Question

The equation e^(4x) + x=2 has a solution near x=0. By replacing the left side of the equation by its linearization, find an approximate value for the solution.
**Write the EXACT answer**
x=_______

Answer 1

Using the linearization mehod, find f'(x) if f(x)=e^(4x)+x, f(x) is approximately =mx+b where m=f'(0). We want to know the value of the x and y intercepts, so we substitute them with 0(vice versa), we will get (0,f(0))=(0,1)

Solve for  x:

f'(x)=(4x)'e^(4x)+(x)'

(4x)'=4 (x)'=1

f'(x)= 5

solve for b:

e^0=1

1= 0+b

b=1

so the approximation for f(x) is y=5x+1

Solve for y, so you substitute the values of x and b

5x+1=2 -1 -1

5x=1 x=1/5

so exact answer is x= 0.2