Posted by Mishaka on Saturday, October 22, 2011 at 9:07pm.
If the following function is continuous, what is the value of a + b?
f(x) = {3x^2  2x +1, if x < 0
a cos(x) + b, if 0 </= x </= pi/3
4sin^2(x), if x > pi/3
A. 0
B. 1
C. 2
D. 3
E. 4
I know that since the function is continuous, it should be equal to 1 at 0 and 3 at pi/3 (To follow the other two pieces of the function). From here, I am having a great deal of difficulty figuring out what coordinates would make the function work in this way. Any help is appreciated.

Calculus (Continuity)  Reiny, Saturday, October 22, 2011 at 10:03pm
To be continuous, the functions should have the same value at the transition values of x
First transition value: x = 0
at x = 0, 3x^2  2x + 1 = 1
and acosx + b = acos(0) + b = a+b
since they wanted the value of a+b and we know its value is 1
we are done: a+b = 1
check:
at x= π/3
acosπ/3 + b = a/2 + b
and 4sin^2 (π/3) = 4(√3/2)^2 = 4(3/4) = 3
then a/2 + b = 3
a+2b = 6
solving with a+b=1, subtract them
b = 5
then a+5 = 1
a = 4
so when x=0 , first function is 1
2nd function is 4cos0 + 5 = 4+5 = 1 , good
when x=π/3
2nd function is 4cos(π/3) + 5 = 4(1/2) + 5 = 3
3rd function is 4sin^2 (π/3) = 4(√3/2)^2 = 3 , good!
So the values of a=4 and b=5
or
a+b=1
make the functions continuous.
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