Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
can anyone tell me the Use the linear approximation of the function f(x)=arctan(e3x) at x=0 to estimate the value of f(0.01).
 10 months ago
 10 months ago
can anyone tell me the Use the linear approximation of the function f(x)=arctan(e3x) at x=0 to estimate the value of f(0.01).
 10 months ago
 10 months ago

This Question is Closed

genius12Best ResponseYou've already chosen the best response.1
Find the equation of the tangent line of f(x) at x = 0 and evaluate f(0.01) using the tangent line. Do you know how to perform these steps? @fozia
 10 months ago

genius12Best ResponseYou've already chosen the best response.1
Ok let's first find the equation of the tangent line. To do this, I will need the slope of the line and a point. To get a point, we find f(0):\[\bf f(0)=\arctan(1)=\frac{ \pi }{ 4}\]So the tangent goes through the point \(\bf (0, \frac{\pi}{4})\). Now to find the slope of the tangent, we evaluate f'(x) at x = 0:\[\bf f'(x)=\frac{ 3e^{3x} }{ 1+e^{6x} } \rightarrow f'(0)=\frac{ 3e^0 }{ 1+e^0 }=3\]So now we have a point and the slope of tangent line. We will use the slopeintercept form (you can use pointslope form as well but I find slopeintercept form easier) to get the tangent line's equation:\[\bf y = mx+b \rightarrow y = 3x + b\]Plug in the point for 'x' and 'y':\[\bf \frac{\pi}{4}=3(0)+b \implies b = \frac{\pi}{4}\]So the equation of the tangent line is:\[\bf g(x)=3x+\frac{\pi}{4}\]Here I called the tangent line g(x). Now to find the linear approximation of f(0.01), we plug in x = 0.01 in to our equation of the tangent line and evaluate:\[\bf g(0.01)=3(0.01) + \frac{\pi}{4} \approx 0.815\]Therefore: \(\bf f(0.01) \approx 0.815\) @fozia
 10 months ago

foziaBest ResponseYou've already chosen the best response.0
oh grt thank you so much
 10 months ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.