Question Video: Finding the Linear Approximation of an Exponential Function Mathematics

Video Transcript

Find the linear approximation of the function 𝑓 of 𝑥 equals two to the 𝑥 power at the point 𝑥 equals zero.

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

Our first step is to solve our original function at 𝑥 equals zero. Two to the zero power equals one. So we need to take one plus the derivative. The derivative of two to the 𝑥 power equals two to the 𝑥 power times the natural log of two. But remember, we want to evaluate this derivative at point 𝑎, 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 don’t wanna use a decimal approximation. So we’ll just write it as the natural log of two.

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