### Video Transcript

By finding the linear approximation
of the function π of π₯ equals π to the power of π₯ at a suitable value of π₯,
estimate the value of π to the power of 0.1.

Weβre told to use the linear
approximation of the function π of π₯ equals π to the power of π₯. So we recall the formula. If π is differentiable at π₯
equals π, then the equation that can be used to find the linear approximation to
the function at π₯ equals π is π of π₯ equals π of π plus π prime of π times
π₯ minus π. In this example, weβre trying to
approximate the value of π to the power of 0.1. This is going to be close to the
value of π to the power of zero. So we let π be equal to zero. This means π of π is equal to π
of zero. And substituting zero into our
function π of π₯ equals π to the power of π₯ gives π to the power of zero which
is one.

Next, we find π prime of π. First, of course, we need to find
an expression for the derivative of our function. So we different π to the power of
π₯ with respect to π₯. The first derivative of π to the
power of π₯ is π to the power of π₯. So π prime of π becomes π prime
of zero which is π to the power of zero. And once again, thatβs one. Substituting what we know into our
formula for our tangent line approximation and we see that π of π₯ is equal to one
plus one times π₯ minus zero. And that simplifies to π₯ plus
one.

Weβll use this to approximate the
value of π to the power of 0.1 by finding π of 0.1. Thatβs 0.1 plus one which is
1.1. And an estimate to the value of π
to the 0.1 is 1.1. And if we type this into our
calculator, π to the 0.1 is 1.10517 and so on. Thatβs very close in value to our
estimation. And thatβs because 0.1 is fairly
close to zero. Had we tried the larger value, our
number might not have been so accurate. Letβs check that.

For example, π of 0.3 is 0.3 plus
one. So according to our approximation,
π to the 0.3 is approximately 1.3. Typing π to the 0.3 into our
calculator and we get 1.349858808, still not a bad approximation but not quite as
close as π to the power of 0.1.