Finding values of a function so that it is differentiable at that point.

Question

The question asks: find the values for m and b so that g(theta) is differentiable at theta=0. Hint Find the rate of change of sin(2theta) at theta approaches 0 from the left. (use the definition and make a table.)

 

g(theta)={sin(2theta) where theta is less than or equal to 0, and m(theta)+b where theta is greater than 0

 

So I used the definition and for .01, .001, -.01, -.001 I got values approaching .035 but I don't know what to do with that or how to use that to find m or b.

 

Please help me, I don't understand and I am so confused on how to use all of this information.

Answer 1

It's finally clear to me that you have a piecewise definition of function g.

 

g'(θ) = 2cos(2θ)   ... for θ < 0

g'(θ) = m      ........... for θ ≥ 0.

 

At 0, these two evaluations must be equal for the function to be differentiable, and also sin 0 = 0 and mx+b must also be 0 in order for g to be continuous at 0, which is one of the requirements for differentiability there.

 

Thus b=0 is required (for continuity) & 2 cos 0 must be equal to m.  Since cos 0 = 1, we conclude that m must be 2.