Calculus family of functions f(t)=Asin(3t)+Acos(3t)+Bsin(8t)+Bcos(8t)?

Question

Find exact values of parameters A and B so that f(0) =2 and f'(0)=1.

f(t)=Asin(3t)+Acos(3t)+Bsin(8t)+Bcos(8t)

The original function got shortened by yahoo full function is in the details

Answer 1

f(0) = A(sin(0) + cos(0)) + B(sin(0) + cos(0)) = 2
A(0 + 1) + B(0 + 1) = 2
A + B = 2
B = 2 - A

f'(t) = A(3cos(3t) - 3sin(3t)) + B(8cos(8t) - 8sin(8t))
f'(0) = 3A + 8B = 1
B = 2 - A
3A + 8(2 - A) = 1
-5A = 1 - 16
A = 15/5 = 3
B = 2 - 3 = -1

Answer 2

2=Asin0+Acos0+B sin0

2=A


f ' (0)=1.
f'(t)=3Acos3t -3Asin3t+8Bcos8t
1=3A-0+8B

1=6+8B
B = - 5/8