Question

Find the linearization L(x) of the function f(x)=e^(3x) at x=0

Answer 1

Isn't linearization pretty much the same thing as the slope at a point?

Answer 2

We need f'(x) and a point to develop the equation of the tangent at x = 0 for f(x), which is also known as a linearisation.\[\bf f'(x)= 3e^{3x}\]Now to find the linearisation at x = 0, we must find f'(0):\[\bf f'(0)=3e^0=3\]We still need to know the y-intercept of this tangent line, so to do that, we must find f(0), since the linearisation of the function is to be taken at x = 0, and coincidentally, x = 0 is where the y-intercept occurs so the f(0) will give us the y-int. If they had asked to find the linearisation at x = 1 let's say, then we would find f(1) and solve for the y-intercept by plugging in the point in y = mx + b form of the tangent line. So f(0) is:\[\bf f(0)=e^0=1\]So the equation of the tangent line to f(x) at x = 0, or the linearisation at x = 0 is:\[\bf y = f'(0)x + f(0) \rightarrow y=3x+1\]
@Invizen

Answer 3

I would've if I wasn't alraedy! LOL you've helped me before apparently =P Thanks again! Just wasn't used to the terminology in the question.

Answer 4

already*