college calc please help!?

Question

find a and b so f is differentiable at pi/3

let f(x)...
1+acosx x b + sin(x/2) x> pi/3

thankss!

if i were to graph it what increments would i use on the x axis...30deg?

Answer 1

f´(x) = -asin(x) for x<= pi/3
(1/2) cos(x/2) for x > pi/3

the left and right derivative must be the same, thus:
-asin(pi/3) = (1/2)cos(pi/6) and sin(pi/3) = cos(p1/6)

a= -1/2

The b value must be avaluated using the fact: the function must be continuous to be differentiable.

limit left = limit right = f(pi/3)

1 + acos(pi/3) = b + sin(pi/6) or 1 + a(1/2) = b + 1/2

b= 1/2 + a/2 .... as a=-1/2.... b = 1/2 -1/4 = 1/4

Answer 2

To be differentiable at pi/3, f must first be continuous there.

So, substitute x=pi/3 into both formulas:

1+a*cos(pi/3) = 1+a/2
b+sin((pi/3)/2) = b+1/2

So you must have 1+a/2=b+1/2, but this is not enough to determine a and b.

Do the same with their derivatives:
f'(x) = -a*sin x, x <= pi/3
= (1/2)cos(x/2), x > pi/3.

Plug in x=pi/3:
-a*sin(pi/3) = -a*sqrt(3)/2
(1/2)cos((pi/3)/2) = sqrt(3)/4.

To be differentiable, you need -a*sqrt(3)/2=sqrt(3)/4, so a=-1/2.

Plug in a=-1/2 into the equation 1+a/2=b+1/2 and get 1-1/4=b+1/2, or b=1/4.