How Can You Find a Function Tangent to Specific Lines at Given Points?

[MC Escher]

Homework Statement

Find a 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).

Homework Equations

I know that the problem involves differentiation and implicit differentiation. I don't really think there are many "equations" that I could give that would help, besides may the limit definition.

The Attempt at a Solution

I haven't made it very far, but I am quite sure you need to somehow separate a, b, and c and than eliminate c altogether. I also have been trying odd and even values for c...
all help is appreciated, thanks...

 

[CarlB]

"Tangent at a point (a,b)" requires two degrees of freedom. The two curves have to both go through (a,b), and the two curves have to have the same deriviative.

This means that you will get two equations for each "tangent at a point" restriction. You've got two such restrictions, so you should be able to get 2x2 = 4 equations from the tangent restrictions.

There are only 3 unknown constants to solve for, a,b,c, so the problem might be over determined. In any case, this turns the problem from a calculus problem, which you are learning (i.e. find the slope of a curve), to an algebra problem (i.e. solve four equations in three unknowns) which you learned the technique in a previous class.

So go for it!

 

[Dick]

The tangency conditions give you requirements on the value of f(x) and f'(x) at various points. Write these equations down and if you are having problems people will try to help you. I'll get you started, f(0)=1.

 

[MC Escher]

thanks for the help so far...

 

[holarjc]

y= 3/2 - (1/2)cos(2x)

 

[Dick]

y= 3/2 - (1/2)cos(2x)

You are i) violating the forum rules by supplying an solution instead of hints on how to solve it (even if it is right, which I haven't checked) and ii) this thread is two years old. What's your point?