find the linearization L(x) of the function at a.

Question

find the linearization f(x)=2^x     a=0

Answer 1

When you find the linearization of a function at a certain point, you are finding the straight line that is tangent to the curve at that point. So the slope of this straight line will be the derivative at that point.

 

f'(x) = ln[2] * 2^x

f'(x = a = 0) = ln[2]

 

So the line will be

 

y = ln[2] * x + b

 

To find b (the y-intercept), remember that the curve must go through the point (a, f(a))

 

(0, f(0)) = (0, 1)

 

1 = ln[2] * 0 + b --> b = 1

 

So the answer is y = ln[2] * x + 1.