### Video Transcript

Find the linear approximation of
the function π of π₯ equals two to the π₯ power at the point π₯ equals zero.

To find a linear approximation for
point π₯ equals π, first, we find the solution at π of π and add that to the
solution of the derivative of π₯ at point π, which we multiply by π₯ minus π. Our π value, the place we want to
evaluate our linear approximation, is zero.

Our first step is to solve our
original function at π₯ equals zero. Two to the zero power equals
one. So we need to take one plus the
derivative. The derivative of two to the π₯
power equals two to the π₯ power times the natural log of two. But remember, we want to evaluate
this derivative at point π, at our point of interest. And that means we wanna calculate
two to the zero power times the natural log of two. Two to the zero power equals one,
times the natural log of two. We can simplify this to just the
natural log of two. We donβt wanna use a decimal
approximation. So weβll just write it as the
natural log of two.

Our last step is then to multiply
the natural log of two by π₯ minus π, π₯ minus our point of interest. π₯ minus π is π₯ minus zero for
us, and that equals π₯. What we now have is our linear
approximation being one plus the natural log of two times π₯. To write it in another way, π₯
times the natural log of two plus one. This linear approximation, π₯ times
the natural log of two plus one, represents the slope of the function two to the π₯
power when π₯ equals zero.