b) If f'(x) = a sinx + bcosx and f' (0) = 4, f (0) = 3,5 (9) = 5, find f(x)
Answers
CORRECT QUESTION.
f'(x) = a sin(x) + b cos(x) and f'(0) = 4, f'(0) = 3, f (π/2) = 5 find f(x).
f(x) = a sin x + b cos x.
f'(0) = 4.
put the value of x = 0 in equation we get,
f'(x) = a sin(0) + b cos(0) = 4.
f'(x) = 0 + b = 4.
f'(x) = b = 4. ........(1)
as we know that,
sin (0°) = 0.
cos (0°) = 1.
f(x) = ∫f'(x)dx.
put the value of f'(x) in equation we get,
f(x) = ∫f'(x)dx = ∫(a sin x + b cos x)dx.
f(x) = ∫f'(x)dx = -a cos x + b sin x + c.
put the value of f(0) = 3 in equation we get,
f(0) = -a cos(0) + b sin(0) + c = 3
f(0) = -a + c = 3. .......(2).
f(π/2) = 5
put the value of x = π/2 in equation we get,
-a cos(π/2) + b sin(π/2) + c = 5.
0 + b + c = 5.
b + c = 5 .............(3).
from equation (1) , (2) and (3) we get,
put the value of b = 4 in equation (3)
4 + c = 5.
c = 1.
put the value of c = 1 in equation (2)
-a + 1 = 3.
-a = 2
a = -2.
value of a = -2, b = 4 and c = 1.
put the value of a , b , c in equation we get,
-(-2)cos x + 4 sin x + 1.
2 cos x + 4 sin x + 1.