### Video Transcript

Consider π of π₯ equals π to the
π₯ power. Thereβs two parts to this
question. The first part says, find the
Maclaurin series of π of π₯. And the second part says, use the
first three terms of this series to find an approximate value of π to the power of
0.4 to two decimal places.

Letβs start by writing out the
Maclaurin series expansion for a general function π of π₯. So, letβs begin by finding some of
the derivatives of our function π of π₯. π of π₯ equals π to the π₯ power,
so π of zero equals π to the zero power, which is one. We differentiate π to the π₯ to
get the first derivative. So, we recall the fact that the
derivative of π to the power of π₯ with respect to π₯ is just π to the power of
π₯. So, π prime of π₯ equals π to the
π₯ power. And so, it follows that π prime of
zero is one. And in fact, we see that all the
derivatives of π of π₯ are π to the power of π₯.

So, now letβs make the
substitutions. As our function is defined as π of
π₯, we can replace π with π and begin to make some substitutions. We found that π of zero is
one. And we found that π prime of zero
is one. And we found that π double prime
of zero is one. And π triple prime of zero is
one. And this will continue in this way
for all the future terms. We can bring the π₯, the π₯
squared, and the π₯ cubed, and so on on to the top of the fraction. And then, we can spot that we can
actually write this as the sum from π equals zero to β of π₯ to the πth power over
π factorial. So, this gives us the Maclaurin
series expansion of π to the π₯ power.

The second part of this question
asks us to use the first three terms of this series to find an approximate value of
π to the 0.4 power to two decimal places. Here are the first three terms of
our series. And we want to approximate π to
the π₯ when π₯ equals 0.4. So, we replace π₯ with 0.4, and we
find that this is one plus 0.4 over one factorial. But we know that one factorial is
just one. So, this is 0.4. And 0.4 squared is 0.16. And thatβs over two factorial,
which is two multiplied by one, which is two. But 0.16 over two is just 0.08. So, adding these up, we find that
our approximation is 1.48. We could then use a calculator to
check how good our approximation is. Using a calculator gives us the π
to the power of 0.4 is 1.4918 to four decimal places. So, we can see that even though we
only use the first three terms of our series, our approximation is actually quite
good.