How do you find the linearization at a=0 of #f(x) = e^(5 x)#?

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Answer 1 · 2015-11-29T04:22:01

Linearization uses the Linear Approximation Method.

#f_T(a) = f(a) + f'(a)(x-a)#

So, let's take the derivative first, since that's one thing we don't have. If we let #u = 5x#...

#f'(x) = d/(dx)[e^(u)]*(du)/(dx) = e^(5x) * 5 = 5e^(5x)#

Finally, let's plug in #a = 0#:

#f_T(0) = f(0) + f'(0)*(x)#

#= e^0 + 5*e^0*(x)#

#= 5x + 1#

So the tangent line is just #color(blue)(f_T(0) = 5x + 1)#.