Find the fourth-degree polynomial
of the function 𝑓 of 𝑥 equals sin of 𝑥 at the point 𝑎 equals 𝜋 over two.
Let’s start by writing out the
general form for a Taylor polynomial which approximates a function 𝑓 of 𝑥 at the
point 𝑥 equals 𝑎. We’ve been asked to find the
fourth-degree Taylor polynomial of this function at the point 𝑎 equals 𝜋 over
two. So let’s write out our general form
for a Taylor polynomial up to the fourth degree and substitute in 𝑎 equals 𝜋 over
We can see here that we’re going to
need to substitute in values for 𝑓 of 𝜋 over two, the first derivative of 𝑓 at 𝜋
over two, the second derivative of 𝑓 at 𝜋 over two, the third derivative of 𝑓 at
𝜋 over two, and the fourth derivative of 𝑓 at 𝜋 over two.
So let’s start by finding the
derivatives that we need. 𝑓 of 𝑥 is the function sin of
𝑥. At this point, we recall this
useful cycle, which shows us what each of these functions differentiates to. So we see that sin of 𝑥
differentiates to cos of 𝑥. So 𝑓 prime of 𝑥 equals cos of
𝑥. Then we differentiate the first
derivative to get the second derivative. And we see that this is going to be
negative sin of 𝑥. We then differentiate again to get
the third derivative, which we see is negative cos of 𝑥. And we differentiate once more to
get the fourth derivative of 𝑓. And we get sin of 𝑥.
But what we actually need to do is
evaluate each of these functions at 𝜋 over two. So we substitute in 𝜋 over two
into each of these functions. We can work these out either on a
calculator or by using the graphs. So we can see that sin of 𝜋 over
two is going to give us one. So negative sin of 𝜋 over two will
be negative one. And we can see that cos of 𝜋 over
two is zero. So negative cos of 𝜋 over two will
also be zero.
So now we can substitute these
values back into our working. When we do this, we find that two
of our terms are actually zero. From here, we just need to simplify
our answer. Remembering that the factorial of a
number is the product of that number and the integers below it down to one. So two factorial is two multiplied
by one, which is two. And four factorial is four
multiplied by three multiplied by two multiplied by one, which is 24. So substituting those values gives
us our final answer.