorial multiplied by 𝑥 minus 𝑎 squared, etcetera.

So, let’s apply this to our question. We’ve been given the derivatives at the point 𝑥 equals negative two. So, for our question, 𝑎 equals negative two. Let’s go ahead and substitute 𝑎 equals negative two into our formula. From here, all we need to do is substitute in the values that we know and simplify. In these brackets, we have 𝑥 minus negative two. And because we’re subtracting a negative, we can write this as 𝑥 add two. We can also calculate these factorials.

One factorial is one. Two factorial is two. Three factorial is six. And four factorial is 24. Let’s now substitute in the derivatives we’ve been given in the question in our new line of working. 𝑓 of negative two is three. 𝑓 dash of negative two is negative one. The second derivative of 𝑓 at negative two is five. The third derivative of 𝑓 at negative two is eight. And the fourth derivative of 𝑓 at negative two is negative six.

From here, we just need to simplify. Negative one over one is just negative one. So, we just need to write a minus before the bracket. Eight over six simplifies to four over three. And negative six over 24 simplifies to negative one over four. And so, that gives us our final answer. The first five terms of the Taylor series of 𝑓 are three minus 𝑥 add two add five over two add two squared add four over three 𝑥 add two cubed minus one over four 𝑥 add two to the power of four.