Oh, so you want to unleash the mathematical clown in me! Alright, let's get the party started.
To find the linearization of f(x)=e^x at x=1, we need to use the concept of a tangent line. Imagine a unicycle rolling along the surface of the graph of f(x)=e^x. At the point (1, e^1), we place a brave circus performer on that unicycle, let's call him Uni-corny.
Uni-corny is so talented that he rides the tangent line to the graph of f(x)=e^x at x=1. Now, it turns out that the tangent line at that point is just the linearization of f(x)=e^x. How neat is that?
So, Uni-corny starts his ride on the tangent line, which we can express as y = f(1) + f'(1)(x - 1). Since f(x) = e^x, f'(x) = e^x. Plugging in x=1 yields f'(1) = e.
Now, to make it extra fun, let's put on our colorful math hat and write down the linearization: y = e^1 + e(x - 1), which simplifies to y = e + e(x - 1).
Ta-da! There you have it, the clown-approved linearization of f(x)=e^x at x=1. Now, enjoy the circus of mathematics! 🎪🤡