1) Find the equation of the tangent line to y=lnx at x=1.
I did this part. The equation of the tangent line would be: y=x-1.
The part I'm having trouble with asks: Based on this information, would ln2=1 (rather, is approximately equal to 1) be a good approximation or a bad approximation? Why?
2) Find the tangent line approximation to sinx at x=pi/6.
Again, I did this, and found f(pi/6)=1/2+(sqrt(3)/2)(x-(pi/6)). I also know that this approximation is an over-estimate. The part I'm having difficulty for this problem asks: To one decimal place, estimate the error on this interval.
The equation I have for error is: E=f(x)-f(a)-f '(a)(x-a)
or E=f(x)-tangent line approximation. So, I had E=sinx-1/2-(sqrt(3)/2)(x-(pi/6)), but it doesn't seem right. What would I plug in for x? The x they gave me? But that would make the error zero. What is the correct way to solve this?
Thanks!