Calculus (Continuity)
posted by Mishaka .
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.

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.
Respond to this Question
Similar Questions

Calculus
COnsider g(x)=(8)/(x6) on (6,13) (a) Is this function continuous on the given interval? 
Calculus
COnsider g(x)=(8)/(x6) on (6,13) (a) Is this function continuous on the given interval? 
Calculus
consider k(t)=(e^t)/(e^t7) on[7,7] Is this function continuous on the given interval? 
Calculus
COnsider g(x)=(8)/(x6) on (6,13) (a) Is this function continuous on the given interval? 
Calculus
consider k(t)=(e^t)/(e^t7) on[7,7] Is this function continuous on the given interval? 
Calculus (Continuity)
If the following function is continuous, what is the value of a + b? 
Calculus
Is this the right explanation for this question 1. Examine the continuity of the function h(x)= x, if x<0 0, if x=0 x, if x>0 Answer: Since h(x)= x, if x<0 0, if x=0 x, if x>0 is h=x, we can conclude that the function … 
Calculus  #6
If the following function is continuous, then what is the value of b? 
Continuity Calculus Help
The given function is defined for all x > 0 except at x = 8. Find the value that should be assigned to f(8), if any, to guarantee that f will be continuous at 8. (If an answer does not exist, enter DNE.) f(x)= Sqrt( x^264/x8) 
Continuity Calculus Help
Find the value of constants c and d that make the function below continuous at x = 4. f(x) = x2 − 3x x < 4 c x = 4 d + x x > 4 c = d = 8 I got d=8, but finding c is giving me trouble.