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March 29, 2017

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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) - ,

    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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