## fozia 2 years 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).

1. genius12

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

2. fozia

im trying

3. genius12

btw, is that e^3x?

4. fozia

yes it is

5. genius12

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 slope-intercept form (you can use point-slope form as well but I find slope-intercept 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

6. fozia

oh grt thank you so much