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=_______

please explain!!! Thank you:)

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=_______

please explain!!! Thank you:)

anonymous · Answer

|dw:1395094405168:dw| in case it wasn't clear when typed:)

freckles · Answer

the linearization method
find f'(x) if f(x)=e^(4x)+x

then the linear equation that f(x) is approximately y=mx+b
where m=f'(0) in case since we want to know what happens near x=0
the y-intercept can be found by using a point on the line which we know to be (0,f(0))=(0,1) correct?

anonymous · Answer

yes:)

anonymous · Answer

so we need to find f'(x) first?

freckles · Answer

yep

anonymous · Answer

so would f'(x)= e^(4x) + 0 = e^(4x)?

freckles · Answer

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

freckles · Answer

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

anonymous · Answer

ohh darn... using the chain rule?

freckles · Answer

Yippie

anonymous · Answer

so f'(x)= 5 ?

freckles · Answer

So we have f'(0) is totally 5 
good job
so we have y=5x+b
we need the y-intercept now

freckles · Answer

we know this line goes through (0,e^(4*0)+0)

freckles · Answer

Use that to find b

anonymous · Answer

so 0=5(0)+b 

b=0?

freckles · Answer

e^0=?

freckles · Answer

a^0=1 whenever a does not equal 0

anonymous · Answer

oh oops e^0=1

so 1= 0+b

b=1?

freckles · Answer

yep
so we have the approximation for f(x) is y=5x+1
So now you solve 5x+1=2
and you are done.

anonymous · Answer

5x+1=2
     -1 -1
--------
5x=1
x=1/5

so exact answer is x= 0.2 ?

freckles · Answer

what that is an approximation since we approximated f(x) to be y.

freckles · Answer

oops well not what*

freckles · Answer

that is the exact approximation they are looking for

anonymous · Answer

ohh okay..

does that mean to 0.2 is the answer to this problem then? :o

freckles · Answer

yep if i have read the question correctly
and by exact answer they mean the exact answer they are looking for and not the exact solution to e^(4x)+x=2

anonymous · Answer

ohh okay i see:) awesome! thank you1! :D

freckles · Answer

Np. Are you doing online homework?

anonymous · Answer

yes:/

my teacher assigned both online and written :( 

but i guess i get more practice! :)

freckles · Answer

fun stuff

anonymous · Answer

haha i know right? all this homework makes me jump with joy :P

anonymous · Answer

^^i wish :P