Find the Function

Question

Find the function of the form f(x)=a+bcos(cx) that is tangent to the line y=1 at the point (0,1), and tangent to the line y = x + (3/2) - (pi/4) at the point (pi/4, 3/2).

Answer 1

f(0)=1, so a+bcos0 = 1, or A) a+b = 1.

 

f(pi/4) = 3/2, so B) a+bcos(cpi/4) = 3/2

 

Let g(x) equal the derivative of f. g(x) = -bcsincx.

 

g(pi/4) = 3/2, so C) -bcsin(cpi/4) = 3/2.

 

Getting ugly. At this point there are 3 equations an 3 unknowns. 

 

From A) we have a = 1 - b.

 

From C) we have b = -1/csin(cpi/4).

 

Should be able to substitute into B) to get one equation with one unknown (c). 

 

After doing that substitution and some simplifying, I got a pretty messy equation which I solved on graphing calculator.

 

Ended up with c = 2 (or -2). This graph give some details.

 

https://www.desmos.com/calculator/mi6rvmbc6s

 

This must have been an extra credit question, right?! :)