Can someone please help me through this problem explaining every step? I will aw
ID: 2847854 • Letter: C
Question
Can someone please help me through this problem explaining every step? I will award all of the points to the most thorough and complete explaination.
"Suppose f(2) = 1, f'(2) = 4, and f''(2) =-5. Find an equation for the osculating circle (circle of curvature) to f at x=2."
I have a solution to which I can follow most, but not all of the work.
The solution is as follows:
Where I get confused is the very beginning specifically, where do the values for K(2) come from? Where did the 5 and (1+16)^3/2 come from? Same goes for the Mtan, where does the value of 4 come from? and N, where does the 1/sqrt17 come from?
if someone could work this through and explain every step, showing the formulas that produced the above values would be incredibly helpful. Thanks!
Explanation / Answer
Let R = radius of osculating circle
This circle passes through point (2, 1) where slope = 4, curve is concave down
So centre of circle is directly R units below point (2, 1). Centre = (2, 1-R)
Curvature of circle is ? = 1/R
We now need to find curvature of y at point (2, 1), then find radius of circle R, so that both the circle and function have the same curvature at this point.
y =1
y' =4
y'' = -5
? = |y''| / (1 + (y')