A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing
 one year 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).
 one year 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).

This Question is Closed

genius12
 one year ago
Best ResponseYou've already chosen the best response.2Find 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

genius12
 one year ago
Best ResponseYou've already chosen the best response.2Ok 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

fozia
 one year ago
Best ResponseYou've already chosen the best response.0oh grt thank you so much
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
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.